publications

Books

2025

1. ESAIM-Proc

   CEMRACS 2023 – Scientific Machine Learning

   Didier Auroux, Manuel Campos Pinto, Bruno Després, Victorita Dolean, Stéphane Lanteri, and Victor Michel-Dansac Eds.

   2025

   Abs DOI Bib

   The CEMRACS (Mathematical Summer Center for Advanced Research in Scientic Computing) is a flagship scientic event organized by the Society of Applied and Industrial Mathematics (SMAI). Established in 1996 by Yvon Maday and Frédéric Coquel from Sorbonne University, it takes place annually at CIRM in Luminy (Marseille, France), spanning six weeks from mid-July. CEMRACS is designed to foster collaboration between academia and industry on a selected cutting-edge scientic theme. The event consists of two key phases: A week-long summer school, featuring introductory lectures by leading experts in the eld, aimed at providing doctoral students, postdocs, and researchers with a state-of-the-art overview of mathematical methodologies; A six-week research program, where young researchers work on supervised projects guided by senior scientists, tackling real-world challenges through mathematical modeling and computational approaches. Over the years, CEMRACS has had a lasting impact, fostering long-term collaborations between researchers and industry, leading to new research directions and innovative projects that extend far beyond the summer session. The CEMRACS 2023 edition was dedicated to Scientic Machine Learning (SciML)a rapidly evolving eld at the intersection of machine learning (ML) and scientic computing. SciML aims to develop robust, reliable, and interpretable methods to solve complex problems such as high-dimensional PDEs, parameter identication, and inverse problems. By naturally integrating data-driven techniques into numerical simulations, SciML is at the forefront of the next wave of scientic discovery in the physical and engineering sciences. This special issue presents a collection of research projects conducted during CEMRACS 2023, showcasing the latest advances in SciML developed at CIRM from July 17th to August 23rd, 2023 (http://smai.emath. fr/cemracs/cemracs23/).

   @proceedings{Auroux:2025:CEM,
     author = {Auroux, Didier and Pinto, Manuel Campos and Despr{\'e}s, Bruno and Dolean, Victorita and Lanteri, St{\'e}phane and Eds., Victor Michel-Dansac},
     journal = {ESAIM Proceedings and Surveys},
     volume = {81},
     title = {CEMRACS 2023 -- Scientific Machine Learning},
     doi = {https://doi.org/10.1051/proc/202581001},
     url = {https://www.esaim-proc.org/articles/proc/abs/2025/04/contents/contents.html},
     year = {2025}
   }

2015

1. SIAM Book

   An introduction to domain decomposition methods: Algorithms, theory, and parallel implementation

   Victorita Dolean, Pierre Jolivet, and Frédéric Nataf

   2015

   Abs DOI Bib

   The purpose of this book is to offer an overview of the most popular domain decomposition methods for partial differential equations (PDEs). These methods are widely used for numerical simulations in solid mechanics, electromagnetism, flow in porous media, etc., on parallel machines from tens to hundreds of thousands of cores. The appealing feature of domain decomposition methods is that, contrary to direct methods, they are naturally parallel. The authors focus on parallel linear solvers. The authors present all popular algorithms, both at the PDE level and at the discrete level in terms of matrices, along with systematic scripts for sequential implementation in a free open-source finite element package as well as some parallel scripts. Also included is a new coarse space construction (two-level method) that adapts to highly heterogeneous problems.

   @book{Dolean:2015:INT,
     author = {Dolean, Victorita and Jolivet, Pierre and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1137/1.9781611974065},
     publisher = {Society for Industrial and Applied Mathematics},
     address = {Philadelphia, PA},
     title = {An introduction to domain decomposition methods: Algorithms, theory, and parallel implementation},
     url = {https://epubs.siam.org/doi/book/10.1137/1.9781611974065},
     year = {2015}
   }

Journal Articles

2026

1. JCAM

   Can symmetric positive definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?

   Victorita Dolean, Mark Fry, and Matthias Langer

   Journal of Computational and Applied Mathematics, 2026

   Abs DOI Bib

   Wave propagation problems governed by the Helmholtz equation remain among the most challenging in scientific computing, due to their indefinite nature. Domain decomposition methods with spectral coarse spaces have emerged as some of the most effective preconditioners, yet their theoretical guarantees often lag behind practical performance. In this work, we introduce and analyse the Delta_k-GenEO coarse space within the two-level additive Schwarz preconditioners for heterogeneous Helmholtz problems. This is an adaptation of the Delta-GenEO coarse space. Our results sharpen the k-explicit conditions for GMRES convergence, reducing the restrictions on the subdomain size and eigenvalue threshold. This narrows the long-standing gap between pessimistic theory and empirical evidence, and reveals why GenEO spaces based on SPD (symmetric positive definite) eigenvalue problems remain surprisingly effective despite their apparent limitations. Numerical experiments confirm the theory, demonstrating scalability, robustness to heterogeneity for low to moderate frequencies (while experiencing limitations in the high frequency cases), and significantly milder coarse-space growth than conservative estimates predict.

   @article{Dolean:2026:CAN,
     title = {Can symmetric positive definite (SPD) coarse spaces perform well for indefinite Helmholtz problems?},
     journal = {Journal of Computational and Applied Mathematics},
     volume = {484},
     pages = {117403},
     year = {2026},
     issn = {0377-0427},
     doi = {https://doi.org/10.1016/j.cam.2026.117403},
     url = {https://www.sciencedirect.com/science/article/pii/S0377042726000683},
     author = {Dolean, Victorita and Fry, Mark and Langer, Matthias},
     keywords = {Helmholtz equation, Domain decomposition method, Two-level method, Coarse space},
   }

2. CMAME

   Local feature filtering for scalable and well-conditioned domain-decomposed Random Feature Methods

   J. W. Beek, Victorita Dolean, and B. Moseley

   Computer Methods in Applied Mechanics and Engineering, 2026

   Abs DOI Bib

   Random Feature Methods (RFMs) [1] and their variants such as extreme learning machine finite-basis physics-informed neural networks (ELM-FBPINNs) [2] offer a scalable approach for solving partial differential equations (PDEs) by using localized, overlapping and randomly initialized neural network basis functions to approximate the PDE solution and training them to minimize PDE residuals through solving structured least-squares problems. This combination leverages the approximation power of randomized neural networks, the parallelism of domain decomposition, and the accuracy and efficiency of least-squares solvers. However, the resulting structured least-squares systems are often severely ill-conditioned, due to local redundancy among random basis functions and correlation introduced by subdomain overlaps, which significantly affects the convergence of standard solvers. In this work, we introduce a block rank-revealing QR (RRQR) filtering and preconditioning strategy that operates directly on the structured least-squares problem. First, local RRQR factorizations identify and remove redundant basis functions while preserving numerically informative ones, reducing problem size, and improving conditioning. Second, we use these factorizations to construct a right preconditioner for the global problem which preserves block-sparsity and numerical stability. Third, we derive deterministic bounds of the condition number of the preconditioned system, with probabilistic refinements for small overlaps. We validate our approach on challenging, multi-scale PDE problems in 1D, 2D, and (2+1)D, demonstrating reductions in condition numbers by up to eleven orders of magnitude, LSQR convergence speedups by factors of 10–1000, and higher accuracy than both unpreconditioned and additive Schwarz-preconditioned baselines, all at significantly lower memory and computational cost. These results establish RRQR-based preconditioning as a scalable, accurate, and efficient enhancement for RFM-based PDE solvers.

   @article{vanBeek:2026:LOC,
     author = {van Beek, J. W. and Dolean, Victorita and Moseley, B.},
     doi = {https://doi.org/10.1016/j.cma.2025.118583},
     journal = {Computer Methods in Applied Mechanics and Engineering},
     pages = {118583},
     title = {Local feature filtering for scalable and well-conditioned domain-decomposed Random Feature Methods},
     url = {https://doi.org/10.1016/j.cma.2025.118583},
     volume = {449},
     year = {2026},
   }

2025

1. IEEE-IUS

   Ultrasonic travel-time tomography for approximating the local elastic tensor in complex media

   J. Ludlam, K. M. M. Tant, A. Curtis, and Victorita Dolean

   IEEE International Ultrasonics Symposium, 2025

   Abs DOI Bib

   Ultrasonic non-destructive evaluation (UNDE) is vital for assessing the structural integrity of safety-critical infrastructure. However, accurate defect detection and characterisation using UNDE is particularly challenging in complex materials like austenitic steel welds, as the heterogeneous and locally anisotropic grain structures distort wave paths, causing traditional imaging methods based on homogeneous and isotropic assumptions to fail. We present a probabilistic framework to reconstruct spatially varying elastic tensor information from ultrasonic travel-time data using stochastic Stein Variational Gradient Descent. Unlike prior approaches, our method relaxes assumptions of uniformity of material properties across the domain and considers uncertainty induced by limited prior knowledge. We show that travel-time data alone cannot fully constrain high-dimensional domains, and accurate imaging requires informed priors on a series of anisotropy parameters we use as a proxy for the stiffness tensor - scale, strength, and orientation.

   @article{Ludlam:2025:ULT,
     author = {Ludlam, J. and Tant, K. M. M. and Curtis, A. and Dolean, Victorita},
     doi = {https://doi.org/10.1109/IUS62464.2025.11201431},
     journal = {IEEE International Ultrasonics Symposium},
     title = {Ultrasonic travel-time tomography for approximating the local elastic tensor in complex media},
     url = {https://ieeexplore.ieee.org/document/11201431},
     year = {2025}
   }

2. JSC

   A robust and adaptive GenEO-type domain decomposition preconditioner for H(curl) problems in three-dimensional general topologies

   N. Bootland, Victorita Dolean, F. Nataf, and P.-H. Tournier

   Journal of Scientific Computing, 2025

   Abs DOI Bib

   In this paper we design, analyse and test domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations, namely for problems. It is well known that convergence of domain decomposition methods rely heavily on the efficiency of the coarse space used in the second level. We design adaptive coarse spaces that complement a near-kernel space made from the gradient of scalar functions. The new class of preconditioner is inspired by the idea of subspace decomposition, but based on spectral coarse spaces, and is specially designed for curl-conforming discretisations of Maxwell’s equations in heterogeneous media on general domains which may have holes. We also address the practical robustness of various solvers in the case of non-trivial topologies and/or high aspect ratio of the domain.

   @article{Bootland:2025:ROB,
     author = {Bootland, N. and Dolean, Victorita and Nataf, F. and Tournier, P.-H.},
     doi = {https://doi.org/10.1007/s10915-025-03061-2},
     journal = {Journal of Scientific Computing},
     number = {3},
     pages = {67},
     title = {A robust and adaptive GenEO-type domain decomposition preconditioner for H(curl) problems in three-dimensional general topologies},
     url = {https://doi.org/10.1007/s10915-025-03061-2},
     volume = {105},
     year = {2025}
   }

2024

1. Numer. Math.

   Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states

   T. Chaumont-Frelet, Victorita Dolean, and M. Ingremeau

   Numerische Mathematik, 2024

   Abs DOI Bib

   We introduce new finite-dimensional spaces specifically designed to approximate the solutions to high-frequency Helmholtz problems with smooth variable coefficients in dimension d. These discretization spaces are spanned by Gaussian coherent states, that have the key property to be localised in phase space. We carefully select the Gaussian coherent states spanning the approximation space by exploiting the (known) micro-localisation properties of the solution. For a large class of source terms (including plane-wave scattering problems), this choice leads to discrete spaces that provide a uniform approximation error for all wavenumber k with a number of degrees of freedom scaling as , which we rigorously establish. In comparison, for discretization spaces based on (piecewise) polynomials, the number of degrees of freedom has to scale at least as to achieve the same property. These theoretical results are illustrated by one-dimensional numerical examples, where the proposed discretization spaces are coupled with a least-squares variational formulation.

   @article{ChaumontFrelet:2024:EFF,
     author = {Chaumont-Frelet, T. and Dolean, Victorita and Ingremeau, M.},
     doi = {https://doi.org/10.1007/s00211-024-01411-0},
     journal = {Numerische Mathematik},
     number = {4},
     pages = {1385--1426},
     title = {Efficient approximation of high-frequency Helmholtz solutions by Gaussian coherent states},
     url = {https://link.springer.com/article/10.1007/s00211-024-01411-0},
     volume = {156},
     year = {2024}
   }

2. ANM

   Robust methods for multiscale coarse approximations of diffusion models in perforated domains

   M. Boutilier, K. Brenner, and Victorita Dolean

   Applied Numerical Mathematics, 2024

   Abs DOI Bib

   For the Poisson equation posed in a domain containing a large number of polygonal perforations, we propose a low-dimensional coarse approximation space based on a coarse polygonal partitioning of the domain. Similarly to other multiscale numerical methods, this coarse space is spanned by locally discrete harmonic basis functions. Along the subdomain boundaries, the basis functions are piecewise polynomial. The main contribution of this article is an error estimate regarding the -projection over the coarse space; this error estimate depends only on the regularity of the solution over the edges of the coarse partitioning. For a specific edge refinement procedure, the error analysis establishes superconvergence of the method even if the true solution has a low general regularity. Additionally, this contribution numerically explores the combination of the coarse space with domain decomposition (DD) methods. This combination leads to an efficient two-level iterative linear solver which reaches the fine-scale finite element error in few iterations. It also bodes well as a preconditioner for Krylov methods and provides scalability with respect to the number of subdomains.

   @article{Boutilier:2024:ROB,
     author = {Boutilier, M. and Brenner, K. and Dolean, Victorita},
     doi = {https://doi.org/10.1016/j.apnum.2024.04.007},
     journal = {Applied Numerical Mathematics},
     pages = {561--578},
     title = {Robust methods for multiscale coarse approximations of diffusion models in perforated domains},
     url = {https://www.sciencedirect.com/science/article/abs/pii/S0168927424000916},
     volume = {201},
     year = {2024}
   }

3. CMAME

   Multilevel domain decomposition-based architectures for physics-informed neural networks

   Victorita Dolean, Alexander Heinlein, Siddhartha Mishra, and B. Moseley

   Computer Methods in Applied Mechanics and Engineering, 2024

   Abs DOI Bib

   Physics-informed neural networks (PINNs) are a powerful approach for solving problems involving differential equations, yet they often struggle to solve problems with high frequency and/or multi-scale solutions. Finite basis physics-informed neural networks (FBPINNs) improve the performance of PINNs in this regime by combining them with an overlapping domain decomposition approach. In this work, FBPINNs are extended by adding multiple levels of domain decompositions to their solution ansatz, inspired by classical multilevel Schwarz domain decomposition methods (DDMs). Analogous to typical tests for classical DDMs, we assess how the accuracy of PINNs, FBPINNs and multilevel FBPINNs scale with respect to computational effort and solution complexity by carrying out strong and weak scaling tests. Our numerical results show that the proposed multilevel FBPINNs consistently and significantly outperform PINNs across a range of problems with high frequency and multi-scale solutions. Furthermore, as expected in classical DDMs, we show that multilevel FBPINNs improve the accuracy of FBPINNs when using large numbers of subdomains by aiding global communication between subdomains.

   @article{Dolean:2024:MUL,
     author = {Dolean, Victorita and Heinlein, Alexander and Mishra, Siddhartha and Moseley, B.},
     doi = {https://doi.org/10.1016/j.cma.2024.117116},
     journal = {Computer Methods in Applied Mechanics and Engineering},
     pages = {117116},
     title = {Multilevel domain decomposition-based architectures for physics-informed neural networks},
     url = {https://doi.org/10.1016/j.cma.2024.117116},
     volume = {429},
     year = {2024}
   }

4. Sensors

   Microwave digital twin prototype for shoulder injury detection

   S. Borzooei, P.-H. Tournier, Victorita Dolean, and C. Migliaccio

   Sensors, 2024

   Abs DOI Bib

   One of the most common shoulder injuries is the rotator cuff tear (RCT). The risk of RCTs increases with age, with a prevalence of 9.7% in those under 20 years old and up to 62% in individuals aged 80 years and older. In this article, we present first a microwave digital twin prototype (MDTP) for RCT detection, based on machine learning (ML) and advanced numerical modeling of the system. We generate a generalizable dataset of scattering parameters through flexible numerical modeling in order to bypass real-world data collection challenges. This involves solving the linear system as a result of finite element discretization of the forward problem with use of the domain decomposition method to accelerate the computations. We use a support vector machine (SVM) to differentiate between injured and healthy shoulder models. This approach is more efficient in terms of required memory resources and computing time compared with traditional imaging methods.

