Martin Gander

• University of Geneva

For optimal control problems there is a classical discussion of whether one should first optimize the problem and then discretize it, or the other way round. We are interested in exploring a similar question related to domain decomposition methods for optimal control problems which have received substantial attention over the past two decades, but...

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 spee...

We propose an approach based on Fourier analysis to wavenumber explicit sharp estimation of absolute and relative errors of finite difference methods for the Helmholtz equation. We use the approach to analyze the classical centred scheme for the Helmholtz equation with a general smooth source term and Dirichlet boundary conditions in 1D. For the Fo...

We are interested in heterogeneous domain decomposition methods to couple partial differential equations in space-time. The coupling can be used to describe the exchange of heat or forces or both, and has important applications like fluid-structure or ocean-atmosphere coupling. Heterogeneous domain decomposition methods permit furthermore the reuse...

We consider the Poisson equation as our model problem.

We are interested in solving the heat equation \(\partial_{t} u - \tilde{v}\partial_{xx}^{2} u = f\,\,on\,\left( { - L,\,L} \right) \times \left( {0,T} \right)\), with an initial condition and with Dirichlet boundary conditions.

In [8] we showed the limits of using Newton’s method to accelerate domain decomposition methods in 1D.

The Neumann-Neumann method (NNM), first introduced in [1] in the case of two subdomains, is among the most popular non-overlapping domain decomposition methods.

Consider the state y (x) governed by the elliptic partial differential equation (PDE).

Wave field simulations have many applications, from seismology over radiation to acoustics.

Approximating transmission conditions is very important for Optimized Schwarz Methods (OSM) [2].

We are interested in solving in parallel anisotropic diffusion problems of the form.

In 1987, Roy Nicolaides introduced what we would now call a coarse space correction for the conjugate gradient method [30].

Parallel-in-time methods, of which parareal [13] and multigrid reduction in time (MGRIT) [3] are well-known examples, are important tools for increasing parallelism beyond traditional spatially parallel methods, see [6, 14] and references therein.

Substructuring domain decomposition methods and iterative substructuring methods referred originally to a few specific methods, see e.g. [13].

Substructured Schwarz methods are interpretations of volume Schwarz methods as algorithms on interface variables.

The Q1 coarse space [7, 8] is based on coarse Q1 bilinear finite element functions on rectangular elements which are here the subdomains.

The mapped tent pitching algorithm (MTP) is a very advanced domain decomposition strategy for the parallel solution of hyperbolic problems.

In order to understand research problems in multi-physics, it is instructive to first look at coupling conditions for mono-physics problems, which arise naturally in domain decomposition.

The Brinkman equations model a combination of Darcy’s law and the Navier-Stokes equations, see [3].

In the times of exascale computing, very efficient algorithms for space parallelism exist and communication between processors has become a bottleneck; parallelism in the time dimension is necessary.

We consider a Stokes flow along a thin fracture coupled to a Darcy flow in the surrounding matrix domain. In order to derive a dimensionally reduced model representing the fracture as an interface coupled to the surrounding matrix, we extend the methodology based on Fourier analysis developed in [1] for a Darcy-Darcy coupling. We show that this app...

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, t...

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 evolu...

Ever since the publication of the first book on domain decomposition methods by Smith, Bjørstad, and Gropp [8], where non-matching grids were used for overlapping Schwarz methods (see on the right), and the methods worked very well, a theoretical understanding of their convergence remained open.

Adaptive Dirichlet–Neumann and Robin–Neumann algorithms for singularlyperturbed advection-diffusion equations were introduced in [2], accounting for transport along characteristics, see also [6] for the discrete setting and damped versions using a modified quadrature rule to recover the hyperbolic limit. Non-overlapping Schwarz DDMs with Robin tran...

We consider an elastic string of length 𝐿, attached at its two end points, vibrating in a plane due to an initial deformation corresponding to a pinch in the middle, see Figure 1.

