Additive Schwarz Domain Decomposition Research Articles

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.

Consistent coupling algorithms for coupled core-edge simulations of plasma turbulence

Two schemes for coupling gyrokinetic simulations of microturbulence in tokamaks are proposed. The first scheme is based on an additive Schwarz domain decomposition. We show that, because the goal of turbulence is long-time averages of the dynamics rather than temporal accuracy, the iteration to self-consistency across domains, which is typically required by Schwarz schemes, can be avoided, thereby accelerating the computation. Second, we propose a coupling scheme that relies entirely on the addition of source terms, leaving the boundary conditions arbitrary. The practical motivations for such a scheme are discussed, and forms of the source terms that ensure consistency and stability are derived. The schemes are tested on a nonlinear, one-dimensional model problem, and the first scheme is further tested on the Hasegawa–Wakatani model.

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 forP1 discontinuous Galerkin multiscale finite element problems

AbstractIn 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.

On the performance of Krylov smoothing for fully coupled AMG preconditioners for VMS resistive MHD

SummaryThis study explores the performance and scaling of a GMRES Krylov method employed as a smoother for an algebraic multigrid preconditioned Newton‐Krylov solution approach applied to a fully implicit variational multiscale finite element resistive magnetohydrodynamics formulation. In this context, a Newton iteration is used for the nonlinear system and a parallel MPI‐based Krylov method is employed for the linear subsystems. The efficiency of this approach is critically dependent on the scalability and performance of the parallel algebraic multigrid preconditioner for the linear solutions and the performance of the multigrid smoothers play a critical role. Krylov multigrid smoothers are considered in an attempt to reduce the time and memory requirements of existing robust smoothers based on additive Schwarz domain decomposition with incomplete LU factorization solves on each subdomain. Three time‐dependent resistive magnetohydrodynamics test cases are considered to evaluate the method. Compared with a domain decomposition incomplete LU smoother, the GMRES smoother can reduce the solve time due to a significant decrease in the preconditioner setup time and often a reduction in outer Krylov solver iterations, and requires less memory, typically 35% less memory.

A coercive heterogeneous media Helmholtz model: formulation, wavenumber-explicit analysis, and preconditioned high-order FEM

We consider a frequency-domain heterogeneous wave propagation model governed by the Helmholtz partial differential equation (PDE) and an impedance boundary condition. The celebrated standard (H1) variational formulation of the model is non-coercive. It is an open problem to establish a coercive variational formulation of the heterogeneous model. The main focus of this article is on solving this continuous model formulation and analysis problem, and hence establishing an efficient preconditioned numerical algorithm for simulating our novel coercive variational formulation. We develop the variational formulation for the heterogeneous model (in a Hilbert space V equipped with a stronger norm than the H1-norm) and prove that the associated sesquilinear form is coercive, with a wavenumber-independent coercivity constant. We use this result to derive a wavenumber-independent bound for solutions of the heterogeneous media wave propagation model in the V -norm. Additionally, we prove continuity of the sesquilinear form, with a wavenumber-explicit continuity constant. Using our analysis-supported coercive formulation, we develop a high-order frequency robust-preconditioned finite element method (FEM)-based heterogeneous media discrete wave model. For demonstrating efficiency and convergence of the coercive high-order FEM model, we use non-convex media comprising curved and non-smooth boundaries and low- to high-frequency input incident waves. For the heterogeneous media, with size varying from tens to hundreds of wavelengths, we demonstrate that our new preconditioned-FEM model requires a very low number of GMRES iterations, and the number of iterations is independent of the wavenumber of the model. We also use a class of additive Schwarz domain decomposition (DD) algorithms to implement the preconditioned-FEM model. The DD-based high-order preconditioned-FEM results and comparisons further demonstrate efficiency of the coercive formulation to simulate wave propagation in heterogenous media.

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

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.