   @article{Borzooei:2024:MIC,
     author = {Borzooei, S. and Tournier, P.-H. and Dolean, Victorita and Migliaccio, C.},
     doi = {https://doi.org/10.3390/s24206663},
     journal = {Sensors},
     number = {20},
     pages = {6663},
     title = {Microwave digital twin prototype for shoulder injury detection},
     url = {https://doi.org/10.3390/s24206663},
     volume = {24},
     year = {2024}
   }

5. IEEE-APP

   A SVM-Based Approach for Detecting Tendon Injury

   S. Borzooei, P.-H. Tournier, Victorita Dolean, and C. Migliaccio

   In IEEE International Symposium on Antennas and Propagation and INC/USNC-URSI Radio Science Meeting, 2024

   Abs DOI Bib

   In this paper, a solution for the fast detection of shoulder’s tendon injury based on numerical modeling and machine learning (ML) algorithm is proposed. The synthetic data for the ML algorithm are the set of scattering parameters which are produced by solving Maxwell’s equations for each transmitting antenna of the microwave imaging (MWI) system. The corresponding data of various healthy and injured models are categorized into two classes. Using support vector machine (SVM) for classification, an accuracy of 100% is achieved.

   @inproceedings{Borzooei:2024:SVM,
     author = {Borzooei, S. and Tournier, P.-H. and Dolean, Victorita and Migliaccio, C.},
     doi = {https://doi.org/10.1109/AP-S/INC-USNC-URSI52054.2024.10685863},
     booktitle = {IEEE International Symposium on Antennas and Propagation and INC/USNC-URSI Radio Science Meeting},
     pages = {1511--1512},
     title = {A SVM-Based Approach for Detecting Tendon Injury},
     url = {https://ieeexplore.ieee.org/abstract/document/10685863},
     volume = {8},
     year = {2024}
   }

6. IEEE-JERM

   Numerical Modeling for Shoulder Injury Detection Using Microwave Imaging

   S. Borzooei, P.-H. Tournier, Victorita Dolean, C. Pichot, N. Joachimowicz, H. Roussel, and C. Migliaccio

   IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology, 2024

   Abs DOI Bib

   Rotator cuff tear (RCT) is one of the most common shoulder injuries, which can be irreparable if it develops to a severe condition. A portable imaging system for the on-site detection of RCT is necessary to identify its extent for early diagnosis. We introduce a microwave tomography system, using state-of-the-art numerical modeling and parallel computing for detection of RCT. The results show that the proposed method is capable of accurately detecting and localizing this injury in different size. In the next step, an efficient design in terms of computing time and complexity is proposed to detect the variations in the injured model with respect to the healthy model. The method is based on finite element discretization and uses parallel preconditioners from the domain decomposition method to accelerate computations. It is implemented using the open source FreeFEM software.

   @article{Borzooei:2024:NUM,
     author = {Borzooei, S. and Tournier, P.-H. and Dolean, Victorita and Pichot, C. and Joachimowicz, N. and Roussel, H. and Migliaccio, C.},
     doi = {https://doi.org/10.1109/JERM.2024.3411799},
     journal = {IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology},
     number = {3},
     pages = {282--289},
     title = {Numerical Modeling for Shoulder Injury Detection Using Microwave Imaging},
     url = {https://ieeexplore.ieee.org/abstract/document/10564578},
     volume = {8},
     year = {2024}
   }

2023

1. FVCA10

   Trefftz Approximation Space for Poisson Equation in Perforated Domains

   M. Boutilier, K. Brenner, and Victorita Dolean

   In Finite Volumes for Complex Applications X—Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, Invited Contributions, 2023

   Abs DOI Bib

   For the Poisson equation posed in a planar domain containing a large number of polygonal perforations, we propose a low-dimensional approximation space based on a coarse polygonal partitioning of the domain. Similar to other multi-scale numerical methods, this coarse space is spanned by basis functions that are locally discrete harmonic. We provide an error estimate in the energy norm that only depends on the regularity of the solution over the edges of the coarse skeleton. For a specific edge refinement procedure, this estimate allows us to establish superconvergence of the method, even if the true solution has low general regularity. Combined with the Restricted Additive Schwarz method, the proposed coarse space leads to an efficient two-level iterative linear solver which achieves the fine-scale finite element error in few iterations. The numerical experiment showcases the use of this coarse space over test cases involving singular solutions and realistic urban geometries.

   @incollection{Boutilier:2023:TRE,
     author = {Boutilier, M. and Brenner, K. and Dolean, Victorita},
     booktitle = {Finite Volumes for Complex Applications X---Volume 1, Elliptic and Parabolic Problems: FVCA10, Strasbourg, France, Invited Contributions},
     doi = {https://doi.org/10.1007/978-3-031-40864-9_14},
     publisher = {Springer Nature},
     title = {Trefftz Approximation Space for Poisson Equation in Perforated Domains},
     url = {https://link.springer.com/chapter/10.1007/978-3-031-40864-9_14},
     year = {2023}
   }

2. DDMSE26

   Optimizing transmission conditions for multiple subdomains in the magnetotelluric approximation of Maxwell’s equations

   Victorita Dolean, Martin J. Gander, and A. Kyriakis

   In Domain decomposition methods in science and engineering XXVI, 2023

   Abs DOI Bib

   Wave propagation phenomena are ubiquitous in science and engineering. In Geophysics, the magnetotelluric approximation of Maxwell’s equations is an important tool to extract information about the spatial variation of electrical conductivity in the Earth’s subsurface.

   @incollection{Dolean:2023:OPT,
     author = {Dolean, Victorita and Gander, Martin J. and Kyriakis, A.},
     booktitle = {Domain decomposition methods in science and engineering XXVI},
     doi = {https://doi.org/10.1007/978-3-030-95025-5_22},
     publisher = {Springer},
     series = {LNCSE},
     title = {Optimizing transmission conditions for multiple subdomains in the magnetotelluric approximation of Maxwell's equations},
     url = {https://doi.org/10.1007/978-3-030-95025-5_22},
     volume = {145},
     year = {2023}
   }

3. DDMSE26

   Inexact subdomain solves using deflated GMRES for Helmholtz problems

   N. Bootland, Victorita Dolean, V. Dwarka, P. Jolivet, and C. Vuik

   In Domain decomposition methods in science and engineering XXVI, 2023

   Abs DOI Bib

   In recent years, domain decomposition based preconditioners have become popular tools to solve the Helmholtz equation. Notorious for causing a variety of convergence issues, the Helmholtz equation remains a challenging PDE to solve numerically. Even for simple model problems, the resulting linear system after discretisation becomes indefinite and tailored iterative solvers are required to obtain the numerical solution efficiently. At the same time, the mesh must be kept fine enough in order to prevent numerical dispersion ‘polluting’ the solution [4]. This leads to very large linear systems, further amplifying the need to develop economical solver methodologies.

   @incollection{Bootland:2023:INE,
     author = {Bootland, N. and Dolean, Victorita and Dwarka, V. and Jolivet, P. and Vuik, C.},
     booktitle = {Domain decomposition methods in science and engineering XXVI},
     doi = {https://doi.org/10.1007/978-3-030-95025-5_11},
     publisher = {Springer},
     series = {LNCSE},
     title = {Inexact subdomain solves using deflated GMRES for Helmholtz problems},
     url = {https://doi.org/10.1007/978-3-030-95025-5_11},
     volume = {145},
     year = {2023}
   }

4. DDMSE26

   Geneo coarse spaces for heterogeneous indefinite elliptic problems

   N. Bootland, Victorita Dolean, I. Graham, C. Ma, and R. Scheichl

   In Domain decomposition methods in science and engineering XXVI, 2023

   Abs DOI Bib

   For domain decomposition preconditioners, the use of a coarse correction as a second level is usually required to provide scalability (in the weak sense), such that the iteration count is independent of the number of subdomains, for subdomains of fixed dimension. In addition, it is desirable to guarantee robustness with respect to strong variations in the physical parameters. Achieving scalability and robustness usually relies on sophisticated tools such as spectral coarse spaces [4, 5]. In particular, we can highlight the GenEO coarse space [9], which has been successfully analysed and applied to highly heterogeneous positive definite elliptic problems. This coarse space relies on the solution of local eigenvalue problems on subdomains and the theory in the SPD case is based on the fact that local eigenfunctions form an orthonormal basis with respect to the energy scalar product induced by the bilinear form

   @incollection{Bootland:2023:GEN,
     author = {Bootland, N. and Dolean, Victorita and Graham, I. and Ma, C. and Scheichl, R.},
     booktitle = {Domain decomposition methods in science and engineering XXVI},
     doi = {https://doi.org/10.1007/978-3-030-95025-5_10},
     publisher = {Springer},
     series = {LNCSE},
     title = {Geneo coarse spaces for heterogeneous indefinite elliptic problems},
     volume = {145},
     year = {2023}
   }

5. DDMSE26

   Several ways to achieve robustness when solving wave propagation problems

   N. Bootland, Victorita Dolean, P. Jolivet, F. Nataf, S. Operto, and P.-H. Tournier

   In Domain decomposition methods in science and engineering XXVI, 2023

   Abs DOI Bib

   Why do we need robust solution methods for wave propagation problems? Very often in applications, as for example in seismic inversion, we need to reconstruct the a priori unknown physical properties of an environment from given measurements.

   @incollection{Bootland:2023:SEV,
     author = {Bootland, N. and Dolean, Victorita and Jolivet, P. and Nataf, F. and Operto, S. and Tournier, P.-H.},
     booktitle = {Domain decomposition methods in science and engineering XXVI},
     doi = {https://doi.org/10.1007/978-3-030-95025-5_2},
     publisher = {Springer},
     series = {LNCSE},
     title = {Several ways to achieve robustness when solving wave propagation problems},
     url = {https://doi.org/10.1007/978-3-030-95025-5_2},
     volume = {145},
     year = {2023}
   }

6. SISC

   Closed form optimized transmission conditions for complex diffusion with many subdomains

   Victorita Dolean, Martin J. Gander, and Alexandros Kyriakis

   SIAM Journal on Scientific Computing, 2023

   Abs DOI Bib

   Optimized transmission conditions in domain decomposition methods have been the focus of intensive research efforts over the past decade. Traditionally, transmission conditions are optimized for two subdomains model configurations, and then used in practice for many subdomains. We optimizetransmission conditions here for the first time directly for many subdomains for a class of complex diffusion problems. Our asymptotic analysis leads to closed form optimized transmission conditions for many subdomains, and shows that the asymptotic best choice in the mesh size only differs from the two subdomain best choice in the constants, for which we derive the dependence on the number of subdomains explicitly, including the limiting case of an infinite number of subdomains, leading to new insight into scalability. Our results include both Robin and Ventcell transmission conditions, and we also optimize for the first time a two-sided Ventcell condition. We illustrate our results with numerical experiments, both for situations covered by our analysis and situations that go beyond.

   @article{Dolean:2023:CLO,
     author = {Dolean, Victorita and Gander, Martin J. and Kyriakis, Alexandros},
     doi = {https://doi.org/10.1137/22M1492386},
     journal = {SIAM Journal on Scientific Computing},
     title = {Closed form optimized transmission conditions for complex diffusion with many subdomains},
     year = {2023}
   }

7. JCP

   Travel times and ray paths for acoustic and elastic waves in generally anisotropic media

   J. Ludlam, K. M. M. Tant, Victorita Dolean, and A. Curtis

   Journal of Computational Physics, 2023

   Abs DOI Bib

   Wavefield travel time tomography is used for a variety of purposes in acoustics, geophysics and non-destructive testing. Since the problem is non-linear, assessing uncertainty in the results requires many forward evaluations. It is therefore important that the forward evaluation of travel times and ray paths is efficient, which is challenging in generally anisotropic media. Given a computed travel time field, ray tracing can be performed to obtain the fastest ray path from any point in the medium to the source of the travel time field. These rays can then be used to speed up gradient based inversion methods. We present a forward modeller for calculating travel time fields by localised estimation of wavefronts, and a novel approach to ray tracing through those travel time fields. These methods have been tested in a complex anisotropic weld and give travel times comparable to those obtained using finite element modelling while being computationally cheaper.

   @article{Ludlam:2023:TRA,
     author = {Ludlam, J. and Tant, K. M. M. and Dolean, Victorita and Curtis, A.},
     doi = {https://doi.org/10.1016/j.jcp.2023.112500},
     journal = {Journal of Computational Physics},
     pages = {112500},
     title = {Travel times and ray paths for acoustic and elastic waves in generally anisotropic media},
     url = {https://doi.org/10.1016/j.jcp.2023.112500},
     volume = {494},
     year = {2023}
   }

8. TLE

   Is 3D frequency-domain FWI of full-azimuth/long-offset OBN data feasible? The Gorgon-data FWI case study

   S. Operto, P. Amestoy, H. S. Aghamiry, S. Beller, A. Buttari, L. Combe, Victorita Dolean, M. Gerest, G. Guo, P. Jolivet, J. Y. L’Excellent, F. Mamfoumbi, T. Mary, C. Puglisi, A. Ribodetti, and P.-H. Tournier

   The Leading Edge, 2023

   Abs DOI Bib

   Frequency-domain full-waveform inversion (FWI) is potentially amenable to efficient processing of full-azimuth long-offset stationary-recording seabed acquisition carried out with a sparse layout of ocean-bottom nodes (OBNs) and broadband sources because the inversion can be performed with a few discrete frequencies. However, computing the solution of the forward (boundary-value) problem efficiently in the frequency domain with linear algebra solvers remains a challenge for large computational domains involving tens to hundreds of millions of parameters. We illustrate the feasibility of 3D frequency-domain FWI with a subset of the 2015/2016 Gorgon OBN data set in the North West Shelf, Australia. We solve the forward problem with the massively parallel multifrontal direct solver MUMPS, which includes four key features to reach high computational efficiency: an efficient parallelism combining message-passing interface and multithreading, block low-rank compression, mixed-precision arithmetic, and efficient processing of sparse sources. The Gorgon subdata set involves 650 OBNs that are processed as reciprocal sources and 400,000 sources. Monoparameter FWI for vertical wavespeed is performed in the viscoacoustic vertically transverse isotropic approximation with a classical frequency continuation approach proceeding from a starting frequency of 1.7 Hz to a final frequency of 13 Hz. The target covers an area ranging from 260 km2 (frequency ≥ 8.5 Hz) to 705 km2 (frequency ≤ 8.5 Hz) for a maximum depth of 8 km. Compared to the starting model, FWI dramatically improves the reconstruction of the bounding faults of the Gorgon horst at reservoir depths as well as several intrahorst faults and several horizons of the Mungaroo Formation down to a depth of 7 km. Seismic modeling reveals a good kinematic agreement between recorded and simulated data, but amplitude mismatches between the recorded and simulated reflection from the reservoir suggest elastic effects. Therefore, future works involve multiparameter reconstruction for density and attenuation before considering elastic FWI from hydrophone and geophone data.

   @article{Operto:2023:IS3,
     author = {Operto, S. and Amestoy, P. and Aghamiry, H. S. and Beller, S. and Buttari, A. and Combe, L. and Dolean, Victorita and Gerest, M. and Guo, G. and Jolivet, P. and L'Excellent, J. Y. and Mamfoumbi, F. and Mary, T. and Puglisi, C. and Ribodetti, A. and Tournier, P.-H.},
     journal = {The Leading Edge},
     doi = {https://doi.org/10.1190/tle42030173.1},
     title = {Is 3D frequency-domain FWI of full-azimuth/long-offset OBN data feasible? The Gorgon-data FWI case study},
     url = {https://arxiv.org/abs/2210.16767},
     year = {2023}
   }

9. MBE

   Scalable computational algorithms for geo-spatial Covid-19 spread in high performance computing

   Sudhi P. V., Victorita Dolean, Pierre Jolivet, Brandon Robinson, Jodi D. Edwards, Tetyana Kendzerska, and Abhijit Sarkar

   Mathematical Biosciences and Engineering, 2023

   Abs DOI Bib

   A nonlinear partial differential equation (PDE) based compartmental model of COVID-19 provides a continuous trace of infection over space and time. Finer resolutions in the spatial discretization, the inclusion of additional model compartments and model stratifications based on clinically relevant categories contribute to an increase in the number of unknowns to the order of millions. We adopt a parallel scalable solver that permits faster solutions for these high fidelity models. The solver combines domain decomposition and algebraic multigrid preconditioners at multiple levels to achieve the desired strong and weak scalabilities. As a numerical illustration of this general methodology, a five-compartment susceptible-exposed-infected-recovered-deceased (SEIRD) model of COVID-19 is used to demonstrate the scalability and effectiveness of the proposed solver for a large geographical domain (Southern Ontario). It is possible to predict the infections for a period of three months for a system size of 186 million (using 3200 processes) within 12 hours saving months of computational effort needed for the conventional solvers.

