arXiv subjects

Pierre Jolivet

Publications and source records attributed to Pierre Jolivet.

17 recordsLinked to original sources

arXiv2026

A Guided Tour of Modern Domain Decomposition: From Schwarz Iterations to Robust Preconditioners and HPC Implementations

Domain decomposition methods (DDMs) provide a unifying framework for the scalable numerical solution of partial differential equations. Originating from Schwarz's alternating method, they have evolved into a rich family of algorithms that combine local robustness with global convergence acceleration and natural parallelism. Over the past decades, domain decomposition has played a central role in enabling large-scale simulations in numerous applications. This chapter presents an overview of modern DDMs, with a particular emphasis on scalable preconditioning techniques for challenging problems, including indefinite and high-frequency regimes. We revisit the fundamental concepts - overlapping decompositions, partition of unity, additive and restricted Schwarz formulations - and explain their algebraic interpretations. We then clarify their role as preconditioners in Krylov subspace solvers and discuss the necessity of coarse space corrections for scalability. Beyond a the survey aspect, the chapter distills key theoretical insights and practical design principles that have emerged over the past twenty years. Special attention is given to robust coarse spaces (GenEO, DtN-based approaches) and high-performance implementations. The goal is to provide both a coherent overview of the field and a concise, practice-oriented guide for readers seeking to understand and apply domain decomposition methods without navigating the entire literature.

math.NA↗

arXiv2026

Efficient Fine-Scale Simulation of Nonlinear Hyperelastic Lattice Structures

With the growing maturity of additive manufacturing, the fabrication of architected or lattice-based metamaterials has become a reality for industrial applications. These materials combine lightweight design with tailored mechanical properties, most of which exhibit pronounced nonlinear, especially large-deformation, behaviors. The main numerical challenge therefore lies in performing nonlinear simulations of such lattice structures, which may contain thousands of geometrically intricate unit cells, while lacking sufficient scale separation for multiscale homogenization schemes to be applicable straightforwardly. In this work, we propose a dedicated solver for the full volumetric fine-scale simulation of nonlinear hyperelastic lattice structures that drastically reduces both memory and computational costs. The key idea is to exploit the intrinsic self-similarity of the cells through a reduced-order modeling strategy applied within a domain-decomposition framework. At each Newton iteration, a limited set of principal cells is identified through a dedicated, weakly intrusive, EIM-like approach, allowing all local tangent operators to be expressed as linear combinations of a few principal ones. This enables fast and memory-efficient operator assembly, and then feeds an efficient inexact FETI-DP based preconditioner at the solution stage, resulting in a quasi matrix-free algorithm for the nonlinear analysis. Numerical experiments in two and three dimensions demonstrate significant computational gains, with runtime reductions from several hours to a few tens of minutes and memory savings by factors of about three, while maintaining full fine-scale accuracy. Notably, the proposed strategy enables the computation of problems involving thousands of cells (i.e., millions of degrees of freedom) within a few minutes on an off-the-shelf laptop.

math.NA↗

arXiv2025

ff-bifbox: A scalable, open-source toolbox for bifurcation analysis of nonlinear PDEs

Nonlinear PDEs give rise to complex dynamics that are often difficult to analyze in state space due to their relatively large numbers of degrees of freedom, ill-conditioned operators, and changing spatial and parameter resolution requirements. This work introduces ff-bifbox: a new open-source toolbox for performing numerical branch tracing, stability/bifurcation analysis, resolvent analysis, and time integration of large, time-dependent nonlinear PDEs discretized on adaptively refined meshes in two and three spatial dimensions. Spatial discretization is handled using finite elements in FreeFEM, with the discretized operators manipulated in a distributed framework via PETSc. Following a summary of the underlying theory and numerics, results from three examples are presented to validate the implementation and demonstrate its capabilities. The considered examples, which are provided with runnable ff-bifbox code, include: a 3-D Brusselator system, a 3-D plate buckling system, and a 2-D compressible Navier--Stokes system. In addition to reproducing results from prior studies, novel results are presented for each system.

math.NA↗

arXiv2025

Comparison of substructured non-overlapping domain decomposition and overlapping additive Schwarz methods for large-scale Helmholtz problems with multiple sources

