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Martin J. Gander

Publications and source records attributed to Martin J. Gander.

At least 19 recordsLinked to original sources
arXiv2026

Dirichlet-Neumann waveform relaxation for heterogeneous heat equations: continuous and time discrete L2 analysis

We consider two coupled linear heat equations on different spatial domains that interact through a lower dimensional interface. This models conjugate heat transfer. The problem is solved using Dirichlet-Neumann waveform relaxation. This allows us to couple separate codes for the subproblems, a so-called partitioned approach. Our overall goal is to develop more efficient partitioned methods, and to this end, we want reliable error estimates. We use an exponentially weighted Fourier technique to derive new error estimates in L2 for finite time T in both continuous and time discrete settings. We identify an optimized relaxation parameter that guarantees superlinear convergence. Our new continuous estimate predicts linear convergence when T is large, and superlinear when T is small. For large T, our new time discrete estimate closely mirrors its continuous counterpart, whereas for small T, superlinear convergence in the time discrete case requires small time step dt. We also show that convergence is fast when the contrast is large, provided that the small physical parameter domain (e.g. air) is using the Dirichlet transmission condition, and the large physical parameter domain (e.g. steel) is using the Neumann transmission condition in the Dirichlet-Neumann waveform relaxation method. Our numerical experiments confirm all these findings.

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arXiv2026

Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet Conditions: Sharp Estimates and Relative Errors

We consider the Dirichlet problem of the indefinite Helmholtz equation in 1D, $u''+k^2u=f$ in $(0,1)$, $u(0)=g_0$, $u(1)=g_1$, with a constant wavenumber $k\in(0,\infty)\backslashπ\mathbb{N}$ and a source term $f\in H^p_0(0,1)$, $p\ge 4$. We propose an approach based on Fourier analysis to derive wavenumber explicit sharp estimates of absolute and relative errors of \emph{finite difference} methods. Such results have been well known for \emph{finite element} methods (FEM). We use the approach to analyze the classical centered finite difference scheme. For the Fourier interpolants of the discrete solution with homogeneous (or inhomogeneous) Dirichlet conditions, we show rigorously, under the two assumptions $k>20$ and $k(kh)^2/σ_k\le4/(π-2)$ with $σ_k:=\operatorname{dist}(k,π\mathbb{N})$, that the worst case attainable convergence order of the absolute error with $\sum_{p=0}^4k^{-p}\|f^{(p)}\|_{L^2}=O(1)$ (or $|g_i|\asymp k^{-1}$) is $(kh)^2/σ_k^2$ in the $L^2$-norm and $k(kh)^2/σ_k^2$ in the $H^1$-semi-norm, and that of the relative error is $k(kh)^2/σ_k$ in both $L^2$- and $H^1$-semi-norms if $\|u^{(p)}\|_{L^2}/\|u^{(p-2)}\|_{L^2}\asymp k^2$ for $p=2,3$. In particular, the lower bounds of these error estimates are established rigorously in the same orders as the upper bounds, which is the main novelty of this work. We show also that the Fourier analysis approach can be used as a convenient visual tool for evaluating finite difference schemes in presence of source terms, which is beyond the scope of dispersion analysis. The results from the theory and visual analysis are corroborated by numerical experiments.

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arXiv2026

Optimized Schwarz Waveform Relaxation for the Damped Wave Equation

The performance of Schwarz Waveform Relaxation is critically dependent on the choice of transmission conditions. While classical absorbing conditions work well for wave propagation, they prove insufficient for damped wave equations, particularly in viscoelastic damping regimes where convergence becomes prohibitively slow. This paper addresses this limitation by introducing a more general transmission operator with two free parameters for the one-dimensional damped wave equation. Through frequency-domain analysis, we derive an explicit expression for the convergence factor governing the convergence rate. We propose and compare two optimization strategies (L-infinity and L-2 minimization) for determining optimal transmission parameters. Numerical experiments demonstrate that our optimized approach significantly accelerates convergence compared to standard absorbing conditions, especially for viscoelastic damping cases. The method provides a computationally efficient alternative to exhaustive parameter search while maintaining robust performance across different damping regimes.

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arXiv2025

Hierarchical Coarse Basis by Randomised SVD: the Helmholtz Problem

The oscillatory waves require sufficient degrees of freedom to resolve. That restriction usually applies also to coarse problems for Schwarz methods. The resulting coarse problem is then too large. To address the issue, a new form of Schwarz methods with coarse correction is proposed for the Helmholtz problem. There are two components in the proposed form: randomised SVD of interface iteration, and hierarchical domain decomposition. The resulting coarse problem is hierarchical and can be solved in a decoupled way at each level of hierarchy.

