Schwarz Domain Decomposition Preconditioner Research Articles

A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

Abstract Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

Three-dimensional finite-difference finite-element frequency-domain wave simulation with multi-level optimized additive Schwarz domain-decomposition preconditioner: A tool for FWI of sparse node datasets

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 the 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 P3 finite-element method on h-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.

Scalable domain decomposition preconditioner for Navier–Stokes equations coupled with the heat equation

In this article, we study the thermal instability that appears from time to time while printing using a 3D printer. To solve the semi-discretized problem at each time-step, we use a scalable parallel algorithm based on a two-level Optimized Restricted Additive Schwarz (ORAS) domain decomposition preconditioner for GMRES. Parallel scalability tests are conducted with comparison against the parallel direct solver MUMPS and the one-level Schwarz method, which show lack of robustness for larger number of processors. 2D numerical tests illustrate that the number of iterations to reach GMRES convergence depends on the state of the physical simulation during time, and that the second level of preconditioning is needed to achieve robustness.

Additive Schwarz preconditioners for a localized orthogonal decomposition method

We investigate a variant of the localized orthogonal decomposition method (Henning and Peterseim, [Multiscale Model. Simul., 11 (2013), pp. 1149–1175] and Målqvist and Peterseim, [Math. Comp., 83 (2014), pp. 2583–2603]) for elliptic problems with rough coefficients. The construction of the basis of the multiscale finite element space is based on domain decomposition techniques, which is motivated by the recent work of Kornhuber, Peterseim, and Yserentant [Math. Comp., 87 (2018), pp. 2765–2774]. We also design and analyze additive Schwarz domain decomposition preconditioners for the resulting discrete problems.

Domain decomposition preconditioners for high-order discretizations of the heterogeneous Helmholtz equation

Abstract We consider one-level additive Schwarz domain decomposition preconditioners for the Helmholtz equation with variable coefficients (modelling wave propagation in heterogeneous media), subject to boundary conditions that include wave scattering problems. Absorption is included as a parameter in the problem. This problem is discretized using $H^1$-conforming nodal finite elements of fixed local degree $p$ on meshes with diameter $h = h(k)$, chosen so that the error remains bounded with increasing $k$. The action of the one-level preconditioner consists of the parallel solution of problems on subdomains (which can be of general geometry), each equipped with an impedance boundary condition. We prove rigorous estimates on the norm and field of values of the left- or right-preconditioned matrix that show explicitly how the absorption, the heterogeneity in the coefficients and the dependence on the degree enter the estimates. These estimates prove rigorously that, with enough absorption and for $k$ large enough, GMRES is guaranteed to converge in a number of iterations that is independent of $k,p$ and the coefficients. The theoretical threshold for $k$ to be large enough depends on $p$ and on the local variation of coefficients in subdomains (and not globally). Extensive numerical experiments are given for both the absorptive and the propagative cases; in the latter case, we investigate examples both when the coefficients are nontrapping and when they are trapping. These experiments support (i) our theory in terms of dependence on polynomial degree and the coefficients; and (ii) the sharpness of our field of values estimates in terms of the level of absorption required.

An adaptively enriched coarse space for Schwarz preconditioners for P1 discontinuous Galerkin multiscale finite element problems

Abstract In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.

Surface Integral Equation-Domain Decomposition Scheme for Solving Multiscale Nanoparticle Assemblies With Repetitions

In this paper, we introduce the use of the multilevel fast multipole algorithm (MLFMA) in synergy with a multilevel nonoverlapping additive Schwarz domain decomposition (DD) preconditioner for the solution of large arrays of nanoparticles presenting multiscale and multiphysics features. The judicious selection of subdomains allows for the isolation of the different scale/physics subproblems, yielding an efficient and effective preconditioner for the surface integral equation matrix system. Furthermore, the MLFMA-method of moments is employed to take advantage of the repetition pattern inherent to these kinds of structures. Numerical experiments solving real-life plasmonic biosensors built up from complicated particle assemblies reveal a great improvement of convergence, testifying to the robustness and versatility of the DD approach.

Parallel subdomain solver strategies for the algebraic additive Schwarz preconditioner

Hybrid domain-decomposition method with two levels of domain partitioning.Solution of the local problem by ShyLU or other parallel approximation techniques.Proposal of an inexact QR factorization to be used within ShyLU.Comparisons of parallel performances between UMFPACK, ShyLU, and inexact QR. Domain-decomposition (DD) methods are used in most, if not all, modern parallel implementations of finite element modeling software. In the solver stage, the algebraic additive Schwarz (AAS) domain-decomposition preconditioner represents a fundamental component and its performance and scalability are key to the overall performance of the solution process. The established approach to construct the preconditioner in a parallel MPI setting is with a 1-to-1 correspondence between the number of MPI processes and the number of AAS subdomains.In this paper, we describe our attempts to extend this paradigm with the possibility to assign more than one MPI process per AAS subdomain, with the goal of improving the overall performance of the AAS preconditioner on supercomputers with multicore nodes.We discuss the implementation of the new AAS preconditioner framework, based on two levels of MPI parallelism, and the performance of different subdomain solver strategies. Finally, we examine the behavior of our novel approach for a series of benchmark problems, performed with the LifeV parallel finite element library.