   @article{Sudhi:2023:SCA,
     author = {V., Sudhi P. and Dolean, Victorita and Jolivet, Pierre and Robinson, Brandon and Edwards, Jodi D. and Kendzerska, Tetyana and Sarkar, Abhijit},
     doi = {https://doi.org/10.3934/mbe.2023655},
     journal = {Mathematical Biosciences and Engineering},
     number = {8},
     pages = {14634--14674},
     title = {Scalable computational algorithms for geo-spatial Covid-19 spread in high performance computing},
     url = {https://arxiv.org/abs/2208.01839},
     volume = {20},
     year = {2023}
   }

10. IEEE-APP

    High-Performance Numerical Modeling for Detection of Rotator Cuff Tear

    S. Borzooei, C. Migliaccio, Victorita Dolean, P.-H. Tournier, and Christian Pichot

    In IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, 2023

    Abs DOI Bib

    In this paper, microwave tomographic imaging for rotator cuff tear by numerical modeling and parallel computing is presented. It requires solving an inverse problem by making use of an iterative method based on a minimization algorithm. To this end, efficient parallel algorithms and high-performance computing is utilized that result in an accurate reconstruction of the electrical properties of the propagation medium as well as fast computations. Results demonstrate the possibility to detect different types of tendon tears in a shoulder model.

    @inproceedings{Borzooei:2023:HIG,
      author = {Borzooei, S. and Migliaccio, C. and Dolean, Victorita and Tournier, P.-H. and Pichot, Christian},
      doi = {https://api.semanticscholar.org/CorpusID:2616135732},
      booktitle = {IEEE International Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting},
      pages = {1879--1880},
      title = {High-Performance Numerical Modeling for Detection of Rotator Cuff Tear},
      url = {https://api.semanticscholar.org/CorpusID:261613573},
      year = {2023}
    }

2022

1. MCA

   Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?

   N. Bootland and Victorita Dolean

   Mathematical and Computational Applications, 2022

   Abs DOI Bib

   Numerical solutions of heterogeneous Helmholtz problems present various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work, we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

   @article{Bootland:2022:CAN,
     author = {Bootland, N. and Dolean, Victorita},
     doi = {https://doi.org/10.3390/mca27030035},
     journal = {Mathematical and Computational Applications},
     number = {3},
     title = {Can DtN and GenEO coarse spaces be sufficiently robust for heterogeneous Helmholtz problems?},
     url = {https://doi.org/10.3390/mca27030035},
     volume = {27},
     year = {2022}
   }

2. IMAJNA

   Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems

   N. Bootland, Victorita Dolean, I. G. Graham, C. Ma, and R. Scheichl

   IMA Journal of Numerical Analysis, 2022

   Abs DOI Bib

   Generalized eigenvalue problems on the overlap(GenEO) is a method for computing an operator-dependent spectral coarse space to be combined with local solves on subdomains to form a robust parallel domain decomposition preconditioner for elliptic PDEs. It has previously been proved, in the self-adjoint and positive-definite case, that this method, when used as a preconditioner for conjugate gradients, yields iteration numbers that are completely independent of the heterogeneity of the coefficient field of the partial differential operator. We extend this theory to the case of convection–diffusion–reaction problems, which may be nonself-adjoint and indefinite, and whose discretizations are solved with preconditioned GMRES. The GenEO coarse space is defined here using a generalized eigenvalue problem based on a self-adjoint and positive-definite subproblem. We prove estimates on GMRES iteration counts that are independent of the variation of the coefficient of the diffusion term in the operator and depend only very mildly on variations of the other coefficients. These are proved under the assumption that the subdomain diameter is sufficiently small and the eigenvalue tolerance for building the coarse space is sufficiently large. While the iteration number estimates do grow as the nonself-adjointness and indefiniteness of the operator increases, practical tests indicate the deterioration is much milder. Thus, we obtain an iterative solver that is efficient in parallel and very effective for a wide range of convection–diffusion–reaction problems.

   @article{Bootland:2022:OVE,
     author = {Bootland, N. and Dolean, Victorita and Graham, I. G. and Ma, C. and Scheichl, R.},
     doi = {https://doi.org/10.1093/imanum/drac036},
     journal = {IMA Journal of Numerical Analysis},
     title = {Overlapping Schwarz methods with GenEO coarse spaces for indefinite and nonself-adjoint problems},
     url = {https://doi.org/10.1093/imanum/drac036},
     year = {2022}
   }

3. ETNA

   Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

   N. Bootland, Victorita Dolean, A. Kyriakis, and J. Pestana

   Electronic Transactions on Numerical Analysis, 2022

   Abs DOI Bib

   In this work we study the convergence properties of the one-level parallel Schwarz method with Robin transmission conditions applied to the one-dimensional and two-dimensional Helmholtz and Maxwell’s equations. One-level methods are not scalable in general. However, it has recently been proven that when impedance transmission conditions are used in the case of the algorithm being applied to the equations with absorption, then, under certain assumptions, scalability can be achieved and no coarse space is required. We show here that this result is also true for the iterative version of the method at the continuous level for strip-wise decompositions into subdomains that are typically encountered when solving wave-guide problems. The convergence proof relies on the particular block Toeplitz structure of the global iteration matrix. Although non-Hermitian, we prove that its limiting spectrum has a near identical form to that of a Hermitian matrix of the same structure. We illustrate our results with numerical experiments.

   @article{Bootland:2022:ANA,
     author = {Bootland, N. and Dolean, Victorita and Kyriakis, A. and Pestana, J.},
     doi = {https://doi.org/10.1553/etna_vol55s112},
     journal = {Electronic Transactions on Numerical Analysis},
     pages = {112--141},
     title = {Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices},
     url = {https://doi.org/10.1553/etna_vol55s112},
     volume = {55},
     year = {2022}
   }

4. Geophysics

   3D finite-difference and finite-element frequency-domain wave simulation with multilevel optimized additive Schwarz domain-decomposition preconditioner: A tool for full-waveform inversion of sparse node data sets

   P.-H. Tournier, P. Jolivet, Victorita Dolean, H. S. Aghamiry, S. Operto, and S. Riffo

   Geophysics, 2022

   Abs DOI Bib

   Efficient frequency-domain full-waveform inversion (FWI) of long-offset node data can be designed with a few discrete frequencies, which lead to modest data volumes to be managed during the inversion process. Moreover, attenuation effects can be straightforwardly implemented in the forward problem without computational overhead. However, 3D frequency-domain seismic modeling is challenging because it requires solving a large and sparse linear indefinite system for each frequency with multiple right-hand sides (RHSs). This linear system can be solved by direct or iterative methods. The former allows efficient processing of multiple RHSs but may suffer from limited scalability for very large problems. Iterative methods equipped with a domain-decomposition preconditioner provide a suitable alternative to process large computational domains for sparse-node acquisition. We have investigated the domain-decomposition preconditioner based on the optimized restricted additive Schwarz (ORAS) method, in which a Robin or perfectly matched layer condition is implemented at the boundaries between the subdomains. The preconditioned system is solved by a Krylov subspace method, whereas a block low-rank lower-upper decomposition of the local matrices is performed at a preprocessing stage. Multiple sources are processed in groups with a pseudoblock method. The accuracy, the computational cost, and the scalability of the ORAS solver are assessed against several realistic benchmarks. In terms of discretization, we compare a compact wavelength-adaptive 27-point finite-difference stencil on a regular Cartesian grid with a finite-element method on -adaptive tetrahedral mesh. Although both schemes have comparable accuracy, the former is more computationally efficient, the latter being beneficial to comply with known boundaries such as bathymetry. The scalability of the method, the block processing of multiple RHSs, and the straightforward implementation of attenuation, which further improves the convergence of the iterative solver, make the method a versatile forward engine for large-scale 3D FWI applications from sparse node data sets.

   @article{Tournier:2022:THR,
     author = {Tournier, P.-H. and Jolivet, P. and Dolean, Victorita and Aghamiry, H. S. and Operto, S. and Riffo, S.},
     doi = {https://doi.org/10.1190/geo2021-0702.1},
     journal = {Geophysics},
     pages = {1--84},
     title = {3D finite-difference and finite-element frequency-domain wave simulation with multilevel optimized additive Schwarz domain-decomposition preconditioner: A tool for full-waveform inversion of sparse node data sets},
     url = {https://doi.org/10.1190/geo2021-0702.1},
     volume = {0},
     year = {2022}
   }

5. BMJ-Open

   Comprehensive compartmental model and calibration algorithm for the study of clinical implications of the population-level spread of COVID-19: a study protocol

   B. Robinson, J. D. Edwards, T. Kendzerska, C. L. Pettit, D. Poirel, J. M. Daly, M. Ammi, M. Khalil, P. J. Taillon, R. Sandhu, S. Mills, S. Mulpuru, T. Walker, V. Percival, Victorita Dolean, and A. Sarkar

   BMJ Open, 2022

   Abs Bib

   We propose a 22-compartment epidemiological model that includes compartments not previously considered concurrently, to account for the effects of vaccination, asymptomatic individuals, inadequate access to hospital care, post-acute COVID-19 and recovery with long-term health complications. Additionally, new connections between compartments introduce new dynamics to the system and provide a framework to study the sensitivity of model outputs to several concurrent effects, including temporary immunity, vaccination rate and vaccine effectiveness. Subject to data availability for a given region, we discuss a means by which population demographics (age, comorbidity, socioeconomic status, sex and geographical location) and clinically relevant information (different variants, different vaccines) can be incorporated within the 22-compartment framework. Considering a probabilistic interpretation of the parameters allows the model’s predictions to reflect the current state of uncertainty about the model parameters and model states. We propose the use of a sparse Bayesian learning algorithm for parameter calibration and model selection. This methodology considers a combination of prescribed parameter prior distributions for parameters that are known to be essential to the modelled dynamics and automatic relevance determination priors for parameters whose relevance is questionable. This is useful as it helps prevent overfitting the available epidemiological data when calibrating the parameters of the proposed model. Population-level administrative health data will serve as partial observations of the model states.

   @article{Robinson:2022:COM,
     author = {Robinson, B. and Edwards, J. D. and Kendzerska, T. and Pettit, C. L. and Poirel, D. and Daly, J. M. and Ammi, M. and Khalil, M. and Taillon, P. J. and Sandhu, R. and Mills, S. and Mulpuru, S. and Walker, T. and Percival, V. and Dolean, Victorita and Sarkar, A.},
     journal = {BMJ Open},
     number = {3},
     title = {Comprehensive compartmental model and calibration algorithm for the study of clinical implications of the population-level spread of COVID-19: a study protocol},
     url = {https://bmjopen.bmj.com/content/12/3/e052681},
     volume = {12},
     year = {2022}
   }

2021

1. CAMWA

   A comparison of coarse spaces for Helmholtz problems in the high frequency regime

   N. Bootland, Victorita Dolean, P. Jolivet, and P.-H. Tournier

   Computers and Mathematics with Applications, 2021

   Abs DOI Bib

   Solving time-harmonic wave propagation problems in the frequency domain and within heterogeneous media brings many mathematical and computational challenges, especially in the high frequency regime. We will focus here on computational challenges and try to identify the best algorithm and numerical strategy for a few well-known benchmark cases arising in applications. The aim is to cover, through numerical experimentation and consideration of the best implementation strategies, the main two-level domain decomposition methods developed in recent years for the Helmholtz equation. The theory for these methods is either out of reach with standard mathematical tools or does not cover all cases of practical interest. More precisely, we will focus on the comparison of three coarse spaces that yield two-level methods: the grid coarse space, DtN coarse space, and GenEO coarse space. We will show that they display different pros and cons, and properties depending on the problem and particular numerical setting.

   @article{Bootland:2021:COM,
     author = {Bootland, N. and Dolean, Victorita and Jolivet, P. and Tournier, P.-H.},
     doi = {https://doi.org/10.1016/j.camwa.2021.07.011},
     journal = {Computers and Mathematics with Applications},
     pages = {239--253},
     title = {A comparison of coarse spaces for Helmholtz problems in the high frequency regime},
     url = {https://doi.org/10.1016/j.camwa.2021.07.011},
     volume = {98},
     year = {2021}
   }

2. ENUMATH2019

   On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition

   N. Bootland and Victorita Dolean

   In Numerical mathematics and advanced applications — ENUMATH 2019, 2021

   Abs DOI Bib

   We examine the use of the Dirichlet-to-Neumann coarse space within an additive Schwarz method to solve the Helmholtz equation in 2D. In particular, we focus on the selection of how many eigenfunctions should go into the coarse space. We find that wave number independent convergence of a preconditioned iterative method can be achieved in certain special cases with an appropriate and novel choice of threshold in the selection criteria. However, this property is lost in a more general setting, including the heterogeneous problem. Nonetheless, the approach converges in a small number of iterations for the homogeneous problem even for relatively large wave numbers and is robust to the number of subdomains used.

   @incollection{Bootland:2021:ONT,
     author = {Bootland, N. and Dolean, Victorita},
     doi = {https://doi.org/10.1007/978-3-030-55874-1_16},
     booktitle = {Numerical mathematics and advanced applications --- ENUMATH 2019},
     pages = {175--184},
     publisher = {Springer},
     series = {LNCSE},
     title = {On the Dirichlet-to-Neumann coarse space for solving the Helmholtz problem using domain decomposition},
     volume = {139},
     year = {2021}
   }

2020

1. SISC

   Natural domain decomposition algorithms for the solution of time-harmonic elastic waves

   R. Brunet, Victorita Dolean, and Martin J. Gander

   SIAM Journal on Scientific Computing, 2020

   Abs DOI Bib

   We study for the first time Schwarz domain decomposition methods for the solution of the Navier equations modeling the propagation of elastic waves. These equations in the time-harmonic regime are difficult to solve by iterative methods, even more so than the Helmholtz equation. We first prove that the classical Schwarz method is not convergent when applied to the Navier equations and can thus not be used as an iterative solver, only as a preconditioner for a Krylov method. We then introduce more natural transmission conditions between the subdomains and show that if the overlap is not too small, this new Schwarz method is convergent. We illustrate our results with numerical experiments, both for situations covered by our technical two subdomain analysis and situations that go far beyond, including many subdomains, cross points, heterogeneous materials in a transmission problem, and Krylov acceleration. Our numerical results show that the Schwarz method with adapted transmission conditions leads systematically to a better solver for the Navier equations than the classical Schwarz method.

   @article{Brunet:2020:NAT,
     author = {Brunet, R. and Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1137/19M125858X},
     journal = {SIAM Journal on Scientific Computing},
     number = {5},
     pages = {A3313--A3339},
     title = {Natural domain decomposition algorithms for the solution of time-harmonic elastic waves},
     url = {https://epubs.siam.org/doi/abs/10.1137/19M125858X},
     volume = {42},
     year = {2020}
   }

2. SEG

   Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners

   Victorita Dolean, P.-H. Tournier, P. Jolivet, and S. Operto

   In , 2020

   Abs DOI Bib

   Frequency-domain full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with a few frequencies and attenuation is easily implemented without computational overhead. In the frequency domain, wave modeling is a Helmholtz-type boundary-value problem which requires to solve a large and sparse system of linear equations per frequency with multiple right-hand sides (sources). This system can be solved with direct or iterative methods. While the former are suitable for FWI application on 3D dense OBC acquisitions covering spatial domains of moderate size, the later should be the approach of choice for sparse node acquisitions covering large domains (more than 50 millions of unknowns). Fast convergence of iterative solvers for Helmholtz problems remains however challenging in high frequency regime due to the non definiteness of the Helmholtz operator, on one side and on the discretization constraints in order to minimize the dispersion error for a given frequency, on the other side, hence requiring efficient preconditioners. In this study, we use the Krylov subspace GMRES iterative solver combined with a two-level domain-decomposition preconditioner. Discretization relies on continuous Lagrange finite elements of order 3 on unstructured tetrahedral meshes to comply with complex geometries and adapt the size of the elements to the local wavelength (h-adaptivity). We assess the accuracy, the convergence and the scalability of our method with the acoustic 3D SEG/EAGE Overthrust model up to a frequency of 20 Hz.