Historically, coarse spaces for domain decomposition methods were based on a coarse grid, like in geometric multigrid methods, see e.g. [17, page 36]: “The subspace 𝑉0 is usually related to a coarse problem, often built on a coarse mesh”.

Method (5)–(8) is an example of methods studied more generally in the Optimized Schwarz literature (e.g., [4, 10]), where Robin (or more sophisticated) transmission conditions are constructed with the aim of optimizing convergence rates. Although the transmission condition (6) above can be justified directly as a first order absorbing condition for...

In [7, 8], we derived and studied an asymptotic model for Darcy flow in fractured porous media, when the fracture aperture 𝛿 approaches zero.We showed that our new, general models coincide in special cases with common models from the literature, such as, for instance [1, 2, 10, 11]. Our general modeling approach leads to coupling conditions, which...

As is well known, domain decomposition methods applied to elliptic problems require in most cases a coarse correction to be scalable (for exceptions, see [5, 6]), the choice of the coarse space being critical to achieve good performance.

One of the attractive features of the DN method for linear problems is that it achieves mesh independent convergence. Does this also hold for the nonlinearDNmethod (2)? We first define the nonlinear DN method for multiple subdomains.

Optimized Schwarz Methods (OSMs) are very versatile: they can be used with or without overlap, converge faster compared to other domain decomposition methods [5], are among the fastest solvers for wave problems [10], and can be robust for heterogeneous problems [7]. This is due to their general transmission conditions, optimized for the problem at...

Our main focus here is on cross points in non-overlapping domain decomposition methods, but our techniques can also be applied to cross points in overlapping domain decomposition methods, which can be an issue as indicated already by P.L. Lions in his seminal paper [17], see Figure 1.

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 strat...

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 method...

https://hal.archives-ouvertes.fr/hal-03837707v1

Cross-points in domain decomposition, i.e., points where more than two subdomains meet, have received substantial attention over the past years, since domain decomposition methods often need special attention in their definition at cross-points, in particular if the transmission conditions of the domain decomposition method contain derivatives, lik...

We analyse parallel overlapping Schwarz domain decomposition methods for the Helmholtz equation, where the exchange of information between subdomains is achieved using first-order absorbing (impedance) transmission conditions, together with a partition of unity. We provide a novel analysis of this method at the PDE level (without discretization). F...

Iterative substructuring Domain Decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. It is less known that classical overlapping DD methods can also be formulated in substructured form, i.e., as iterative methods acting on variables defined exclusively on the interfaces of the...

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 th...

In Ciaramella et al. (2020) we defined a new partition of unity for the Bank–Jimack domain decomposition method in 1D and proved that with the new partition of unity, the Bank–Jimack method is an optimal Schwarz method in 1D and thus converges in two iterations for two subdomains: it becomes a direct solver, and this independently of the outer coar...

Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodi...

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...

Parallel-in-time integration has been the focus of intensive research efforts over the past two decades due to the advent of massively parallel computer architectures and the scaling limits of purely spatial parallelization. Various iterative parallel-in-time (PinT) algorithms have been proposed, like Parareal, PFASST, MGRIT, and Space-Time Multi-G...

Discrete Duality Finite Volume (DDFV) methods are very well suited to discretize anisotropic diffusion problems, even on meshes with low mesh quality. Their performance stems from an accurate reconstruction of the gradients between mesh cell boundaries, which comes however at the cost of using both a primal (cell centered) and a dual (vertex center...

We introduce a new non-overlapping optimized Schwarz method for fully anisotropic diffusion problems. Optimized Schwarz methods take into account the underlying physical properties of the problem at hand in the transmission conditions, and are thus ideally suited for solving anisotropic diffusion problems. We first study the new method at the conti...

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 d...