Multilevel Schwarz methods for elliptic partial differential equations

We investigate multilevel Schwarz domain decomposition preconditioners, to efficiently solve linear systems arising from numerical discretizations of elliptic partial differential equations by the finite element method. In our analysis we deal with unstructured mesh partitions and with subdomain boundaries resulting from using the mesh partitioner. We start from two-level preconditioners with either aggregative or interpolative coarse level components, then we focus on a strategy to increase the number of levels. For all preconditioners, we consider the additive residual update and its multiplicative variants within and between levels. Moreover, we compare the preconditioners behaviour, regarding scalability and rate of convergence. Numerical results are provided for elliptic boundary value problems, including a convection–diffusion problem when suitable stabilization becomes necessary.

Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems

Overlapping Schwarz domain decomposition preconditioners for the local discontinuous Galerkin method for elliptic problems

A parallel fully coupled algebraic multilevel preconditioner applied to multiphysics PDE applications: Drift‐diffusion, flow/transport/reaction, resistive MHD

AbstractThis study considers the performance of a fully coupled algebraic multilevel preconditioner for Newton–Krylov solution methods. The performance of the preconditioner is demonstrated on a set of challenging multiphysics partial differential equation (PDE) applications: a drift‐diffusion approximation for semiconductor devices; a low Mach number formulation for the simulation of coupled flow, transport and non‐equilibrium chemical reactions; and a low Mach number formulation for visco‐resistive magnetohydrodynamics (MHD) systems. These systems contain multiple physical mechanisms that are strongly coupled, highly nonlinear, non‐symmetric and produce solutions with multiple length‐ and time‐scales. In the context of this study the governing PDEs for these systems are discretized in space by a stabilized finite element (FE) method that collocates all unknowns at each node of the FE mesh. The algebraic multilevel preconditioner is based on an aggressive‐coarsening graph‐partitioning of the non‐zero block structure of the Jacobian matrix. The performance of the algebraic multilevel preconditioner is compared with a standard variable overlap additive Schwarz domain decomposition preconditioner. Representative performance and parallel scaling results are presented for a set of direct‐to‐steady‐state and fully implicit transient solutions. The performance studies include parallel weak scaling studies on up to 4096 cores and also includes the solution of systems as large as two billion unknowns carried out on 24000 cores of a Cray XT3/4. In general, the results of this study indicate that on this reasonably diverse set of challenging multiphysics applications the algebraic multilevel preconditioner performs very well. Copyright © 2010 John Wiley & Sons, Ltd.

Domain decomposition for multiscale PDEs

We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (Monte–Carlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains, when the classical method fails to be robust. In particular our estimates prove very precisely the previously made empirical observation that the use of low-energy coarse spaces can lead to robust preconditioners. We go on to consider coarse spaces constructed from multiscale finite elements and prove that preconditioners using this type of coarsening lead to robust preconditioners for a variety of binary (i.e., two-scale) media model problems. Moreover numerical experiments show that the new preconditioner has greatly improved performance over standard preconditioners even in the random coefficient case. We show also how the analysis extends in a straightforward way to multiplicative versions of the Schwarz method.

Schwarz domain decomposition preconditioners for discontinuous Galerkin approximations of elliptic problems: non-overlapping case

We propose and study some new additive, two-level non-overlapping Schwarz preconditioners for the solution of the algebraic linear systems arising from a wide class of discontinuous Galerkin approximations of elliptic problems that have been proposed up to now. In particular, two-level methods for both symmetric and non-symmetric schemes are introduced and some interesting features, which have no analog in the conforming case, are discussed. Both the construction and analysis of the proposed domain decomposition methods are presented in a unified framework. For symmetric schemes, it is shown that the condition number of the preconditioned system is of order O(H/h) , where H and h are the mesh sizes of the coarse and fine grids respectively, which are assumed to be nested. For non-symmetric schemes, we show by numerical computations that the Eisenstat et al. [SIAM J. Numer. Anal. 20 (1983) 345–357] GMRES convergence theory, generally used in the analysis of Schwarz methods for non-symmetric problems, cannot be applied even if the numerical results show that the GMRES applied to the preconditioned systems converges in a finite number of steps and the proposed preconditioners seem to be scalable. Extensive numerical experiments to validate our theory and to illustrate the performance and robustness of the proposed two-level methods are presented.