   @inproceedings{Dolean:2020:LAR,
     author = {Dolean, Victorita and Tournier, P.-H. and Jolivet, P. and Operto, S.},
     doi = {https://doi.org/10.1190/segam2020-3427414.1},
     journal = {SEG Technical Program Expanded Abstracts 2020},
     pages = {2683--2688},
     title = {Large-scale frequency-domain seismic wave modeling on h-adaptive tetrahedral meshes with iterative solver and multi-level domain-decomposition preconditioners},
     year = {2020}
   }

3. SEG

   Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners

   Victorita Dolean, P. Jolivet, P. Tournier, and S. Operto

   EAGE 2020 Annual Conference & Exhibition Online, 2020

   Abs DOI Bib

   Frequency-domain full-waveform inversion (FWI) is suitable for long-offset stationary-recording acquisition, since reliable subsurface models can be reconstructed with a few frequencies and attenuation is easily implemented without computational overhead. In the frequency domain, wave modelling is a Helmholtz-type boundary-value problem which requires to solve a large and sparse linear system per frequency with multiple right-hand sides. This system can be solved with direct or iterative methods. While the former are suitable for FWI application on 3D dense OBC acquisitions covering spatial domains of moderate size, the later should be the approach of choice for sparse node acquisitions covering large domains (> 50 millions of unknowns). Fast convergence of iterative solvers for Helmholtz problems remains however challenging due to the non definiteness of the Helmholtz operator, hence requiring efficient preconditioners. Here, we use the Krylov subspace GMRES iterative solver combined with a multi-level domain-decomposition preconditioner. Discretization relies on finite elements on tetrahedral meshes to comply with complex geometries and adapt the size of the elements to the local wavelength (h-adaptivity). We assess the convergence and the scalability of our method with the 3D SEG/EAGE Overthrust model up to a frequency of 20Hz and discuss its efficiency for multi right-hand side processing.

   @article{Dolean:2020:ITE,
     author = {Dolean, Victorita and Jolivet, P. and Tournier, P. and Operto, S.},
     doi = {https://doi.org/10.3997/2214-4609.202011328},
     journal = {EAGE 2020 Annual Conference & Exhibition Online},
     pages = {1--5},
     title = {Iterative frequency-domain seismic wave solvers based on multi-level domain-decomposition preconditioners},
     year = {2020}
   }

4. DDMSE25

   Can classical Schwarz methods for time-harmonic elastic waves converge?

   R. Brunet, Victorita Dolean, and Martin J. Gander

   In Domain decomposition methods in science and engineering XXV, 2020

   Abs DOI Bib

   The propagation of waves in elastic media is a problem of undeniable practical importance in geophysics. in several important applications - e.g. seismic exploration or earthquake prediction - one seeks to infer unknown material properties of the earth’s subsurface by sending seismic waves down and measuring the scattered field which comes back, implying the solution of inverse problems.

   @incollection{Brunet:2020:CAN,
     author = {Brunet, R. and Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1007/978-3-030-56750-7_49},
     booktitle = {Domain decomposition methods in science and engineering XXV},
     pages = {425--432},
     publisher = {Springer},
     series = {LNCSE},
     title = {Can classical Schwarz methods for time-harmonic elastic waves converge?},
     volume = {138},
     year = {2020}
   }

5. ICOSAHOM2018

   Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions

   G. Barrenechea, M. Bosy, and Victorita Dolean

   In Spectral and high order methods for partial differential equations — ICOSAHOM 2018, 2020

   Abs DOI Bib

   In several studies it has been observed that, when using stabilised elements for both velocity and pressure, the error for the pressure is smaller, or even of a higher order in some cases, than the one obtained when using inf-sup stable (although no formal proof of either of these facts has been given). This increase in polynomial order requires the introduction of stabilising terms, since the finite element pairs used do not guarantee the inf-sup condition. With this motivation, we apply the stabilisation approach to the hybrid discontinuous Galerkin discretisation for the Stokes problem with non-standard boundary conditions.

   @incollection{Barrenechea:2020:STA,
     author = {Barrenechea, G. and Bosy, M. and Dolean, Victorita},
     doi = {https://doi.org/10.1007/978-3-030-39647-3_13},
     booktitle = {Spectral and high order methods for partial differential equations --- ICOSAHOM 2018},
     pages = {179--189},
     publisher = {Springer},
     series = {Lecture Notes in Computational Science and Engineering},
     title = {Stabilised hybrid discontinuous Galerkin methods for the Stokes problem with non-standard boundary conditions},
     volume = {134},
     year = {2020}
   }

2019

1. PARCO

   Microwave tomographic imaging of cerebrovascular accidents by using high-performance computing

   P.-H. Tournier, I. Aliferis, M. Bonazzoli, M. Buhan, M. Darbas, Victorita Dolean, F. Hecht, P. Jolivet, I. El Kanfoud, C. Migliaccio, F. Nataf, Ch. Pichot, and S. Semenov

   Parallel Computing, 2019

   Abs DOI Bib

   The motivation of this work is the detection of cerebrovascular accidents by microwave tomographic imaging. This requires the solution of an inverse problem relying on a minimization algorithm (for example, gradient-based), where successive iterations consist in repeated solutions of a direct problem. The reconstruction algorithm is extremely computationally intensive and makes use of efficient parallel algorithms and high-performance computing. The feasibility of this type of imaging is conditioned on one hand by an accurate reconstruction of the material properties of the propagation medium and on the other hand by a considerable reduction in simulation time. Fulfilling these two requirements will enable a very rapid and accurate diagnosis. From the mathematical and numerical point of view, this means solving Maxwell’s equations in time-harmonic regime by appropriate domain decomposition methods, which are naturally adapted to parallel architectures.

   @article{Tournier:2019:MIC,
     author = {Tournier, P.-H. and Aliferis, I. and Bonazzoli, M. and de Buhan, M. and Darbas, M. and Dolean, Victorita and Hecht, F. and Jolivet, P. and Kanfoud, I. El and Migliaccio, C. and Nataf, F. and Pichot, Ch. and Semenov, S.},
     doi = {https://doi.org/10.1016/j.parco.2019.02.004},
     journal = {Parallel Computing},
     pages = {88--97},
     title = {Microwave tomographic imaging of cerebrovascular accidents by using high-performance computing},
     url = {https://doi.org/10.1016/j.parco.2019.02.004},
     volume = {85},
     year = {2019}
   }

2. CMAM

   Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem

   G. Barrenechea, M. Bosy, Victorita Dolean, F. Nataf, and Pierre-Henri Tournier

   Computational Methods in Applied Mathematics, 2019

   Abs DOI Bib

   Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of nonstandard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the nonstandard interface conditions are naturally defined at the boundary between elements. In this paper, we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with nonstandard boundary conditions. The full stability and convergence analysis of the discretisation method is presented, and the results are corroborated by numerical experiments. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.

   @article{Barrenechea:2019:HYB,
     author = {Barrenechea, G. and Bosy, M. and Dolean, Victorita and Nataf, F. and Tournier, Pierre-Henri},
     doi = {https://doi.org/10.1515/cmam-2018-0005},
     journal = {Computational Methods in Applied Mathematics},
     number = {4},
     pages = {703--722},
     title = {Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem},
     url = {https://doi.org/10.1515/cmam-2018-0005},
     volume = {19},
     year = {2019}
   }

3. MCOM

   Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption

   M. Bonazzoli, Victorita Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier

   Mathematics of Computation, 2019

   Abs DOI Bib

   This paper rigorously analyses preconditioners for the time- harmonic Maxwell equations with absorption, where the PDE is discretised using curl-conforming finite-element methods of fixed, arbitrary order and the preconditioner is constructed using additive Schwarz domain decomposition methods. The theory developed here shows that if the absorption is large enough, and if the subdomain and coarse mesh diameters and overlap are chosen appropriately, then the classical two-level overlapping additive Schwarz preconditioner (with PEC boundary conditions on the subdomains) performs optimally—in the sense that GMRES converges in a wavenumber-independent number of iterations—for the problem with absorption. An important feature of the theory is that it allows the coarse space to be built from low-order elements even if the PDE is discretised using high-order elements. It also shows that additive methods with minimal overlap can be robust. Numerical experiments are given that illustrate the theory and its dependence on various parameters. These experiments motivate some extensions of the preconditioners which have better robustness for problems with less absorption, including the propagative case. At the end of the paper we illustrate the performance of these on two substantial applications; the first (a problem with absorption arising from medical imaging) shows the empirical robustness of the preconditioner against heterogeneity, and the second (scattering by a COBRA cavity) shows good scalability of the preconditioner with up to 3,000 processors.

   @article{Bonazzoli:2019:DOM,
     author = {Bonazzoli, M. and Dolean, Victorita and Graham, I. G. and Spence, E. A. and Tournier, P.-H.},
     doi = {https://doi.org/10.1090/mcom/3447},
     journal = {Mathematics of Computation},
     number = {320},
     pages = {2559--2604},
     title = {Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption},
     url = {https://doi.org/10.1090/mcom/3447},
     volume = {88},
     year = {2019}
   }

4. APNUM

   An introduction to multi-trace formulations and associated domain decomposition solvers

   X. Claeys, Victorita Dolean, and Martin J. Gander

   Applied Numerical Mathematics, 2019

   Abs DOI Bib

   Multi-trace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. The first aim of the present contribution is to provide a gentle introduction to MTFs. We introduce these formulations on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multi-trace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calderón projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments.

   @article{Claeys:2019:INT,
     author = {Claeys, X. and Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1016/j.apnum.2018.07.006},
     journal = {Applied Numerical Mathematics},
     pages = {69--86},
     title = {An introduction to multi-trace formulations and associated domain decomposition solvers},
     url = {https://doi.org/10.1016/j.apnum.2018.07.006},
     volume = {135},
     year = {2019}
   }

5. IEEE-JERM

   Detection of simulated brain strokes using microwave tomography

   V. L. Coli, P.-H. Tournier, Victorita Dolean, I. El Kanfoud, C. Pichot, C. Migliaccio, and L. Blanc-Féraud

   IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology, 2019

   Abs Bib

   Brain strokes are one of the leading causes of disability and mortality in adults in developed countries. Ischemic stroke (85% of total cases) and hemorrhagic stroke (15%) must be treated with opposing therapies, and thus, the nature of the stroke must be determined quickly in order to apply the appropriate treatment. Recent studies in biomedical imaging have shown that strokes produce variations in the complex electric permittivity of brain tissues, which can be detected by means of microwave tomography. Here, we present some synthetic results obtained with an experimental microwave tomography-based portable system for the early detection and monitoring of brain strokes. The determination of electric permittivity first requires the solution of a coupled forward-inverse problem. We make use of massive parallel computation from domain decomposition method and regularization techniques for optimization methods. Synthetic data are obtained with electromagnetic simulations corrupted by noise, which have been derived from measurements errors of the experimental imaging system. Results demonstrate the possibility to detect hemorrhagic strokes with microwave systems when applying the proposed reconstruction algorithm with edge preserving regularization.

   @article{Coli:2019:DET,
     author = {Coli, V. L. and Tournier, P.-H. and Dolean, Victorita and Kanfoud, I. El and Pichot, C. and Migliaccio, C. and Blanc-F{\'e}raud, L.},
     journal = {IEEE Journal of Electromagnetics, RF and Microwaves in Medicine and Biology},
     number = {4},
     pages = {254--260},
     title = {Detection of simulated brain strokes using microwave tomography},
     url = {https://ieeexplore.ieee.org/document/8731640},
     volume = {3},
     year = {2019}
   }

2018

1. M2AN

   Asymptotic analysis of optimized Schwarz methods for Maxwell’s equations with discontinuous coefficients

   Victorita Dolean, Martin J. Gander, and Erwin Veneros

   ESAIM: Mathematical Modelling and Numerical Analysis, 2018

   Abs DOI Bib

   Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.

   @article{Dolean:2018:ASY,
     author = {Dolean, Victorita and Gander, Martin J. and Veneros, Erwin},
     doi = {https://doi.org/10.1051/m2an/2018041},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
     number = {6},
     pages = {2457--2477},
     title = {Asymptotic analysis of optimized Schwarz methods for Maxwell's equations with discontinuous coefficients},
     url = {https://doi.org/10.1051/m2an/2018041},
     volume = {52},
     year = {2018}
   }

2. ETNA

   Numerical assessment of two-level domain decomposition preconditioners for incompressible Stokes and elasticity equations

   G. Barrenechea, M. Bosy, and Victorita Dolean

   Electronic Transactions on Numerical Analysis, 2018

   Abs Bib

   Solving the linear elasticity and Stokes equations by an optimal domain decomposition method derived algebraically involves the use of non-standard interface conditions. The one-level domain decomposition preconditioners are based on the solution of local problems. This has the undesired consequence that the results are not scalable, which means that the number of iterations needed to reach convergence increases with the number of subdomains. This is the reason why in this work we introduce, and test numerically, two-level preconditioners. Such preconditioners use a coarse space in their construction. We consider the nearly incompressible elasticity problems and Stokes equations, and discretise them by using two finite element methods, namely, the hybrid discontinuous Galerkin and Taylor-Hood discretisations.

   @article{Barrenechea:2018:NUM,
     author = {Barrenechea, G. and Bosy, M. and Dolean, Victorita},
     journal = {Electronic Transactions on Numerical Analysis},
     pages = {41--63},
     title = {Numerical assessment of two-level domain decomposition preconditioners for incompressible Stokes and elasticity equations},
     url = {https://epub.oeaw.ac.at/?arp=0x003892f2},
     volume = {49},
     year = {2018}
   }

3. CAMWA

   An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell’s equations

   M. Bonazzoli, Victorita Dolean, F. Hecht, and F. Rapetti

   Computers and Mathematics with Applications, 2018

   Abs DOI Bib

   In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell’s equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell’s equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.

   @article{Bonazzoli:2018:EXA,
     author = {Bonazzoli, M. and Dolean, Victorita and Hecht, F. and Rapetti, F.},
     doi = {https://doi.org/10.1016/j.camwa.2017.11.013},
     journal = {Computers and Mathematics with Applications},
     number = {5},
     pages = {1498--1514},
     title = {An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations},
     url = {https://doi.org/10.1016/j.camwa.2017.11.013},
     volume = {75},
     year = {2018}
   }

4. IJNM

   Parallel preconditioners for high-order discretizations arising from full system modeling for brain microwave imaging

   M. Bonazzoli, Victorita Dolean, F. Rapetti, and P.-H. Tournier

   International Journal of Numerical Modelling: Electronic Networks, Devices and Fields, 2018

   Abs DOI Bib

   This paper combines the use of high-order finite element methods with parallel preconditioners of domain decomposition type for solving electromagnetic problems arising from brain microwave imaging. The numerical algorithms involved in such complex imaging systems are computationally expensive since they require solving the direct problem of Maxwell equations several times. Moreover, wave propagation problems in the high-frequency regime are challenging because a sufficiently high number of unknowns are required to accurately represent the solution. To use these algorithms in practice for brain stroke diagnosis, running time should be reasonable. The method presented in this paper, coupling high-order finite elements and parallel preconditioners, makes it possible to reduce the overall computational cost and simulation time while maintaining accuracy.

   @article{Bonazzoli:2018:PAR,
     author = {Bonazzoli, M. and Dolean, Victorita and Rapetti, F. and Tournier, P.-H.},
     doi = {https://doi.org/10.1002/jnm.2229},
     journal = {International Journal of Numerical Modelling: Electronic Networks, Devices and Fields},
     number = {2},
     pages = {e2229},
     title = {Parallel preconditioners for high-order discretizations arising from full system modeling for brain microwave imaging},
     url = {https://doi.org/10.1002/jnm.2229},
     volume = {31},
     year = {2018}
   }

5. DDMSE24

   A two-level domain-decomposition preconditioner for the time-harmonic Maxwell’s equations

   M. Bonazzoli, Victorita Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier

   In Domain decomposition methods in science and engineering XXIV, 2018

   Abs DOI Bib

   The construction of fast iterative solvers for the indefinite time-harmonic Maxwell’s system at mid- to high-frequency is a problem of great current interest. Some of the difficulties that arise are similar to those encountered in the case of the mid- to high-frequency Helmholtz equation. Here we investigate how two-level domain-decomposition preconditioners recently proposed for the Helmholtz equation work in the Maxwell case, both from the theoretical and numerical points of view.

   @incollection{Bonazzoli:2018:TWO,
     author = {Bonazzoli, M. and Dolean, Victorita and Graham, I. G. and Spence, E. A. and Tournier, P.-H.},
     doi = {https://doi.org/10.1007/978-3-319-93873-8_12},
     booktitle = {Domain decomposition methods in science and engineering XXIV},
     pages = {149--157},
     publisher = {Springer},
     series = {LNCSE},
     title = {A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations},
     volume = {125},
     year = {2018}
   }

6. DDMSE24

   Two-level preconditioners for the Helmholtz equation

   M. Bonazzoli, Victorita Dolean, I. G. Graham, E. A. Spence, and P.-H. Tournier

   In Domain decomposition methods in science and engineering XXIV, 2018

   Abs DOI Bib

   In this paper we compare numerically two different coarse space definitions for two-level domain decomposition preconditioners for the Helmholtz equation, both in two and three dimensions. While we solve the pure Helmholtz problem without absorption, the preconditioners are built from problems with absorption. In the first method, the coarse space is based on the discretization of the problem with absorption on a coarse mesh, with diameter constrained by the wavenumber. In the second method, the coarse space is built by solving local eigenproblems involving the Dirichlet-to-Neumann (DtN) operator.