Substructured domain decomposition (DD) methods have been extensively studied, and they are usually associated with nonoverlapping decompositions. We introduce here a substructured version of Restricted Additive Schwarz (RAS) which we call SRAS, and we discuss its advantages compared to the standard volume formulation of the Schwarz method when the...

The Dirichlet-Neumann (DN) method has been extensively studied for linear partial differential equations, while little attention has been devoted to the nonlinear case. In this paper, we analyze the DN method both as a nonlinear iterative method and as a preconditioner for Newton's method. We discuss the nilpotent property and prove that under spec...

In this paper we revisit the Restricted Additive Schwarz method for solving discretized Helmholtz problems, using impedance boundary conditions on subdomains (sometimes called ORAS). We present this method in its variational form and show that it can be seen as a finite element discretization of a parallel overlapping domain decomposition method de...

Optimized Schwarz Methods (OSMs) are based on optimized transmission conditions along the interfaces between the subdomains. Optimized transmission conditions are derived at the theoretical level, using techniques developed in the last decades. The hypothesis behind these analyses are quite strong, so that the applicability of OSMs is still limited...

Classically transmission conditions between subdomains are optimized for a simplified two subdomain decomposition to obtain optimized Schwarz methods for many subdomains. We investigate here if such a simplified optimization suffices for the magnetotelluric approximation of Maxwell's equation which leads to a complex diffusion problem. We start wit...

Over the past decade, partial differential equation models in elliptical geometries have become a focus of interest in several scientific and engineering applications: the classical studies of flow past a cylinder, the spherical particles in nano-fluids and spherical water filled domains are replaced by elliptical geometries which more accurately d...

We develop new error estimates for the one-dimensional advection equation, considering general space-time discretization schemes based on Runge–Kutta methods and finite difference discretizations. We then derive conditions on the number of points per wavelength for a given error tolerance from these new estimates. Our analysis also shows the existe...

In 2001 Randolph E. Bank and Peter K. Jimack [1] introduced a new domain decomposition method for the adaptive solution of elliptic partial differential equations, see also [2].

Recently a lot of attention has been devoted to the Stokes-Darcy coupling which is a system of equations used to model the flow of fluids in porous media. In [2, 1[ a non standard behaviour of the optimized Schwarz method (OSM) has been observed: the optimized parameters obtained solving the classical min-max problems do not lead to an optimized co...

We define a new two-level optimized Schwarz method (OSM), and we provide a convergence analysis both for overlapping and nonoverlapping decompositions. The two-level analysis suggests how to choose the optimized parameters. We also discuss an optimization procedure which relies only on the already studied one-level min-max problems, and we show tha...

“Vous n’avez vraiment rien à faire”!1 This was the smiling reaction of Laurence Halpern when the first author told her about our wish to accurately estimate the convergence rate of the Schwarz method for the solution of the ddm logo2, see Figure 1 (left).

Among many applications of parallel computing, solving large systems of ordinary differential equations (ODEs) which arise from large scale electronic circuits, or discretizations of partial differential equations (PDEs), form an important part.

Time-periodic problems appear naturally in engineering applications. For instance,the time-periodic steady-state behavior of an electromagnetic device is often the main interest in electrical engineering, because devices are operated most of their life-time in this state.

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, im...

Neumann-Neumann methods (NNMs) are among the best parallel solvers for discretized partial differential equations, see [12] and references therein. Their common polylogarithmic condition number estimate shows their effectiveness for many discretized elliptic problems, see [9, 10, 5].

Maxwell’s equations can be used to model the propagation of electro-magneticwaves in the subsurface of the Earth. The interaction of such waves with the material in the subsurface produces response waves, which carry information about the physical properties of the Earth’s subsurface, and their measurement allows geophysicists to detect the presenc...

The classical alternating Schwarz method does not need a partition of unity in its definition [3]: one solves one subdomain after the other, stores subdomain solutions, and always uses the newest data available from the neighboring subdomains. In the parallel Schwarz method introduced by Lions in [10], where all subdomains are solved simultaneously...