Solving large-scale Helmholtz problems discretized with high-order finite elements is notoriously difficult, especially in 3D where direct factorization of the system matrix is very expensive and memory demanding, and robust convergence of iterative methods is difficult to obtain. Domain decomposition methods (DDM) constitute one of the most promising strategy so far, by combining direct and iterative approaches: using direct solvers on overlapping or non-overlapping subdomains, as a preconditioner for a Krylov subspace method on the original Helmholtz system or as an iterative solver on a substructured problem involving field values or Lagrange multipliers on the interfaces between the subdomains. In this work we compare the computational performance of non-overlapping substructured DDM and Optimized Restricted Additive Schwarz (ORAS) preconditioners for solving large-scale Helmholtz problems with multiple sources, as is encountered, e.g., in frequency-domain Full Waveform Inversion. We show on a realistic geophysical test-case that, when appropriately tuned, the non-overlapping methods can reduce the convergence gap sufficiently to significantly outperform the overlapping methods.

math.NA↗

arXiv2024

Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFem for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFem with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFem with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature.

cond-mat.quant-gas↗

arXiv2024

A Robust Two-Level Schwarz Preconditioner For Sparse Matrices

This paper introduces a fully algebraic two-level additive Schwarz preconditioner for general sparse large-scale matrices. The preconditioner is analyzed for symmetric positive definite (SPD) matrices. For those matrices, the coarse space is constructed based on approximating two local subspaces in each subdomain. These subspaces are obtained by approximating a number of eigenvectors corresponding to dominant eigenvalues of two judiciously posed generalized eigenvalue problems. The number of eigenvectors can be chosen to control the condition number. For general sparse matrices, the coarse space is constructed by approximating the image of a local operator that can be defined from information in the coefficient matrix. The connection between the coarse spaces for SPD and general matrices is also discussed. Numerical experiments show the great effectiveness of the proposed preconditioners on matrices arising from a wide range of applications. The set of matrices includes SPD, symmetric indefinite, nonsymmetric, and saddle-point matrices. In addition, we compare the proposed preconditioners to the state-of-the-art domain decomposition preconditioners.

math.NA↗

arXiv2023

Multilevel Scalable Solvers for Stochastic Linear and Nonlinear Problems

This article discusses the uncertainty quantification (UQ) for time-independent linear and nonlinear partial differential equation (PDE)-based systems with random model parameters carried out using sampling-free intrusive stochastic Galerkin method leveraging multilevel scalable solvers constructed combining two-grid Schwarz method and AMG. High-resolution spatial meshes along with a large number of stochastic expansion terms increase the system size leading to significant memory consumption and computational costs. Domain decomposition (DD)-based parallel scalable solvers are developed to this end for linear and nonlinear stochastic PDEs. A generalized minimum residual (GMRES) iterative solver equipped with a multilevel preconditioner consisting of restricted additive Schwarz (RAS) for the fine grid and algebraic multigrid (AMG) for the coarse grid is constructed to improve scalability. Numerical experiments illustrate the scalabilities of the proposed solver for stochastic linear and nonlinear Poisson problems.

cs.CE↗

arXiv2022

Scalable Computational Algorithms for Geo-spatial Covid-19 Spread in High Performance Computing

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 allowing 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 scalability. 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 up to three months for a system size of 92 million (using 1780 processes) within 7 hours saving months of computational effort needed for the conventional solvers.

cs.CE↗

arXiv2022

Efficient Algebraic Two-Level Schwarz Preconditioner For Sparse Matrices

Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.

math.NA↗

arXiv2022

A Robust Algebraic Domain Decomposition Preconditioner for Sparse Normal Equations

Solving the normal equations corresponding to large sparse linear least-squares problems is an important and challenging problem. For very large problems, an iterative solver is needed and, in general, a preconditioner is required to achieve good convergence. In recent years, a number of preconditioners have been proposed. These are largely serial and reported results demonstrate that none of the commonly used preconditioners for the normal equations matrix is capable of solving all sparse least-squares problems. Our interest is thus in designing new preconditioners for the normal equations that are efficient, robust, and can be implemented in parallel. Our proposed preconditioners can be constructed efficiently and algebraically without any knowledge of the problem and without any assumption on the least-squares matrix except that it is sparse. We exploit the structure of the symmetric positive definite normal equations matrix and use the concept of algebraic local symmetric positive semi-definite splittings to introduce two-level Schwarz preconditioners for least-squares problems. The condition number of the preconditioned normal equations is shown to be theoretically bounded independently of the number of subdomains in the splitting. This upper bound can be adjusted using a single parameter $τ$ that the user can specify. We discuss how the new preconditioners can be implemented on top of the PETSc library using only 150 lines of Fortran, C, or Python code. Problems arising from practical applications are used to compare the performance of the proposed new preconditioner with that of other preconditioners.