   @incollection{Bonazzoli:2018:HEL,
     author = {Bonazzoli, M. and Dolean, Victorita and Graham, I. G. and Spence, E. A. and Tournier, P.-H.},
     doi = {https://doi.org/10.1007/978-3-319-93873-8_11},
     booktitle = {Domain decomposition methods in science and engineering XXIV},
     pages = {139--147},
     publisher = {Springer},
     series = {LNCSE},
     title = {Two-level preconditioners for the Helmholtz equation},
     volume = {125},
     year = {2018}
   }

2017

1. IEEE-APM

   Numerical modeling and high-speed parallel computing: New perspectives on tomographic microwave imaging for brain stroke detection and monitoring

   P.-H. Tournier, M. Bonazzoli, Victorita Dolean, F. Rapetti, F. Hecht, F. Nataf, I. Aliferis, I. El Kanfoud, C. Migliaccio, M. Buhan, M. Darbas, S. Semenov, and C. Pichot

   IEEE Antennas and Propagation Magazine, 2017

   Abs DOI Bib

   This article deals with microwave tomography for brain stroke imaging using state-of-the-art numerical modeling and massively parallel computing. Iterative microwave tomographic imaging requires the solution of an inverse problem based on a minimization algorithm (e.g., gradient based) with successive solutions of a direct problem such as the accurate modeling of a whole-microwave measurement system. Moreover, a sufficiently high number of unknowns is required to accurately represent the solution. As the system will be used for detecting a brain stroke (ischemic or hemorrhagic) as well as for monitoring during the treatment, the running times for the reconstructions should be reasonable. The method used is based on high-order finite elements, parallel preconditioners from the domain decomposition method and domain-specific language with the opensource FreeFEM++ solver.

   @article{Tournier:2017:NUM,
     author = {Tournier, P.-H. and Bonazzoli, M. and Dolean, Victorita and Rapetti, F. and Hecht, F. and Nataf, F. and Aliferis, I. and Kanfoud, I. El and Migliaccio, C. and de Buhan, M. and Darbas, M. and Semenov, S. and Pichot, C.},
     doi = {https://ieeexplore.ieee.org/document/8014422},
     journal = {IEEE Antennas and Propagation Magazine},
     number = {5},
     pages = {98--110},
     title = {Numerical modeling and high-speed parallel computing: New perspectives on tomographic microwave imaging for brain stroke detection and monitoring},
     url = {https://ieeexplore.ieee.org/document/8014422},
     volume = {59},
     year = {2017}
   }

2. DDMSE23

   Optimized Schwarz methods for heterogeneous Helmholtz and Maxwell’s equations

   Victorita Dolean, Martin J. Gander, E. Veneros, and H. Zhang

   In Domain decomposition methods in science and engineering XXIII, 2017

   Abs DOI Bib

   Both the Helmholtz equation and the time-harmonic Maxwell’s equations are difficult to solve by iterative methods in the intermediate to high frequency regime, and domain decomposition methods are among the most promising techniques for this task. We focus here on the case of dissipative and conductive media with strongly heterogeneous coefficients, and develop optimized transmission conditions for this case. We establish a link for the use of such conditions between the case of Helmholtz and Maxwell’s equations, and show that in both cases jumps aligned with the interfaces of the subdomains can improve the convergence of the subdomain iteration.

   @incollection{Dolean:2017:OPT,
     author = {Dolean, Victorita and Gander, Martin J. and Veneros, E. and Zhang, H.},
     doi = {https://doi.org/10.1007/978-3-319-52389-7_13},
     booktitle = {Domain decomposition methods in science and engineering XXIII},
     pages = {145--152},
     publisher = {Springer},
     series = {LNCSE},
     title = {Optimized Schwarz methods for heterogeneous Helmholtz and Maxwell's equations},
     volume = {116},
     year = {2017}
   }

3. DDMSE23

   Schwarz preconditioning for high order edge element discretizations of the time-harmonic Maxwell’s equations

   M. Bonazzoli, Victorita Dolean, R. Pasquetti, and F. Rapetti

   In Domain decomposition methods in science and engineering XXIII, 2017

   Abs DOI Bib

   We focus on high order edge element approximations of waveguide problems. For the associated linear systems, we analyze the impact of two Schwarz preconditioners, the Optimized Additive Schwarz (OAS) and the Optimized Restricted Additive Schwarz (ORAS), on the convergence of the iterative solver.

   @incollection{Bonazzoli:2017:SCH,
     author = {Bonazzoli, M. and Dolean, Victorita and Pasquetti, R. and Rapetti, F.},
     doi = {https://doi.org/10.1007/978-3-319-52389-7_10},
     booktitle = {Domain decomposition methods in science and engineering XXIII},
     pages = {117--124},
     publisher = {Springer},
     series = {LNCSE},
     title = {Schwarz preconditioning for high order edge element discretizations of the time-harmonic Maxwell's equations},
     volume = {116},
     year = {2017}
   }

4. DDMSE23

   Closed form inverse of local multi-trace operators

   A. Ayala, X. Claeys, Victorita Dolean, and Martin J. Gander

   In Domain decomposition methods in science and engineering XXIII, 2017

   Abs DOI Bib

   Local multi-trace formulations are a way to express transmission problems. They are based on integral formulations of the solution on each subdomain, and between the subdomains both the known jumps in the traces and the fluxes of the transmission problem are imposed, and thus the method contains multiple traces in its formulation. We show in this paper that it is possible to derive a closed form inverse for local multi-trace operators for an elliptic model problem.

   @incollection{Ayala:2017:CLO,
     author = {Ayala, A. and Claeys, X. and Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1007/978-3-319-52389-7_9},
     booktitle = {Domain decomposition methods in science and engineering XXIII},
     pages = {107--115},
     publisher = {Springer},
     series = {LNCSE},
     title = {Closed form inverse of local multi-trace operators},
     volume = {116},
     year = {2017}
   }

2016

1. SISC

   Nonlinear preconditioning: how to use a nonlinear Schwarz method to precondition Newton’s method

   Victorita Dolean, Martin J. Gander, W. Kheriji, F. Kwok, and R. Masson

   SIAM Journal on Scientific Computing, 2016

   Abs DOI Bib

   For linear problems, domain decomposition methods can be used directly as iterative solvers but also as preconditioners for Krylov methods. In practice, Krylov acceleration is almost always used, since the Krylov method finds a much better residual polynomial than the stationary iteration and thus converges much faster. We show in this paper that also for nonlinear problems, domain decomposition methods can be used either directly as iterative solvers or as preconditioners for Newton’s method. For the concrete case of the parallel Schwarz method, we show that we obtain a preconditioner we call RASPEN (restricted additive Schwarz preconditioned exact Newton), which is similar to ASPIN (additive Schwarz preconditioned inexact Newton) but with all components directly defined by the iterative method. This has the advantage that RASPEN already converges when used as an iterative solver, in contrast to ASPIN, and we thus get a substantially better preconditioner for Newton’s method. The iterative construction also allows us to naturally define a coarse correction using the multigrid full approximation scheme, which leads to a convergent two-level nonlinear iterative domain decomposition method and a two level RASPEN nonlinear preconditioner. We illustrate our findings with numerical results on the Forchheimer equation and a nonlinear diffusion problem.

   @article{Dolean:2016:NON,
     author = {Dolean, Victorita and Gander, Martin J. and Kheriji, W. and Kwok, F. and Masson, R.},
     doi = {https://doi.org/10.1137/15M102887X},
     journal = {SIAM Journal on Scientific Computing},
     number = {6},
     pages = {A3357--A3380},
     title = {Nonlinear preconditioning: how to use a nonlinear Schwarz method to precondition Newton's method},
     url = {https://doi.org/10.1137/15M102887X},
     volume = {38},
     year = {2016}
   }

2. DDMSE22

   Schwarz methods for second order Maxwell equations in 3D with coefficient jumps

   Victorita Dolean, Martin J. Gander, and E. Veneros

   In Domain decomposition methods in science and engineering XXII, 2016

   Abs DOI Bib

   We study non-overlapping Schwarz Methods for solving second order time-harmonic 3D Maxwell equations in heterogeneous media. Choosing the interfaces between the subdomains to be aligned with the discontinuities in the coefficients, we show for a model problem that while the classical Schwarz method is not convergent, optimized transmission conditions dependent on the discontinuities of the coefficients lead to convergent methods. We prove asymptotically that the resulting methods converge in certain cases independently of the mesh parameter, and convergence can even become better as the coefficient jumps increase.

   @incollection{Dolean:2016:SCH,
     author = {Dolean, Victorita and Gander, Martin J. and Veneros, E.},
     doi = {https://doi.org/10.1007/978-3-319-18827-0_48},
     booktitle = {Domain decomposition methods in science and engineering XXII},
     pages = {471--479},
     publisher = {Springer},
     series = {LNCSE},
     title = {Schwarz methods for second order Maxwell equations in 3D with coefficient jumps},
     volume = {104},
     year = {2016}
   }

3. DDMSE22

   Multitrace formulations and Dirichlet-Neumann algorithms

   Victorita Dolean and Martin J. Gander

   In Domain decomposition methods in science and engineering XXII, 2016

   Abs DOI Bib

   Multitrace formulations (MTF) for boundary integral equations (BIE) were developed over the last few years in [1, 2, 4] for the simulation of electromagnetic problems in piecewise constant media, see also [3] for associated boundary integral methods. The MTFs are naturally adapted to the developments of new block preconditioners, as indicated in [5], but very little is known so far about such associated iterative solvers. The goal of our presentation is to give an elementary introduction to MTFs, and also to establish a natural connection with the more classical Dirichlet-Neumann algorithms that are well understood in the domain decomposition literature, see for example [6, 7]. We present for a model problem a convergence analysis for a naturally arising block iterative method associated with the MTF, and also first numerical results to illustrate what performance one can expect from such an iterative solver.

   @incollection{Dolean:2016:MUL,
     author = {Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1007/978-3-319-18827-0_13},
     booktitle = {Domain decomposition methods in science and engineering XXII},
     pages = {147--155},
     publisher = {Springer},
     series = {LNCSE},
     title = {Multitrace formulations and Dirichlet-Neumann algorithms},
     volume = {104},
     year = {2016}
   }

2015

1. JCAM

   Addendum to “A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator”

   L. Conen, Victorita Dolean, R. Krause, and F. Nataf

   Journal of Computational and Applied Mathematics, 2015

   Abs DOI Bib

   This communication gives an addendum to the paper Conen et al. (2014).

   @article{Conen:2015:ADD,
     author = {Conen, L. and Dolean, Victorita and Krause, R. and Nataf, F.},
     doi = {https://doi.org/10.1016/j.cam.2015.04.031},
     journal = {Journal of Computational and Applied Mathematics},
     pages = {670--674},
     title = {Addendum to ``A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator''},
     url = {https://doi.org/10.1016/j.cam.2015.04.031},
     volume = {290},
     year = {2015}
   }

2. JCP

   Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell’s equations

   Victorita Dolean, Martin J. Gander, Stéphane Lanteri, Jin-Fa Lee, and Zhen Peng

   Journal of Computational Physics, 2015

   Abs DOI Bib

   The time-harmonic Maxwell equations describe the propagation of electromagnetic waves and are therefore fundamental for the simulation of many modern devices we have become used to in everyday life. The numerical solution of these equations is hampered by two fundamental problems: first, in the high frequency regime, very fine meshes need to be used in order to avoid the pollution effect well known for the Helmholtz equation, and second the large scale systems obtained from the vector valued equations in three spatial dimensions need to be solved by iterative methods, since direct factorizations are not feasible any more at that scale. As for the Helmholtz equation, classical iterative methods applied to discretized Maxwell equations have severe convergence problems. We explain in this paper a family of domain decomposition methods based on well chosen transmission conditions. We show that all transmission conditions proposed so far in the literature, both for the first and second order formulation of Maxwell’s equations, can be written and optimized in the common framework of optimized Schwarz methods, independently of the first or second order formulation one uses, and the performance of the corresponding algorithms is identical. We use a decomposition into transverse electric and transverse magnetic fields to describe these algorithms, which greatly simplifies the convergence analysis of the methods. We illustrate the performance of our algorithms with large scale numerical simulations.

   @article{Dolean:2015:EFF,
     author = {Dolean, Victorita and Gander, Martin J. and Lanteri, St{\'e}phane and Lee, Jin-Fa and Peng, Zhen},
     doi = {https://doi.org/10.1016/j.jcp.2014.09.024},
     journal = {Journal of Computational Physics},
     pages = {232--247},
     title = {Effective transmission conditions for domain decomposition methods applied to the time-harmonic curl-curl Maxwell's equations},
     url = {https://doi.org/10.1016/j.jcp.2014.09.024},
     volume = {280},
     year = {2015}
   }

3. ETNA

   Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell’s equations

   M. El Bouajaji, Victorita Dolean, Martin J. Gander, Stéphane Lanteri, and R. Perrussel

   Electronic Transactions on Numerical Analysis, 2015

   Abs DOI Bib

   We show in this paper how to properly discretize optimized Schwarz methods for the time-harmonic Maxwell’s equations in two and three spatial dimensions using a discontinuous Galerkin (DG) method. Due to the multiple traces between elements in the DG formulation, it is not clear a priori how the more sophisticated transmission conditions in optimized Schwarz methods should be discretized, and the most natural approach, at convergence of the Schwarz method, does not lead to the monodomain DG solution, which implies that for such discretizations, the DG error estimates do not hold when the Schwarz method has converged. We present here a consistent discretization of the transmission conditions in the framework of a DG weak formulation, for which we prove that the multidomain and monodomain solutions for the Maxwell’s equations are the same. We illustrate our results with several numerical experiments of propagation problems in homogeneous and heterogeneous media.

   @article{ElBouajaji:2015:DIS,
     author = {Bouajaji, M. El and Dolean, Victorita and Gander, Martin J. and Lanteri, St{\'e}phane and Perrussel, R.},
     doi = {https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol44/abstract.php?pages=572-592},
     journal = {Electronic Transactions on Numerical Analysis},
     pages = {572--592},
     title = {Discontinuous Galerkin discretizations of optimized Schwarz methods for solving the time-harmonic Maxwell's equations},
     url = {https://etna.ricam.oeaw.ac.at/volumes/2011-2020/vol44/abstract.php?pages=572-592},
     volume = {44},
     year = {2015}
   }

2014

1. Numer. Math.

   Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps

   N. Spillane, Victorita Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl

   Numerische Mathematik, 2014

   Abs DOI Bib

   Coarse spaces are instrumental in obtaining scalability for domain decomposition methods for partial differential equations (PDEs). However, it is known that most popular choices of coarse spaces perform rather weakly in the presence of heterogeneities in the PDE coefficients, especially for systems of PDEs. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems in the overlaps of subdomains that isolate the terms responsible for slow convergence. We prove a general theoretical result that rigorously establishes the robustness of the new coarse space and give some numerical examples on two and three dimensional heterogeneous PDEs and systems of PDEs that confirm this property.

   @article{Spillane:2014:ABS,
     author = {Spillane, N. and Dolean, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R.},
     doi = {https://doi.org/10.1007/s00211-013-0576-y},
     journal = {Numerische Mathematik},
     number = {4},
     pages = {741--770},
     title = {Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps},
     url = {https://link.springer.com/article/10.1007/s00211-013-0576-y},
     volume = {126},
     year = {2014}
   }

2. JCAM

   A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator

   L. Conen, Victorita Dolean, R. Krause, and F. Nataf

   Journal of Computational and Applied Mathematics, 2014

   Abs DOI Bib

   The Helmholtz equation governing wave propagation and scattering phenomena is difficult to solve numerically. Its discretization with piecewise linear finite elements results in typically large linear systems of equations. The inherently parallel domain decomposition methods constitute hence a promising class of preconditioners. An essential element of these methods is a good coarse space. Here, the Helmholtz equation presents a particular challenge, as even slight deviations from the optimal choice can be devastating. In this paper, we present a coarse space that is based on local eigenproblems involving the Dirichlet-to-Neumann operator. Our construction is completely automatic, ensuring good convergence rates without the need for parameter tuning. Moreover, it naturally respects local variations in the wave number and is hence suited also for heterogeneous Helmholtz problems. The resulting method is parallel by design and its efficiency is demonstrated on 2D homogeneous and heterogeneous numerical examples.