A characteristic nonoverlapping domain decomposition method for multidimensional convection‐diffusion equations

AbstractWe develop a quasi‐two‐level, coarse‐mesh‐free characteristic nonoverlapping domain decomposition method for unsteady‐state convection‐diffusion partial differential equations in multidimensional spaces. The development of the domain decomposition method is carried out by utilizing an additive Schwarz domain decomposition preconditioner, by using an Eulerian‐Lagrangian method for convection‐diffusion equations and by delicately choosing appropriate interface conditions that fully respect and utilize the hyperbolic nature of the governing equations. Numerical experiments are presented to illustrate the method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

A Weakly Overlapping Domain Decomposition Preconditioner for the Finite Element Solution of Elliptic Partial Differential Equations

We present a new two-level additive Schwarz domain decomposition preconditioner which is appropriate for use in the parallel finite element solution of elliptic partial differential equations (PDEs). As with most parallel domain decomposition methods each processor may be assigned one or more subdomains, and the preconditioner is such that the processors are able to solve their own subproblem(s) concurrently. The novel feature of the technique proposed here is that it requires just a single layer of overlap in the elements which make up each subdomain at each level of refinement, and it is shown that this amount of overlap is sufficient to yield an optimal preconditioner. Some numerical experiments---posed in both two and three space dimensions---are included to confirm that the condition number when using the new preconditioner is indeed independent of the level of mesh refinement on the test problems considered.

An Aggregation-Based Domain Decomposition Preconditioner for Groundwater Flow

We consider theoretical and computational issues associated with an aggregation-based domain decomposition preconditioner applied to a Bi-CGSTAB iterative solver used to solve both Laplace's equation and an important nonlinear model from hydrology used to simulate unsaturated flow, Richards' equation. Theoretical results for Laplace's equation provide estimates of the condition number and the rate of convergence for a two-level Schwarz domain decomposition preconditioner. Computational results for Laplace's equation and Richards' equation show excellent scalability, although no theory is yet available to support the results for the difficult nonlinear problem.

Parallel Computation of Flow in Heterogeneous Media Modelled by Mixed Finite Elements

In this paper we describe a fast parallel method for solving highly ill-conditioned saddle-point systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of the linear first-order velocity–pressure system comprising Darcy's law coupled with an incompressibility constraint. The chief difficulty is that the permeability may be highly variable, especially when the statistical model has a large variance and a small correlation length. For reasonable accuracy, the discretisation has to be extremely fine. We solve these problems by first reducing the saddle-point formulation to a symmetric positive definite (SPD) problem using a suitable basis for the space of divergence-free velocities. The reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive Schwarz domain decomposition preconditioner. The result is a solver which exhibits a good degree of robustness with respect to the mesh size as well as to the variance and to physically relevant values of the correlation length of the underlying permeability field. Numerical experiments exhibit almost optimal levels of parallel efficiency. The domain decomposition solver (DOUG, http://www.maths.bath.ac.uk/~parsoft) used here not only is applicable to this problem but can be used to solve general unstructured finite element systems on a wide range of parallel architectures.

Unstructured Additive Schwarz--Conjugate Gradient Method for Elliptic Problems with Highly Discontinuous Coefficients

This paper concerns the iterative solution of symmetric elliptic problems with piecewise constant coefficients in two or three space dimensions discretized by linear finite element methods on unstructured triangular or tetrahedral meshes. The effect of the discontinuous coefficients is studied by first postulating that there are d fixed regions of the domain where the coefficient takes constant positive values a = (a1, . . . , ad)and then considering certain (positive) coefficient sequences {a(m)} in which some of these values approach $0$ or $\infty$ as $m \rightarrow \infty$. We consider the performance of additive Schwarz domain decomposition preconditioners constructed from local solves on automatically generated subdomains together with a global solve on some coarser grid. Assuming no relationship between the regions on which the coefficient function is constant and either the subdomains or the coarse grid, we show that the preconditioned conjugate gradient method converges (in both the energy and the Euclidean norms) with a number of iterations which grows only logarithmically in the size of the maximum jump ${\cal J}^{(m)} := \max\{a_k^{(m)}/a_l^{(m)} : k, l = 1, \ldots , d\} $, as $m\rightarrow \infty$. The result is obtained by a careful analysis of the preconditioned matrix. It cannot be obtained by the usual procedure of estimating condition numbers: a simple example is given in which the condition number of both the original and the preconditioned stiffness matrices degrade linearly in ${\cal J}^{(m)}$. Recent results of Chan, Smith, and Zou have shown that by using this preconditioner together with the conjugate gradient method, the number of iterations can be bounded independently of the mesh diameter provided the subdomains have overlap commensurate with the size of the coarse mesh. Our results now show that this method is also highly resilient to discontinuous coefficients, even if no attention is paid to the coefficient discontinuity in the construction of the solver.