math.NA↗

arXiv2021

A Robust Algebraic Multilevel Domain Decomposition Preconditioner For Sparse Symmetric Positive Definite Matrices

Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices from discretized partial differential equations (PDEs). Therefore, DD methods were mainly successful when solving systems stemming from PDEs. In this paper, we present a fully algebraic multilevel DD method where the coarse space can be constructed locally and efficiently without any information besides the coefficient matrix. The condition number of the preconditioned matrix can be bounded by a user-prescribed number. Numerical experiments illustrate the effectiveness of the preconditioner on a range of problems arising from different applications.

math.NA↗

arXiv2021

Inexact subdomain solves using deflated GMRES for Helmholtz problems

We examine the use of a two-level deflation preconditioner combined with GMRES to locally solve the subdomain systems arising from applying domain decomposition methods to Helmholtz problems. Our results show that the direct solution method can be replaced with an iterative approach. This will be particularly important when solving large 3D high-frequency problems as subdomain problems can be too large for direct inversion or otherwise become inefficient. We additionally show that, even with a relatively low tolerance, inexact solution of the subdomain systems does not lead to a drastic increase in the number of outer iterations. As a result, it is promising that a combination of a two-level domain decomposition preconditioner with inexact subdomain solves could provide more economical and memory efficient numerical solutions to large-scale Helmholtz problems.

math.NA↗

arXiv2021

Several ways to achieve robustness when solving wave propagation problems

Wave propagation problems are notoriously difficult to solve. Time-harmonic problems are especially challenging in mid and high frequency regimes. The main reason is the oscillatory nature of solutions, meaning that the number of degrees of freedom after discretisation increases drastically with the wave number, giving rise to large complex-valued problems to solve. Additional difficulties occur when the problem is defined in a highly heterogeneous medium, as is often the case in realistic physical applications. For time-discretised problems of Maxwell type, the main challenge remains the significant kernel in curl-conforming spaces, an issue that impacts on the design of robust preconditioners. This has already been addressed theoretically for a homogeneous medium but not yet in the presence of heterogeneities. In this review we provide a big-picture view of the main difficulties encountered when solving wave propagation problems, from the first step of their discretisation through to their parallel solution using two-level methods, by showing their limitations on a few realistic examples. We also propose a new preconditioner inspired by the idea of subspace decomposition, but based on spectral coarse spaces, for curl-conforming discretisations of Maxwell's equations in heterogeneous media.

math.NA↗

arXiv2021

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

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.

math.NA↗

arXiv2021

Large-scale finite-difference and finite-element frequency-domain seismic wave modelling with multi-level domain-decomposition preconditioner

The emergence of long-offset sparse stationary-recording surveys carried out with ocean bottom nodes (OBN) makes frequency-domain full waveform inversion (FWI) attractive to manage compact volume of data and perform attenuation imaging. One challenge of frequency-domain FWI is the forward problem, which requires the solution of large and sparse linear systems with multiple right-hand sides. While direct methods are suitable for dense acquisitions and problems involving less than 100 million unknowns, iterative solver are more suitable for large computational domains covered by sparse OBN surveys. Here, we solve these linear systems with a Krylov subspace method preconditioned with the two-level Optimized Restricted Additive Schwarz (ORAS) domain decomposition preconditioner, the prefix optimized referring to the use of absorbing conditions at the subdomain interfaces. We implement this method with finite differences on uniform grid and finite elements on unstructured tetrahedral meshes. A simulation in a model where the velocity linearly increases with depth allows us to validate the accuracy of the two schemes against an analytical solution while highlighting how their relative cost varies with the band of propagated wavelengths. A simulation in the overthrust model involving up to 2 billions of parameters allows us to tune the method and highlights its scalability.

physics.comp-ph↗

arXiv2020

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

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.

physics.comp-ph↗

arXiv2020

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

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 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 due to the non definiteness of the Helmholtz operator, hence requiring efficient preconditioners. In this study, we use the Krylov subspace GMRES iterative solver combined with a multi-level domain-decomposition preconditioner. Discretization relies on continuous finite elements 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 convergence and the scalability of our method with the acoustic 3D SEG/EAGE Overthrust model up to a frequency of 20~Hz and discuss its efficiency for multi right-hand side processing.

physics.comp-ph↗