   @article{Conen:2014:COA,
     author = {Conen, L. and Dolean, Victorita and Krause, R. and Nataf, F.},
     doi = {https://doi.org/10.1016/j.cam.2014.03.031},
     journal = {Journal of Computational and Applied Mathematics},
     pages = {83--99},
     title = {A coarse space for heterogeneous Helmholtz problems based on the Dirichlet-to-Neumann operator},
     url = {https://doi.org/10.1016/j.cam.2014.03.031},
     volume = {271},
     year = {2014}
   }

3. OGST

   Two-level domain decomposition methods for highly heterogeneous Darcy equations. Connections with multiscale methods

   Victorita Dolean, Frédéric Nataf, N. Spillane, and H. Xiang

   Oil & Gas Science and Technology – Revue d’IFP Energies nouvelles, 2014

   Abs DOI Bib

   Multiphase, compositional porous media flow models lead to the solution of highly heterogeneous systems of Partial Differential Equations (PDE). We focus on overlapping Schwarz type methods on parallel computers and on multiscale methods. We present a coarse space [Nataf F., Xiang H., Dolean V., Spillane N. (2011) SIAM J. Sci. Comput. 33, 4, 1623-1642] that is robust even when there are such heterogeneities. The two-level domain decomposition approach is compared to multiscale methods.

   @article{Dolean:2014:TWO,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Spillane, N. and Xiang, H.},
     doi = {https://doi.org/10.2516/ogst/2013206},
     journal = {Oil \& Gas Science and Technology -- Revue d'IFP Energies nouvelles},
     number = {4},
     pages = {731--752},
     title = {Two-level domain decomposition methods for highly heterogeneous Darcy equations. Connections with multiscale methods},
     url = {https://ogst.ifpenergiesnouvelles.fr/articles/ogst/abs/2014/04/ogst130025/ogst130025.html},
     volume = {69},
     year = {2014}
   }

4. DDMSE21

   Optimized Schwarz methods for Maxwell equations with discontinuous coefficients

   Victorita Dolean, Martin J. Gander, and E. Veneros

   In Domain decomposition methods in science and engineering XXI, 2014

   Abs DOI Bib

   We study non-overlapping Schwarz methods for solving time-harmonic Maxwell’s equations in heterogeneous media. We show that the classical Schwarz algorithm is always divergent when coefficient jumps are present along the interface. In the case of transverse magnetic or transverse electric two dimensional formulations, convergence can be achieved in specific configurations only. We then develop optimized Schwarz methods which can take coefficient jumps into account in their transmission conditions. These methods exhibit rapid convergence, and sometimes converge independently of the mesh parameter, even without overlap. We illustrate our analysis with numerical experiments.

   @incollection{Dolean:2014:OPTM,
     author = {Dolean, Victorita and Gander, Martin J. and Veneros, E.},
     doi = {https://doi.org/10.1007/978-3-319-05789-7_49},
     booktitle = {Domain decomposition methods in science and engineering XXI},
     pages = {517--525},
     publisher = {Springer},
     series = {LNCSE},
     title = {Optimized Schwarz methods for Maxwell equations with discontinuous coefficients},
     volume = {98},
     year = {2014}
   }

5. DDMSE21

   Achieving robustness through coarse space enrichment in the two level Schwarz framework

   N. Spillane, Victorita Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl

   In Domain decomposition methods in science and engineering XXI, 2014

   Abs DOI Bib

   As many DD methods the two level Additive Schwarz method may suffer from a lack of robustness with respect to coefficient variation. This is the case in particular if the partition into is not aligned with all jumps in the coefficients. The theoretical analysis traces this lack of robustness back to the so called stable splitting property. In this work we propose to solve a generalized eigenvalue problem in each subdomain which identifies which vectors are responsible for violating the stable splitting property. These vectors are used to span the coarse space and taken care of by a direct solve while all remaining components behave well. The result is a condition number estimate for the two level method which does not depend on the number of subdomains or any jumps in the coefficients.

   @incollection{Spillane:2014:ACH,
     author = {Spillane, N. and Dolean, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R.},
     doi = {https://doi.org/10.1007/978-3-319-05789-7_42},
     booktitle = {Domain decomposition methods in science and engineering XXI},
     pages = {447--455},
     publisher = {Springer},
     series = {LNCSE},
     title = {Achieving robustness through coarse space enrichment in the two level Schwarz framework},
     volume = {98},
     year = {2014}
   }

6. DDMSE21

   DG discretization of optimized Schwarz methods for Maxwell’s equations

   M. El Bouajaji, Victorita Dolean, Martin J. Gander, Stéphane Lanteri, and R. Perrussel

   In Domain decomposition methods in science and engineering XXI, 2014

   Abs DOI Bib

   We study here optimized Schwarz domain decomposition methods for solving the time-harmonic Maxwell equations discretized by a discontinuous Galerkin (DG) method. Due to the particularity of the latter, a discretization of a more sophisticated Schwarz method is not straightforward. A strategy of discretization is shown in the framework of a DG weak formulation, and the equivalence between multi-domain and single-domain solutions is proved. The proposed discrete framework is then illustrated by some numerical results through the simulation of two-dimensional propagation problems.

   @incollection{ElBouajaji:2014:DGD,
     author = {Bouajaji, M. El and Dolean, Victorita and Gander, Martin J. and Lanteri, St{\'e}phane and Perrussel, R.},
     doi = {https://doi.org/10.1007/978-3-319-05789-7_18},
     booktitle = {Domain decomposition methods in science and engineering XXI},
     pages = {217--225},
     publisher = {Springer},
     series = {LNCSE},
     title = {DG discretization of optimized Schwarz methods for Maxwell's equations},
     volume = {98},
     year = {2014}
   }

2013

1. CRMA

   Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method

   N. Spillane, Victorita Dolean, P. Hauret, F. Nataf, and D. Rixen

   Comptes Rendus Mathématique, 2013

   Abs DOI Bib

   FETI is a very popular method, which has proved to be extremely efficient on many large-scale industrial problems. One drawback is that it performs best when the decomposition of the global problem is closely related to the parameters in equations. This is somewhat confirmed by the fact that the theoretical analysis goes through only if some assumptions on the coefficients are satisfied. We propose here to build a coarse space for which the convergence rate of the two-level method is guaranteed regardless of any additional assumptions. We do this by identifying the problematic modes using generalized eigenvalue problems.

   @article{Spillane:2013:SOL,
     author = {Spillane, N. and Dolean, Victorita and Hauret, P. and Nataf, F. and Rixen, D.},
     doi = {https://doi.org/10.1016/j.crma.2013.03.010},
     journal = {Comptes Rendus Math{\'e}matique},
     number = {5-6},
     pages = {197--201},
     title = {Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method},
     url = {https://doi.org/10.1016/j.crma.2013.03.010},
     volume = {351},
     year = {2013}
   }

2. DDMSE20

   Comparison of a one and two parameter family of transmission conditions for Maxwell’s equations with damping

   M. El Bouajaji, Victorita Dolean, Martin J. Gander, and Stéphane Lanteri

   In Domain decomposition methods in science and engineering XX, 2013

   Abs DOI Bib

   Transmission conditions between subdomains have a substantial influence on the convergence of iterative domain decomposition algorithms. For Maxwell’s equations, transmission conditionswhich lead to rapidly converging algorithms have been developed both for the curl-curl formulation of Maxwell’s equation, see [1–3], and also for first order formulations, see [6, 7]. These methods have well found their way into applications, see for example [9] and the references therein. It turns out that good transmission conditions are approximations of transparent boundary conditions.

   @incollection{ElBouajaji:2013:COM,
     author = {Bouajaji, M. El and Dolean, Victorita and Gander, Martin J. and Lanteri, St{\'e}phane},
     doi = {https://doi.org/10.1007/978-3-642-35275-1_31},
     booktitle = {Domain decomposition methods in science and engineering XX},
     pages = {271--278},
     publisher = {Springer},
     series = {LNCSE},
     title = {Comparison of a one and two parameter family of transmission conditions for Maxwell's equations with damping},
     volume = {91},
     year = {2013}
   }

3. DDMSE20

   A two-level Schwarz preconditioner for heterogeneous problems

   Victorita Dolean, Frédéric Nataf, R. Scheichl, and N. Spillane

   In Domain decomposition methods in science and engineering XX, 2013

   Abs DOI Bib

   Coarse space correction is essential to achieve algorithmic scalability in domain decomposition methods. Our goal here is to build a robust coarse space for Schwarz– type preconditioners for elliptic problems with highly heterogeneous coefficients when the discontinuities are not just across but also along subdomain interfaces, where classical results break down [3, 6, 9, 15].

   @incollection{Dolean:2013:TWO,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Scheichl, R. and Spillane, N.},
     doi = {https://doi.org/10.1007/978-3-642-35275-1_8},
     booktitle = {Domain decomposition methods in science and engineering XX},
     pages = {87--94},
     publisher = {Springer},
     series = {LNCSE},
     title = {A two-level Schwarz preconditioner for heterogeneous problems},
     volume = {91},
     year = {2013}
   }

4. DDMSE20

   Symbolic techniques for domain decomposition methods

   T. Cluzeau, Victorita Dolean, Frédéric Nataf, and A. Quadrat

   In Domain decomposition methods in science and engineering XX, 2013

   Abs DOI Bib

   Some algorithmic aspects of systems of PDEs based simulations can be better clarified by means of symbolic computation techniques. This is very important since numerical simulations heavily rely on solving systems of PDEs. For the large-scale problems we deal with in today’s standard applications, it is necessary to rely on iterative Krylov methods that are scalable (i.e., weakly dependent on the number of degrees on freedom and number of subdomains) and have limited memory requirements.

   @incollection{Cluzeau:2013:SYM,
     author = {Cluzeau, T. and Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Quadrat, A.},
     doi = {https://doi.org/10.1007/978-3-642-35275-1_3},
     booktitle = {Domain decomposition methods in science and engineering XX},
     pages = {27--38},
     publisher = {Springer},
     series = {LNCSE},
     title = {Symbolic techniques for domain decomposition methods},
     volume = {91},
     year = {2013}
   }

2012

1. CMAM

   Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps

   Victorita Dolean, Frédéric Nataf, R. Scheichl, and N. Spillane

   Computational Methods in Applied Mathematics, 2012

   Abs DOI Bib

   Coarse grid correction is a key ingredient in order to have scalable domain decomposition methods. For smooth problems, the theory and practice of such two-level methods is well established, but this is not the case for problems with complicated variation and high contrasts in the coefficients. In a previous study, two of the authors introduced a coarse space adapted to highly heterogeneous coefficients using the low frequency modes of the subdomain DtN maps. In this work, we present a rigorous analysis of a two-level overlapping additive Schwarz method with this coarse space, which provides an automatic criterion for the number of modes that need to be added per subdomain to obtain a convergence rate of the order of the constant coefficient case. Our method is suitable for parallel implementation and its efficiency is demonstrated by numerical examples on some challenging problems with high heterogeneities for automatic partitionings.

   @article{Dolean:2012:ANA,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Scheichl, R. and Spillane, N.},
     doi = {https://doi.org/10.2478/cmam-2012-0027},
     journal = {Computational Methods in Applied Mathematics},
     number = {4},
     pages = {391--414},
     title = {Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps},
     url = {https://doi.org/10.2478/cmam-2012-0027},
     volume = {12},
     year = {2012}
   }

2. JNUM

   High performance domain decomposition methods on massively parallel architectures with FreeFem++

   P. Jolivet, V. Dolean, F. Hecht, F. Nataf, C. Prud’Homme, and N. Spillane

   Journal of Numerical Mathematics, 2012

   Abs DOI Bib

   In this document, we present a parallel implementation in freefem++ of scalable two-level domain decomposition methods. Numerical studies with highly heterogeneous problems are then performed on large clusters in order to assert the performance of our code.

   @article{Jolivet:2012:HIG,
     author = {Jolivet, P. and Dolean, V. and Hecht, F. and Nataf, F. and Prud'Homme, C. and Spillane, N.},
     doi = {https://doi.org/10.1515/jnum-2012-0015},
     journal = {Journal of Numerical Mathematics},
     number = {3-4},
     pages = {287--302},
     title = {High performance domain decomposition methods on massively parallel architectures with FreeFem++},
     url = {https://doi.org/10.1515/jnum-2012-0015},
     volume = {20},
     year = {2012}
   }

3. SISC

   Optimized Schwarz methods for the time-harmonic Maxwell equations with damping

   M. El Bouajaji, V. Dolean, M. J. Gander, and S. Lanteri

   SIAM Journal on Scientific Computing, 2012

   Abs DOI Bib

   In a previous paper, two of the authors have proposed and analyzed an entire hierarchy of optimized Schwarz methods for Maxwell’s equations in both the time-harmonic and the time-domain case. The optimization process has been performed in a particular situation where the electric conductivity was neglected. Here, we take into account this physical parameter which leads to a fundamentally different analysis and a new class of algorithms for this more general case. From the mathematical point of view, the approach is different, since the algorithm does not encounter the pathological situations of the zero-conductivity case and thus the optimization problems are different. We analyze one of the algorithms in this class in detail and provide asymptotic results for the remaining ones. We illustrate our analysis with numerical results.

   @article{ElBouajaji:2012:OPT,
     author = {Bouajaji, M. El and Dolean, V. and Gander, M. J. and Lanteri, S.},
     doi = {https://doi.org/10.1137/110842995},
     journal = {SIAM Journal on Scientific Computing},
     number = {4},
     pages = {A2048--A2071},
     title = {Optimized Schwarz methods for the time-harmonic Maxwell equations with damping},
     url = {https://doi.org/10.1137/110842995},
     volume = {34},
     year = {2012}
   }

2011

1. SISC

   A coarse space construction based on local Dirichlet-to-Neumann maps

   Frédéric Nataf, H. Xiang, Victorita Dolean, and N. Spillane

   SIAM Journal on Scientific Computing, 2011

   Abs DOI Bib

   Coarse-grid correction is a key ingredient of scalable domain decomposition methods. In this work we construct coarse-grid space using the low-frequency modes of the subdomain Dirichlet-to-Neumann maps and apply the obtained two-level preconditioners to the extended or the original linear system arising from an overlapping domain decomposition. Our method is suitable for parallel implementation, and its efficiency is demonstrated by numerical examples on problems with large heterogeneities for both manual and automatic partitionings.

   @article{Nataf:2011:ACO,
     author = {Nataf, Fr{\'e}d{\'e}ric and Xiang, H. and Dolean, Victorita and Spillane, N.},
     doi = {https://doi.org/10.1137/100796376},
     journal = {SIAM Journal on Scientific Computing},
     number = {4},
     pages = {1623--1642},
     title = {A coarse space construction based on local Dirichlet-to-Neumann maps},
     url = {https://doi.org/10.1137/100796376},
     volume = {33},
     year = {2011}
   }

2. CRMA

   A robust two-level domain decomposition preconditioner for systems of PDEs

   N. Spillane, Victorita Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl

   Comptes Rendus Mathématique, 2011

   Abs DOI Bib

   Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.

   @article{Spillane:2011:ROB,
     author = {Spillane, N. and Dolean, Victorita and Hauret, P. and Nataf, F. and Pechstein, C. and Scheichl, R.},
     doi = {https://doi.org/10.1016/j.crma.2011.10.021},
     journal = {Comptes Rendus Math{\'e}matique},
     number = {23-24},
     pages = {1255--1259},
     title = {A robust two-level domain decomposition preconditioner for systems of PDEs},
     url = {https://doi.org/10.1016/j.crma.2011.10.021},
     volume = {349},
     year = {2011}
   }

3. ICOSAHOM09

   Hybrid explicit-implicit time integration for grid-induced stiffness in a DGTD method for time domain electromagnetics

   Victorita Dolean, H. Fahs, L. Fezoui, and S. Lanteri

   In Spectral and high order methods for partial differential equations, 2011

   Abs DOI Bib

   In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the numerical modeling of electromagnetic wave propagation. Such methods most often rely on explicit time integration schemes which are constrained by a stability condition that can be very restrictive on highly refined meshes. In this paper, we report on some efforts to design a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on locally refined simplicial meshes. The proposed method consists in applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part.

   @incollection{Dolean:2011:HYB,
     author = {Dolean, Victorita and Fahs, H. and Fezoui, L. and Lanteri, S.},
     doi = {https://doi.org/10.1007/978-3-642-15337-2_13},
     booktitle = {Spectral and high order methods for partial differential equations},
     pages = {163--170},
     publisher = {Springer},
     series = {LNCSE},
     title = {Hybrid explicit-implicit time integration for grid-induced stiffness in a DGTD method for time domain electromagnetics},
     volume = {76},
     year = {2011}
   }

4. DDMSE19

   Schwarz waveform relaxation methods for systems of semi-linear reaction-diffusion equations

   S. Descombes, Victorita Dolean, and Martin J. Gander

   In Domain decomposition methods in science and engineering XIX, 2011

   Abs DOI Bib

   Schwarz waveform relaxation methods have been studied for a wide range of scalar linear partial differential equations (PDEs) of parabolic and hyperbolic type. They are based on a space-time decomposition of the computational domain and the subdomain iteration uses an overlapping decomposition in space. There are only few convergence studies for non-linear PDEs. We analyze in this paper the convergence of Schwarz waveform relaxation applied to systems of semi-linear reaction-diffusion equations. We show that the algorithm converges linearly under certain conditions over long time intervals. We illustrate our results, and further possible convergence behavior, with numerical experiments.