Additive Schwarz Methods (ASM) are implemented in the PETSc library [2, 1, 3] within its PCASM preconditioning option. By default this applies the Restricted Additive Schwarz (RAS) method of Cai and Sarkis [4].

There has been substantial attention on coarse correction in the domain decomposition community over the last decade, sparked by the interest of solving high contrast and multiscale problems, since in this case, the convergence of two-level domain decomposition methods is deteriorating when the contrast becomes large, see [1, 10, 16, 17, 11, 9, 8]...

We study the behavior of solutions of PDE models on domains containing a heterogeneous layer of aperture tending to zero. We consider general second order differential operators on the outer domains and elliptic operators inside the layer. Our study is motivated by the modeling of flow through fractured porous media, when one represents the fractur...

Time parallel time integration has received substained attention over the last decades, for a review, see [2]. More recently, renewed interest in this area was sparked by the invention of the Parareal algorithm [5] for solving initial value problems like

The Helmholtz equation is the simplest model for time harmonic wave propagation, and it contains already all the fundamental difficulties such problems pose when trying to compute their solution numerically.

thm uses a single implicit Runge-Kutta (IRK) method with the same small step-size for both the F and G propagators in parareal, and would thus converge in one iteration when used directly like this, however without any speedup due to the sequential way parareal uses G. We then approximate G with a head-tail coupled condition such that G can be para...

In this paper, a new waveform relaxation variant of the Dirichlet–Neumann algorithm is introduced for general parabolic problems as well as for the second-order wave equation for decompositions with multiple subdomains. The method is based on a non-overlapping decomposition of the domain in space, and the iteration involves subdomain solves in spac...

Among many applications of parallel computing, solving large systems of ordinary differential equations (ODEs) which arise from large scale electronic circuits, or discretizations of partial differential equations (PDEs), form an important part. A systematic approach to their parallel solution are Waveform Relaxation (WR) tech- niques, which were i...

Waveform relaxation (WR) methods are based on partitioning large circuits into sub-circuits which then are solved separately for multiple time steps in so called time windows, and an iteration is used to converge to the global circuit solution in each time window. Classical WR converges quite slowly, especially when long time windows are used. To o...

The computation of Gauss quadrature rules for arbitrary weight functions using the Stieltjes algorithm is a purely sequential process, and the computational cost significantly increases when high accuracy is required. ParaStieltjes is a new algorithm to compute the recurrence coefficients of the associated orthogonal polynomials in parallel, from w...

We introduce an overlapping optimized Schwarz methods in the DDFV framework for an anisotropic diffusion equation, and we show that a discrete and bounded domain convergence analysis is important to get best performance for strong anisotropy.

In 2008, Maday and R{\o}nquist introduce{d} an interesting new approach for the direct parallel-in-time (PinT) solution of time-dependent PDEs. The idea is to diagonalize the time stepping matrix, keeping the matrices for the space discretization unchanged, and then to solve all time steps in parallel. Since then, several variants appeared, and we...

BDF formulas are among the most efficient methods for numerical integration, in particular of stiff equations (see e.g. Gear in Numerical initial value problems in ordinary differential equations, Prentice Hall, Upper Saddle River, 1971). Their excellent stability properties are known for precisely half a century, from the first calculation of thei...

Waveform relaxation (WR) methods are based on partitioning large circuits into sub-circuits which then are solved separately for multiple time steps in so-called time windows, and an iteration is used to converge to the global circuit solution in each time window. Classical WR converges quite slowly, especially when long time windows are used. To o...

Recently a lot of attention has been devoted to the Stokes-Darcy couplingwhich is a system of equations used to model the flow of fluids in porousmedia. In [2, 1] a non standard behaviour of the optimized Schwarz method(OSM) has been observed: the optimized parameters obtained solving theclassical min-max problems do not lead to an optimized conver...