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arXiv2025

A Parareal Algorithm with Low-Rank Coarse Solvers

We consider a new class of Parareal algorithms, which use ideas from localized reduced basis methods to construct the coarse solver from truncated SVD approximations of the transfer operators mapping initial values for a given time interval to the solution at the end of the interval. By leveraging randomized singular value decompositions, these low-rank approximations are obtained embarrassingly parallel by computing local fine solutions for random initial values. We show a priori and a posteriori error bounds in terms of the computed singular values of the transfer operators. Our numerical experiments demonstrate that our approach can significantly outperform Parareal with single-step coarse solvers. At the same time, it permits to further increase parallelism in Parareal by trading global iterations for a larger number of independent local solves.

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arXiv2025

Optimized Schwarz methods for heterogeneous heat transfer problems

We present here nonoverlapping optimized Schwarz methods applied to heat transfer problems with heterogeneous diffusion coefficients. After a Laplace transform in time, we derive the error equation and obtain the convergence factor. The optimal transmission operators are nonlocal, and thus inconvenient to use in practice. We introduce three versions of local approximations for the transmission parameter, and provide a detailed analysis at the continuous level in each case to identify the best local transmission conditions. Numerical experiments are presented to illustrate the performance of each local transmission condition. As shown in our analysis, local transmission conditions, which are scaled appropriately with respect to the heterogeneous diffusion coefficients, are more efficient and robust especially when the discontinuity of the diffusion coefficient is large.

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arXiv2025

Discontinuous Galerkin time integration for second-order differential problems: formulations, analysis, and analogies

We thoroughly investigate Discontinuous Galerkin (DG) discretizations as time integrators for second-order oscillatory systems, considering both second-order and first-order formulations of the original problem. Key contributions include new convergence analyses for the second-order formulation and equivalence proofs between DG and classical time-stepping schemes (such as Newmark schemes and general linear methods). In addition, the chapter provides a detailed review and convergence analysis for the first-order formulation, alongside comparisons of the proposed schemes in terms of accuracy, consistency, and computational cost.

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arXiv2025

Time parallelization for hyperbolic and parabolic problems

Time parallelization, also known as PinT (Parallel-in-Time) is a new research direction for the development of algorithms used for solving very large scale evolution problems on highly parallel computing architectures. Despite the fact that interesting theoretical work on PinT appeared as early 1964, it was not until 2004, when processor clock speeds reached their physical limit, that research in PinT took off. A distinctive characteristic of parallelization in time is that information flow only goes forward in time, meaning that time evolution processes seem necessarily to be sequential. Nevertheless, many algorithms have been developed over the last two decades to do PinT computations, and they are often grouped into four basic classes according to how the techniques work and are used: shooting-type methods; waveform relaxation methods based on domain decomposition; multigrid methods in space-time; and direct time parallel methods. However, over the past few years, it has been recognized that highly successful PinT algorithms for parabolic problems struggle when applied to hyperbolic problems. We focus in this survey therefore on this important aspect, by first providing a summary of the fundamental differences between parabolic and hyperbolic problems for time parallelization. We then group PinT algorithms into two basic groups: the first group contains four effective PinT techniques for hyperbolic problems, namely Schwarz Waveform Relaxation with its relation to Tent Pitching; Parallel Integral Deferred Correction; ParaExp; and ParaDiag. While the methods in the first group also work well for parabolic problems, we then present PinT methods especially designed for parabolic problems in the second group: Parareal: the Parallel Full Approximation Scheme in Space-Time; Multigrid Reduction in Time; and Space-Time Multigrid.

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arXiv2024

A Parareal algorithm without Coarse Propagator?

The Parareal algorithm was invented in 2001 in order to parallelize the solution of evolution problems in the time direction. It is based on parallel fine time propagators called F and sequential coarse time propagators called G, which alternatingly solve the evolution problem and iteratively converge to the fine solution. The coarse propagator G is a very important component of Parareal, as one sees in the convergence analyses. We present here for the first time a Parareal algorithm without coarse propagator, and explain why this can work very well for parabolic problems. We give a new convergence proof for coarse propagators approximating in space, in contrast to the more classical coarse propagators which are approximations in time, and our proof also applies in the absence of the coarse propagator. We illustrate our theoretical results with numerical experiments, and also explain why this approach can not work for hyperbolic problems.

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arXiv2023

A new ParaDiag time-parallel time integration method

Time-parallel time integration has received a lot of attention in the high performance computing community over the past two decades. Indeed, it has been shown that parallel-in-time techniques have the potential to remedy one of the main computational drawbacks of parallel-in-space solvers. In particular, it is well-known that for large-scale evolution problems space parallelization saturates long before all processing cores are effectively used on today's large scale parallel computers. Among the many approaches for time-parallel time integration, ParaDiag schemes have proved themselves to be a very effective approach. In this framework, the time stepping matrix or an approximation thereof is diagonalized by Fourier techniques, so that computations taking place at different time steps can be indeed carried out in parallel. We propose here a new ParaDiag algorithm combining the Sherman-Morrison-Woodbury formula and Krylov techniques. A panel of diverse numerical examples illustrates the potential of our new solver. In particular, we show that it performs very well compared to different ParaDiag algorithms recently proposed in the literature.