   @incollection{Descombes:2011:SCH,
     author = {Descombes, S. and Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1007/978-3-642-11304-8_49},
     booktitle = {Domain decomposition methods in science and engineering XIX},
     pages = {423--430},
     publisher = {Springer},
     series = {LNCSE},
     title = {Schwarz waveform relaxation methods for systems of semi-linear reaction-diffusion equations},
     volume = {78},
     year = {2011}
   }

5. DDMSE19

   Optimized Schwarz methods for Maxwell’s equations with non-zero electric conductivity

   Victorita Dolean, M. El Bouajaji, M. J. Gander, and S. Lanteri

   In Domain decomposition methods in science and engineering XIX, 2011

   Abs DOI Bib

   The study of optimized Schwarz methods for Maxwell’s equations started with the Helmholtz equation, see [2–4, 11]. For the rot-rot formulation of Maxwell’s equations, optimized Schwarz methods were developed in [1], and for the more general form in [9, 10]. An entire hierarchy of families of optimized Schwarz methods was analyzed in [8], see also [5] for discontinuous Galerkin discretizations and large scale experiments. We present in this paper a first analysis of optimized Schwarz methods for Maxwell’s equations with non-zero electric conductivity. This is an important case for real applications, and requires a new, and fundamentally different optimization of the transmission conditions. We illustrate our analysis with numerical experiments.

   @incollection{Dolean:2011:OPT,
     author = {Dolean, Victorita and Bouajaji, M. El and Gander, M. J. and Lanteri, S.},
     doi = {https://doi.org/10.1007/978-3-642-11304-8_30},
     booktitle = {Domain decomposition methods in science and engineering XIX},
     pages = {269--276},
     publisher = {Springer},
     series = {LNCSE},
     title = {Optimized Schwarz methods for Maxwell's equations with non-zero electric conductivity},
     volume = {78},
     year = {2011}
   }

6. DDMSE19

   Can the discretization modify the performance of Schwarz methods?

   Victorita Dolean and M. J. Gander

   In Domain decomposition methods in science and engineering XIX, 2011

   Abs DOI Bib

   Schwarz domain decomposition methods can be analyzed both at the continuous and discrete level. For consistent discretizations, one would naturally expect that the discretized method performs as predicted by the continuous analysis. We show in this short note for two model problems that this is not always the case, and that the discretization can both increase and decrease the convergence speed predicted by the continuous analysis.

   @incollection{Dolean:2011:CAN,
     author = {Dolean, Victorita and Gander, M. J.},
     doi = {https://doi.org/10.1007/978-3-642-11304-8_11},
     booktitle = {Domain decomposition methods in science and engineering XIX},
     pages = {117--124},
     publisher = {Springer},
     series = {LNCSE},
     title = {Can the discretization modify the performance of Schwarz methods?},
     volume = {78},
     year = {2011}
   }

7. DDMSE19

   Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains

   Victorita Dolean, M. El Bouajaji, M. J. Gander, S. Lanteri, and R. Perrussel

   In Domain decomposition methods in science and engineering XIX, 2011

   Abs DOI Bib

   We are interested here in the numerical modeling of time-harmonic electromagnetic wave propagation problems in irregularly shaped domains and heterogeneous media. In this context, we are naturally led to consider volume discretization methods (i.e. finite element method) as opposed to surface discretization methods (i.e. boundary element method). Most of the related existing work deals with the second order form of the time-harmonic Maxwell equations discretized by a conforming finite element method [14]. More recently, discontinuous Galerkin (DG) methods have also been considered for this purpose. While the DG method keeps almost all the advantages of a conforming finite element method (large spectrum of applications, complex geometries, etc.), the DG method has other nice properties which explain the renewed interest it gains in various domains in scientific computing: easy extension to higher order interpolation (one may increase the degree of the polynomials in the whole mesh as easily as for spectral methods and this can also be done locally), no global mass matrix to invert when solving time-domain systems of partial differential equations using an explicit time discretization scheme, easy handling of complex meshes (the mesh may be a classical conforming finite element mesh, a non-conforming one or even a mesh made of various types of elements), natural treatment of discontinuous solutions and coefficient heterogeneities and nice parallelization properties.

   @incollection{Dolean:2011:DOM,
     author = {Dolean, Victorita and Bouajaji, M. El and Gander, M. J. and Lanteri, S. and Perrussel, R.},
     doi = {https://doi.org/10.1007/978-3-642-11304-8_2},
     booktitle = {Domain decomposition methods in science and engineering XIX},
     pages = {15--26},
     publisher = {Springer},
     series = {LNCSE},
     title = {Domain decomposition methods for electromagnetic wave propagation problems in heterogeneous media and complex domains},
     volume = {78},
     year = {2011}
   }

2010

1. CRMA

   A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps

   Frédéric Nataf, H. Xiang, and Victorita Dolean

   Comptes Rendus Mathématique, 2010

   Abs DOI Bib

   Coarse spaces are instrumental in obtaining scalability for domain decomposition methods. However, it is known that most popular choices of coarse spaces perform rather weakly in presence of heterogeneities in the coefficients in the partial differential equations, especially for systems. Here, we introduce in a variational setting a new coarse space that is robust even when there are such heterogeneities. We achieve this by solving local generalized eigenvalue problems which isolate the terms responsible for slow convergence. We give a general theoretical result and then some numerical examples on a heterogeneous elasticity problem.

   @article{Nataf:2010:TWO,
     author = {Nataf, Fr{\'e}d{\'e}ric and Xiang, H. and Dolean, Victorita},
     doi = {https://doi.org/10.1016/j.crma.2011.10.021},
     journal = {Comptes Rendus Math{\'e}matique},
     number = {21-22},
     pages = {1163--1167},
     title = {A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps},
     url = {https://doi.org/10.1016/j.crma.2011.10.021},
     volume = {348},
     year = {2010}
   }

2. JCP

   Locally implicit discontinuous Galerkin method for time domain electromagnetics

   Victorita Dolean, H. Fahs, L. Fezoui, and S. Lanteri

   Journal of Computational Physics, 2010

   Abs DOI Bib

   In the recent years, there has been an increasing interest in discontinuous Galerkin time domain (DGTD) methods for the solution of the unsteady Maxwell equations modeling electromagnetic wave propagation. One of the main features of DGTD methods is their ability to deal with unstructured meshes which are particularly well suited to the discretization of the geometrical details and heterogeneous media that characterize realistic propagation problems. Such DGTD methods most often rely on explicit time integration schemes and lead to block diagonal mass matrices. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. An implicit time integration scheme is a natural way to obtain a time domain method which is unconditionally stable but at the expense of the inversion of a global linear system at each time step. A more viable approach consists of applying an implicit time integration scheme locally in the refined regions of the mesh while preserving an explicit time scheme in the complementary part, resulting in an hybrid explicit–implicit (or locally implicit) time integration strategy. In this paper, we report on our recent efforts towards the development of such a hybrid explicit–implicit DGTD method for solving the time domain Maxwell equations on unstructured simplicial meshes. Numerical experiments for 3D propagation problems in homogeneous and heterogeneous media illustrate the possibilities of the method for simulations involving locally refined meshes.

   @article{Dolean:2010:LOC,
     author = {Dolean, Victorita and Fahs, H. and Fezoui, L. and Lanteri, S.},
     doi = {https://doi.org/10.1016/j.jcp.2009.09.038},
     journal = {Journal of Computational Physics},
     number = {2},
     pages = {512--526},
     title = {Locally implicit discontinuous Galerkin method for time domain electromagnetics},
     url = {https://doi.org/10.1016/j.jcp.2009.09.038},
     volume = {229},
     year = {2010}
   }

3. COMPEL

   An implicit discontinuous Galerkin time-domain method for two-dimensional electromagnetic wave propagation

   A. Catella, Victorita Dolean, and S. Lanteri

   COMPEL, 2010

   Abs DOI Bib

   The purpose of this paper is to develop a time implicit discontinuous Galerkin method for the simulation of two‐dimensional time‐domain electromagnetic wave propagation on non‐uniform triangular meshes.

   @article{Catella:2010:IMP,
     author = {Catella, A. and Dolean, Victorita and Lanteri, S.},
     doi = {https://doi.org/10.1108/03321641011028215},
     journal = {COMPEL},
     number = {3},
     pages = {602--625},
     title = {An implicit discontinuous Galerkin time-domain method for two-dimensional electromagnetic wave propagation},
     url = {https://doi.org/10.1108/03321641011028215},
     volume = {29},
     year = {2010}
   }

4. IEEE-TMAG

   Recent achievements on a DGTD method for time-domain electromagnetics

   H. Fahs, L. Fezoui, S. Lanteri, Victorita Dolean, and F. Rapetti

   IEEE Transactions on Magnetics, 2010

   Abs DOI Bib

   We report on results concerning a discontinuous Galerkin time domain (DGTD) method for the solution of Maxwell equations. This DGTD method is formulated on unstructured simplicial meshes (triangles in 2-D and tetrahedra in 3-D). Within each mesh element, the electromagnetic field components are approximated by an arbitrarily high order nodal polynomial while, in the original formulation of the method, time integration is achieved by a second order Leap-Frog scheme. Here, we discuss about several recent developments aiming at improving the accuracy and the computational efficiency of this DGTD method in view of the simulation of problems involving general domains and heterogeneous media.

   @article{Fahs:2010:REC,
     author = {Fahs, H. and Fezoui, L. and Lanteri, S. and Dolean, Victorita and Rapetti, F.},
     doi = {https://doi.org/10.1109/TMAG.2010.2043342},
     journal = {IEEE Transactions on Magnetics},
     number = {8},
     pages = {3061--3064},
     title = {Recent achievements on a DGTD method for time-domain electromagnetics},
     url = {https://doi.org/10.1109/TMAG.2010.2043342},
     volume = {46},
     year = {2010}
   }

2009

1. SISC

   Optimized Schwarz methods for Maxwell’s equations

   Victorita Dolean, Martin J. Gander, and L. Gerardo-Giorda

   SIAM Journal on Scientific Computing, 2009

   Abs DOI Bib

   Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell’s equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell’s equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments.

   @article{Dolean:2009:OPT,
     author = {Dolean, Victorita and Gander, Martin J. and Gerardo-Giorda, L.},
     doi = {https://doi.org/10.1137/080728536},
     journal = {SIAM Journal on Scientific Computing},
     number = {3},
     pages = {2193--2213},
     title = {Optimized Schwarz methods for Maxwell's equations},
     url = {https://doi.org/10.1137/080728536},
     volume = {31},
     year = {2009}
   }

2. MCOM

   Deriving a new domain decomposition method for the Stokes equations using the Smith factorization

   Victorita Dolean, Frédéric Nataf, and G. Rapin

   Mathematics of Computation, 2009

   Abs DOI Bib

   In this paper the Smith factorization is used systematically to derive a new domain decomposition method for the Stokes problem. In two dimensions the key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show, how a proposed domain decomposition method for the bi-harmonic problem leads to a domain decomposition method for the Stokes equations which inherits the convergence behavior of the scalar problem. Thus, it is sufficient to study the convergence of the scalar algorithm. The same procedure can also be applied to the three-dimensional Stokes problem. As transmission conditions for the resulting domain decomposition method of the Stokes problem we obtain natural boundary conditions. Therefore it can be implemented easily. A Fourier analysis and some numerical experiments show very fast convergence of the proposed algorithm. Our algorithm shows a more robust behavior than Neumann-Neumann or FETI type methods.

   @article{Dolean:2009:DER,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Rapin, G.},
     doi = {https://doi.org/10.1090/S0025-5718-08-02172-8},
     journal = {Mathematics of Computation},
     number = {266},
     pages = {789--814},
     title = {Deriving a new domain decomposition method for the Stokes equations using the Smith factorization},
     url = {https://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-02172-8/},
     volume = {78},
     year = {2009}
   }

3. DDMSE18

   A domain decomposition preconditioner of Neumann-Neumann type for the Stokes equations

   Victorita Dolean, Frédéric Nataf, and G. Rapin

   In Domain decomposition methods in science and engineering XVIII, 2009

   Abs DOI Bib

   In this paper we recall a new domain decomposition method for the Stokes problem obtained via the Smith factorization. From the theoretical point of view, this domain decomposition method is optimal in the sense that it converges in two iterations for a decomposition into two equal domains. Previous results illustrated the fast convergence of the proposed algorithm in some cases. Our algorithm has shown a more robust behavior than Neumann-Neumann or FETI type methods for particular decompositions; as far as general decompositions are concerned, the performances of the three algorithms are similar. Nevertheless, the computations of the singular values of the interface preconditioned problem have shown that one needs a coarse space whose dimension is less than the one needed for the Neumann-Neumann algorithm. In this work we present a new strategy in order to improve the convergence of the new algorithm in the presence of cross points.

   @incollection{Dolean:2009:NEU,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Rapin, G.},
     doi = {https://doi.org/10.1007/978-3-642-02677-5_16},
     booktitle = {Domain decomposition methods in science and engineering XVIII},
     pages = {161--168},
     publisher = {Springer},
     series = {LNCSE},
     title = {A domain decomposition preconditioner of Neumann-Neumann type for the Stokes equations},
     volume = {70},
     year = {2009}
   }

2008

1. JCP

   A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods

   Victorita Dolean, S. Lanteri, and R. Perrussel

   Journal of Computational Physics, 2008

   Abs DOI Bib

   We present here a domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by a discontinuous Galerkin method. In order to allow the treatment of irregularly shaped geometries, the discontinuous Galerkin method is formulated on unstructured tetrahedral meshes. The domain decomposition strategy takes the form of a Schwarz-type algorithm where a continuity condition on the incoming characteristic variables is imposed at the interfaces between neighboring subdomains. A multifrontal sparse direct solver is used at the subdomain level. The resulting domain decomposition strategy can be viewed as a hybrid iterative/direct solution method for the large, sparse and complex coefficients algebraic system resulting from the discretization of the time-harmonic Maxwell equations by a discontinuous Galerkin method.

   @article{Dolean:2008:DOM,
     author = {Dolean, Victorita and Lanteri, S. and Perrussel, R.},
     doi = {https://doi.org/10.1016/j.jcp.2007.10.004},
     journal = {Journal of Computational Physics},
     number = {3},
     pages = {2044--2072},
     title = {A domain decomposition method for solving the three-dimensional time-harmonic Maxwell equations discretized by discontinuous Galerkin methods},
     url = {https://doi.org/10.1016/j.jcp.2007.10.004},
     volume = {227},
     year = {2008}
   }

2. JCAM

   Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods

   Victorita Dolean, H. Fol, S. Lanteri, and R. Perrussel

   Journal of Computational and Applied Mathematics, 2008

   Abs DOI Bib

   We present numerical results concerning the solution of the time-harmonic Maxwell equations discretized by discontinuous Galerkin methods. In particular, a numerical study of the convergence, which compares different strategies proposed in the literature for the elliptic Maxwell equations, is performed in the two-dimensional case.

   @article{Dolean:2008:SOL,
     author = {Dolean, Victorita and Fol, H. and Lanteri, S. and Perrussel, R.},
     doi = {https://doi.org/10.1016/j.cam.2007.05.026},
     journal = {Journal of Computational and Applied Mathematics},
     number = {2},
     pages = {435--445},
     title = {Solution of the time-harmonic Maxwell equations using discontinuous Galerkin methods},
     url = {https://doi.org/10.1016/j.cam.2007.05.026},
     volume = {218},
     year = {2008}
   }

3. IEEE-TMAG

   An unconditionally stable discontinuous Galerkin method for solving the 2D time-domain Maxwell equations on unstructured triangular meshes

   A. Catella, Victorita Dolean, and S. Lanteri

   IEEE Transactions on Magnetics, 2008

   Abs DOI Bib

   Numerical methods for solving the time-domain Maxwell equations often rely on Cartesian meshes and are variants of the finite-difference time-domain (FDTD) method due to Yee (1966). In recent years, there has been an increasing interest in discontinuous Galerkin time-domain (DGTD) methods dealing with unstructured meshes since the latter are particularly well adapted to the discretization of geometrical details that characterize applications of practical relevance. However, similarly to Yee’s finite difference time-domain method, existing DGTD methods generally rely on explicit time integration schemes and are therefore constrained by a stability condition that can be very restrictive on locally refined unstructured meshes. An implicit time integration scheme is a possible strategy to overcome this limitation. The present study aims at investigating such an implicit DGTD method for solving the 2-D time-domain Maxwell equations on nonuniform triangular meshes.

   @article{Catella:2008:UNC,
     author = {Catella, A. and Dolean, Victorita and Lanteri, S.},
     doi = {https://doi.org/10.1109/TMAG.2007.916578},
     journal = {IEEE Transactions on Magnetics},
     number = {6},
     pages = {1250--1253},
     title = {An unconditionally stable discontinuous Galerkin method for solving the 2D time-domain Maxwell equations on unstructured triangular meshes},
     url = {https://ieeexplore.ieee.org/document/4526821},
     volume = {44},
     year = {2008}
   }

4. IEEE-TMAG

   Optimized Schwarz algorithms for solving time-harmonic Maxwell’s equations discretized by a discontinuous Galerkin method

   Victorita Dolean, S. Lanteri, and R. Perrussel

   IEEE Transactions on Magnetics, 2008

   Abs DOI Bib

   The numerical solution of the three-dimensional time-harmonic Maxwell equations using high order methods such as discontinuous Galerkin formulations require efficient solvers. A domain decomposition strategy is introduced for this purpose. This strategy is based on optimized Schwarz methods applied to the first order form of the Maxwell system and leads to the best possible convergence of these algorithms. The principles are explained for a 2D model problem and numerical simulations confirm the predicted theoretical behavior. The efficiency is further demonstrated on more realistic 3D geometries including a bioelectromagnetism application.

   @article{Dolean:2008:OPTM2,
     author = {Dolean, Victorita and Lanteri, S. and Perrussel, R.},
     doi = {https://doi.org/10.1109/TMAG.2008.915830},
     journal = {IEEE Transactions on Magnetics},
     number = {6},
     pages = {954--957},
     title = {Optimized Schwarz algorithms for solving time-harmonic Maxwell's equations discretized by a discontinuous Galerkin method},
     url = {https://ieeexplore.ieee.org/abstract/document/4526850},
     volume = {44},
     year = {2008}
   }

5. DDMSE17

   p-multigrid for Fekete spectral element method

   Victorita Dolean, R. Pasquetti, and F. Rapetti

   In Domain decomposition methods in science and engineering XVII, 2008

   Abs DOI Bib

   Spectral element approximations based on triangular elements and on the so-called Fekete points of the triangle have been recently developed. p-multigrid methods offer an interesting way to resolve efficiently the resulting ill-conditioned algebraic systems. For elliptic problems, it is shown that a well chosen restriction operator and a good set up of the coarse grid matrices may lead to valuable results, even with a standard Gauss-Seidel smoother.

   @incollection{Dolean:2008:PMU,
     author = {Dolean, Victorita and Pasquetti, R. and Rapetti, F.},
     doi = {https://doi.org/10.1007/978-3-540-75199-1_61},
     booktitle = {Domain decomposition methods in science and engineering XVII},
     pages = {485--492},
     publisher = {Springer},
     series = {LNCSE},
     title = {p-multigrid for Fekete spectral element method},
     volume = {60},
     year = {2008}
   }

6. DDMSE17

   How to use the Smith factorization for domain decomposition methods applied to the Stokes equations

   Victorita Dolean, Frédéric Nataf, and G. Rapin

   In Domain decomposition methods in science and engineering XVII, 2008

   Abs DOI Bib

   In this paper we demonstrate that the Smith factorization is a powerful tool to derive new domain decomposition methods for vector valued problems. Here, the factorization is applied to the two-dimensional Stokes system. The key idea is the transformation of the Stokes problem into a scalar bi-harmonic problem. We show how a proposed domain decomposition method for the bi-harmonic problem leads to an algorithm for the Stokes equations which inherits the convergence behavior of the scalar problem.

   @incollection{Dolean:2008:HOW,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Rapin, G.},
     doi = {https://doi.org/10.1007/978-3-540-75199-1_60},
     booktitle = {Domain decomposition methods in science and engineering XVII},
     pages = {477--484},
     publisher = {Springer},
     series = {LNCSE},
     title = {How to use the Smith factorization for domain decomposition methods applied to the Stokes equations},
     volume = {60},
     year = {2008}
   }

7. DDMSE17

   Why classical Schwarz methods applied to certain hyperbolic systems converge even without overlap

   Victorita Dolean and Martin J. Gander

   In Domain decomposition methods in science and engineering XVII, 2008

   Abs DOI Bib

   Overlap is essential for the classical Schwarz method to be convergent when solving elliptic problems. Over the last decade, it was however observed that when solving systems of hyperbolic partial differential equations, the classical Schwarz method can be convergent even without overlap. We show that the classical Schwarz method without overlap applied to the Cauchy-Riemann equations which represent the discretization in time of such a system, is equivalent to an optimized Schwarz method for a related elliptic problem, and thus must be convergent, since optimized Schwarz methods are well known to be convergent without overlap.

   @incollection{Dolean:2008:WHY,
     author = {Dolean, Victorita and Gander, Martin J.},
     doi = {https://doi.org/10.1007/978-3-540-75199-1_59},
     booktitle = {Domain decomposition methods in science and engineering XVII},
     pages = {467--475},
     publisher = {Springer},
     series = {LNCSE},
     title = {Why classical Schwarz methods applied to certain hyperbolic systems converge even without overlap},
     volume = {60},
     year = {2008}
   }

8. DDMSE17

   A new domain decomposition method for the compressible Euler equations using Smith factorization

   Victorita Dolean and Frédéric Nataf

   In Domain decomposition methods in science and engineering XVII, 2008

   Abs DOI Bib

   In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [1]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, …).

   @incollection{Dolean:2008:NEW,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1007/978-3-540-75199-1_40},
     booktitle = {Domain decomposition methods in science and engineering XVII},
     pages = {331--338},
     publisher = {Springer},
     series = {LNCSE},
     title = {A new domain decomposition method for the compressible Euler equations using Smith factorization},
     volume = {60},
     year = {2008}
   }

2006

1. M2AN

   A new domain decomposition method for the compressible Euler equations

   Victorita Dolean and Frédéric Nataf

   M2AN: Mathematical Modelling and Numerical Analysis, 2006

   Abs DOI Bib

   In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf, C. R. Acad. Sci. Paris Sér. I 325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ...).

   @article{Dolean:2006:NEW,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1051/m2an:2006026},
     journal = {M2AN: Mathematical Modelling and Numerical Analysis},
     number = {4},
     pages = {689--703},
     title = {A new domain decomposition method for the compressible Euler equations},
     url = {https://doi.org/10.1051/m2an:2006026},
     volume = {40},
     year = {2006}
   }

2005

1. CRMA

   New constructions of domain decomposition methods for systems of PDEs

   Victorita Dolean, Frédéric Nataf, and G. Rapin

   Comptes Rendus Mathématique, 2005

   Abs DOI Bib

   We propose new domain decomposition methods for systems of partial differential equations in two and three dimensions. The algorithms are derived with the help of the Smith factorization. This could also be validated by numerical experiments.

   @article{Dolean:2005:NEW,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Rapin, G.},
     doi = {https://doi.org/10.1016/j.crma.2005.03.026},
     journal = {Comptes Rendus Math{\'e}matique},
     number = {9},
     pages = {693--696},
     title = {New constructions of domain decomposition methods for systems of PDEs},
     url = {https://doi.org/10.1016/j.crma.2005.03.026},
     volume = {340},
     year = {2005}
   }

2004

1. PARCO

   Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs

   Victorita Dolean and S. Lanteri

   Parallel Computing, 2004

   Abs DOI Bib

   We report on our efforts towards the design of efficient parallel hierarchical iterative methods for the solution of sparse and irregularly structured linear systems resulting from CFD applications. The solution strategies considered here share a central numerical kernel which consists in a linear multigrid by volume agglomeration method. Starting from this method, we study two parallel solution strategies. The first variant results from a direct intra-grid parallelization of multigrid operations on coarse grids. The second variant is based on an additive Schwarz domain decomposition algorithm which is formulated at the continuous level through the introduction of specific interface conditions. In this variant, the linear multigrid by volume agglomeration method is used to approximately solve the local systems obtained at each iteration of the Schwarz algorithm. As a result, the proposed hybrid domain decomposition/multigrid method can be viewed as a particular form of parallel multigrid in which multigrid acceleration is applied on a subdomain basis, these local calculations being coordinated by an appropriate domain decomposition iteration at the global level. The parallel performances of these two parallel multigrid methods are evaluated through numerical experiments that are performed on several clusters of PCs with different computational nodes and interconnection networks.

   @article{Dolean:2004:PAR,
     author = {Dolean, Victorita and Lanteri, S.},
     doi = {https://doi.org/10.1016/j.parco.2004.03.002},
     journal = {Parallel Computing},
     number = {4},
     pages = {503--525},
     title = {Parallel multigrid methods for the calculation of unsteady flows on unstructured grids: algorithmic aspects and parallel performances on clusters of PCs},
     url = {https://doi.org/10.1016/j.parco.2004.03.002},
     volume = {30},
     year = {2004}
   }

2. APNUM

   Convergence analysis of additive Schwarz for the Euler equations

   Victorita Dolean, Stéphane Lanteri, and Frédéric Nataf

   Applied Numerical Mathematics, 2004

   Abs DOI Bib

   In a previous paper [Internat. J. Numer. Math. Fluids 37 (2001) 625], we reported on numerical experiments with a non-overlapping domain decomposition method that has been specifically designed for the calculation of steady compressible inviscid flows governed by the two-dimensional Euler equations. In the present work, we study this method from the theoretical point of view. The proposed method relies on the formulation of an additive Schwarz algorithm which involves interface conditions that are Dirichlet conditions for the characteristic variables corresponding to incoming waves (often referred to as natural or classical interface conditions), thus taking into account the hyperbolic nature of the Euler equations. In the first part of this paper, the convergence of the additive Schwarz algorithm is analyzed in the two- and three-dimensional continuous cases by considering the linearized equations and applying a Fourier analysis. We limit ourselves to the cases of two and three-subdomain decompositions with or without overlap and we obtain analytical expressions of the convergence rate of the Schwarz algorithm. Besides the fact that the algorithm is always convergent, surprisingly, there exist flow conditions for which the asymptotic convergence rate is equal to zero. Moreover, this behavior is independent of the space dimension. In the second part, we study the discrete counterpart of the non-overlapping additive Schwarz algorithm based on the implementation adopted in [Internat. J. Numer. Math. Fluids 37 (2001) 625] but assuming a finite volume formulation on a quadrangular mesh. We find out that the expression of the convergence rate is actually more characteristic of an overlapping additive Schwarz algorithm. We conclude by presenting numerical results that confirm qualitatively the convergence behavior found analytically.

   @article{Dolean:2004:CON,
     author = {Dolean, Victorita and Lanteri, St{\'e}phane and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1016/j.apnum.2003.09.006},
     journal = {Applied Numerical Mathematics},
     number = {2},
     pages = {153--186},
     title = {Convergence analysis of additive Schwarz for the Euler equations},
     url = {https://doi.org/10.1016/j.apnum.2003.09.006},
     volume = {49},
     year = {2004}
   }

2002

1. JCFD

   Optimized interface conditions for domain decomposition methods in fluid dynamics

   Victorita Dolean, Stéphane Lanteri, and Frédéric Nataf

   International Journal for Numerical Methods in Fluids, 2002

   Abs DOI Bib

   Interface conditions (IC) between subdomains have an important impact on the convergence rate of domain decomposition algorithms. We first recall the Schwarz method which is based on the use of Dirichlet conditions on the boundaries of the subdomains and overlapping subdomains. We explain how it is possible to replace them by more efficient ICs with normal and tangential derivatives so that overlapping is not necessary. It is possible to optimize the coefficients of the IC in order to achieve the best convergence rate. Results are given for the convection—diffusion equation. Then we consider the compressible Euler equations which form a system of equations. We present a new analysis of the use of interface conditions based on the flux splitting. We compute the convergence rate in the Fourier space. We find a dependence of their effectiveness on the Mach number M. For M=⅓, the convergence rate tends to zero as the wavenumber of the error goes to infinity. We stress the differences with the scalar equations. We present numerical results in agreement with the theoretical results.

   @article{Dolean:2002:OPT,
     author = {Dolean, Victorita and Lanteri, St{\'e}phane and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1002/fld.410},
     journal = {International Journal for Numerical Methods in Fluids},
     number = {12},
     pages = {1539--1550},
     title = {Optimized interface conditions for domain decomposition methods in fluid dynamics},
     url = {https://doi.org/10.1002/fld.410},
     volume = {40},
     year = {2002}
   }

2. JCFD

   Construction of interface conditions for solving the compressible Euler equations by non-overlapping domain decomposition methods

   Victorita Dolean, Frédéric Nataf, and Stéphane Lanteri

   International Journal for Numerical Methods in Fluids, 2002

   Abs DOI Bib

   In this work we examine the acceleration of the convergence of a non-overlapping additive Schwarz-type algorithm by modifying the transmission conditions applied to the subdomain interfaces. We have built generalized zero-order interface conditions using the Smith theory of diagonalizing polynomial matrices. The numerical experiments confirmed qualitatively the behaviour in accordance with the theory, but we could not reproduce identically the results obtained in the continuous case. The preliminary results are very encouraging since they lead to a very good convergence rate for certain Mach numbers.

   @article{Dolean:2002:CONS,
     author = {Dolean, Victorita and Nataf, Fr{\'e}d{\'e}ric and Lanteri, St{\'e}phane},
     doi = {https://doi.org/10.1002/fld.406},
     journal = {International Journal for Numerical Methods in Fluids},
     number = {12},
     pages = {1485--1492},
     title = {Construction of interface conditions for solving the compressible Euler equations by non-overlapping domain decomposition methods},
     url = {https://doi.org/10.1002/fld.406},
     volume = {40},
     year = {2002}
   }

3. ASFD

   Nonoverlapping domain decomposition algorithms for the system of Euler equations

   Victorita Dolean, S. Lanteri, and Frédéric Nataf

   In Domain decomposition methods in science and engineering (Lyon, 2000), 2002

   Abs DOI Bib

   In this work we present an overview of some classical and new domain decomposition methods for the resolution of the Euler equations. The classical Schwarz methods are formulated and analyzed in the framework of first order hyperbolic systems and the differences with respect to the scalar problems are presented. This kind of algorithms behave quite well for bigger Mach numbers but we can further improve their performances in the case of lower Mach numbers. There are two possible ways to achieve this goal. The first one implies the use of the optimized interface conditions depending on a few parameters that generalize the classical ones. The second is inspired from the Robin-Robin preconditioner for the convection-diffusion equation by using the equivalence via the Smith factorization with a third order scalar equation.

   @incollection{Dolean:2002:NON,
     author = {Dolean, Victorita and Lanteri, S. and Nataf, Fr{\'e}d{\'e}ric},
     doi = {https://doi.org/10.1007/978-3-7643-7742-7_5},
     booktitle = {Domain decomposition methods in science and engineering (Lyon, 2000)},
     pages = {455--462},
     publisher = {Internat. Center Numer. Methods Eng. (CIMNE)},
     title = {Nonoverlapping domain decomposition algorithms for the system of Euler equations},
     year = {2002}
   }

2001

1. JCFD

   A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

   Victorita Dolean and S. Lanteri

   International Journal for Numerical Methods in Fluids, 2001

   Abs DOI Bib

   We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two-dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo-time step requires the solution of a sparse linear system for the flow variables. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson-type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In our approach, the interface unknowns are numerical (normal) fluxes.

   @article{Dolean:2001:DOM,
     author = {Dolean, Victorita and Lanteri, S.},
     doi = {https://doi.org/10.1002/fld.184},
     journal = {International Journal for Numerical Methods in Fluids},
     number = {6},
     pages = {625--656},
     title = {A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes},
     url = {https://doi.org/10.1002/fld.184},
     volume = {37},
     year = {2001}
   }

2. JCFD

   A hybrid domain decomposition and multigrid method for the acceleration of compressible viscous flow calculations on unstructured triangular meshes

   Victorita Dolean and S. Lanteri

   International Journal of Computational Fluid Dynamics, 2001

   Abs DOI Bib

   This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.

   @article{Dolean:2001:HYB,
     author = {Dolean, Victorita and Lanteri, S.},
     doi = {https://doi.org/10.1080/10618560108940730},
     journal = {International Journal of Computational Fluid Dynamics},
     number = {4},
     pages = {287--304},
     title = {A hybrid domain decomposition and multigrid method for the acceleration of compressible viscous flow calculations on unstructured triangular meshes},
     url = {https://doi.org/10.1080/10618560108940730},
     volume = {14},
     year = {2001}
   }