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arXiv2023

Cross-points in the Dirichlet-Neumann method II: a geometrically convergent variant

When considered as a standalone iterative solver for elliptic boundary value problems, the Dirichlet-Neumann (DN) method is known to converge geometrically for domain decompositions into strips, even for a large number of subdomains. However, whenever the domain decomposition includes cross-points, i.e.$\!$ points where more than two subdomains meet, the convergence proof does not hold anymore as the method generates subproblems that might not be well-posed. Focusing on a simple two-dimensional example involving one cross-point, we proposed in a previous work a decomposition of the solution into two parts: an even symmetric part and an odd symmetric part. Based on this decomposition, we proved that the DN method was geometrically convergent for the even symmetric part and that it was not well-posed for the odd symmetric part. Here, we introduce a new variant of the DN method which generates subproblems that remain well-posed for the odd symmetric part as well. Taking advantage of the symmetry properties of the domain decomposition considered, we manage to prove that our new method converges geometrically in the presence of cross-points. We also extend our results to the three-dimensional case, and present numerical experiments that illustrate our theoretical findings.

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arXiv2023

Cross-points in the Neumann-Neumann method

In this work, we focus on the Neumann-Neumann method (NNM), which is one of the most popular non-overlapping domain decomposition methods. Even though the NNM is widely used and proves itself very efficient when applied to discrete problems in practical applications, it is in general not well defined at the continuous level when the geometric decomposition involves cross-points. Our goals are to investigate this well-posedness issue and to provide a complete analysis of the method at the continuous level, when applied to a simple elliptic problem on a configuration involving one cross-point. More specifically, we prove that the algorithm generates solutions that are singular near the cross-points. We also exhibit the type of singularity introduced by the method, and show how it propagates through the iterations. Then, based on this analysis, we design a new set of transmission conditions that makes the new NNM geometrically convergent for this simple configuration. Finally, we illustrate our results with numerical experiments.

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arXiv2023

An optimized Space-Time Multigrid algorithm for parabolic PDEs

We investigate three directions to further improve the highly efficient Space-Time Multigrid algorithm with block-Jacobi smoother introduced in [GanNeu16]. First, we derive an analytical expression for the optimal smoothing parameter in the case of a full space-time coarsening strategy; second, we propose a new and efficient direct coarsening strategy which simplifies the code by preventing changes of coarsening regimes; and third, we also optimize the entire two cycle to investigate if further efficiency gains are possible. Especially, we show that our new coarsening strategy leads to a significant efficiency gain when the ratio $τ/h^2$ is small, where $τ$ and $h$ represent the time and space steps. Our analysis is performed for the heat equation in one spatial dimension, using centered finite differences in space and Backward Euler in time, but could be generalized to other situations. We also present numerical experiments that confirm our theoretical findings.

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arXiv2022

Low-rank Parareal: a low-rank parallel-in-time integrator

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge--Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

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arXiv2022

Convergence of parallel overlapping domain decomposition methods for the Helmholtz equation

We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the subdomain problems satisfy first-order absorbing (impedance) transmission conditions, and exchange of information between subdomains is achieved using a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). First, we formulate the method as a fixed point iteration, and show (in dimensions 1,2,3) that it is well-defined in a tensor product of appropriate local function spaces, each with $L^2$ impedance boundary data. Given this, we then obtain a bound on the norm of the fixed point operator in terms of the local norms of certain impedance-to-impedance maps arising from local interactions between subdomains. These bounds provide conditions under which (some power of) the fixed point operator is a contraction. In 2-d, for rectangular domains and strip-wise domain decompositions (with each subdomain only overlapping its immediate neighbours), we present two techniques for verifying the assumptions on the impedance-to-impedance maps which ensure power contractivity of the fixed point operator. The first is through semiclassical analysis, which gives rigorous estimates valid as the frequency tends to infinity. These results verify the required assumptions for sufficiently large overlap. For more realistic domain decompositions, we directly compute the norms of the impedance-to-impedance maps by solving certain canonical (local) eigenvalue problems. We give numerical experiments that illustrate the theory. These also show that the iterative method remains convergent and/or provides a good preconditioner in cases not covered by the theory, including for general domain decompositions, such as those obtained via automatic graph-partitioning software.

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arXiv2022

Schwarz methods by domain truncation

Schwarz methods use a decomposition of the computational domain into subdomains and need to put boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and also needs to put boundary conditions on the computational domain boundaries. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as truncation of the domain onto subdomains. The first truly optimal Schwarz method that converges in a finite number of steps was proposed in 1994 and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods -- a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions. This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviors of optimized Schwarz methods based on Fourier analysis taking into account the original boundary conditions, many subdomain decompositions and layered media. The transmission conditions we study include the lowest order absorbing conditions (Robin), and also more advanced perfectly matched layers (PML), both developed first for domain truncation.

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arXiv2022

Closed form optimized transmission conditions for complex diffusion with many subdomains

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 subdomain model configurations, and then used in practice for many subdomains. We optimize here transmission conditions 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.

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