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 Domain Decomposition Based Distributed State Estimation for Large-scale Power Systems

Growing system sizes and complexity, along with the large amount of data provided by phasor measurement units (PMUs), are the drivers to accurate state estimation algorithms for online monitoring and operation of power systems. In this paper, a distributed weighted-least-square state estimation method using an additive Schwarz domain decomposition technique is proposed to reduce the computational execution time. The proposed approach divides a data set into several subsets to be processed in parallel using a multiprocessor architecture considering data exchange among distributed areas. The slow coherency method and balanced partitioning are utilized to reduce the communication overhead and increase accuracy. Moreover, bad data analysis is also investigated in a distributed manner. The performance of the proposed distributed state estimator, along with the speed-up for several test systems, was compared with the traditional centralized state estimator. The simulation results show a speed-up of 6.5 for a 4992-bus system.

Stabilization and scalable block preconditioning for the Navier–Stokes equations

This study compares several block-oriented preconditioners for the stabilized finite element discretization of the incompressible Navier–Stokes equations. This includes standard additive Schwarz domain decomposition methods, aggressive coarsening multigrid, and three preconditioners based on an approximate block LU factorization, specifically SIMPLEC, LSC, and PCD. Robustness is considered with a particular focus on the impact that different stabilization methods have on preconditioner performance. Additionally, parallel scaling studies are undertaken. The numerical results indicate that aggressive coarsening multigrid, LSC and PCD all have good algorithmic scalability. Coupling this with the fact that block methods can be applied to systems arising from stable mixed discretizations implies that these techniques are a promising direction for developing scalable methods for Navier–Stokes.

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.

MLD2P4

Domain decomposition ideas have long been an essential tool for the solution of PDEs on parallel computers. In recent years many research efforts have been focused on recursively employing domain decomposition methods to obtain multilevel preconditioners to be used with Krylov solvers. In this context, we developed MLD2P4 (MultiLevel Domain Decomposition Parallel Preconditioners Package based on PSBLAS), a package of parallel multilevel preconditioners that combines additive Schwarz domain decomposition methods with a smoothed aggregation technique to build a hierarchy of coarse-level corrections in an algebraic way. The design of MLD2P4 was guided by objectives such as extensibility, flexibility, performance, portability, and ease of use. They were achieved by following an object-based approach while using the Fortran 95 language, as well as by employing the PSBLAS library as a basic framework. In this article, we present MLD2P4 focusing on its design principles, software architecture, and use.

Additive Schwarz algorithms for parabolic problems

In this paper, we consider the solution of linear systems of algebraic equations that arise from parabolic finite element problems. We introduce additive Schwarz domain decomposition methods for parabolic differential equations and also consider the dependence of convergence rates of these algorithms on parameters of time step and space-mesh. The resulting preconditioned linear system of equations is solved by the conjugate gradient method. Finally we give a numerical example.

COMPARISON BETWEEN GLOBAL, CLASSICAL DOMAIN DECOMPOSITION AND LOCAL, SINGLE AND DOUBLE COLLOCATION METHODS BASED ON RBF INTERPOLATION FOR SOLVING CONVECTION-DIFFUSION EQUATION

This work presents a performance comparison of several meshless RBF formulations for convection-diffusion equation with moderate-to-high Peclet number regimes. For the solution of convection-diffusion problems, several comparisons between global (full-domain) meshless RBF methods and mesh-based methods have been presented in the literature. However, in depth studies between new local RBF collocation methods and full-domain symmetric RBF collocation methods are not reported yet. The RBF formulations included: global symmetric method, symmetric double boundary collocation method, additive Schwarz domain decomposition method (DDM) when it is incorporated into two anterior approaches, and local single and double collocation methods. It can be found that the accuracy of solutions deteriorates as Pe increases, if no special treatment is used. From the numerical tests, it seems that the local methods, especially the derived double collocation technique incorporating PDE operator, are more effective than full domain approaches even with iterative DDM in solving moderate-to-high Pe convection-diffusion problems subject to mixed boundary conditions.

A taxonomy and comparison of parallel block multi-level preconditioners for the incompressible Navier–Stokes equations

In recent years, considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible Navier–Stokes equations based on preconditioned Krylov methods. These include physics-based methods, such as SIMPLE, and purely algebraic preconditioners based on the approximation of the Schur complement. All these techniques can be represented as approximate block factorization (ABF) type preconditioners. The goal is to decompose the application of the preconditioner into simplified sub-systems in which scalable multi-level type solvers can be applied. In this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the Navier–Stokes system first presented in [A. Quarteroni, F. Saleri, A. Veneziani, Factorization methods for the numerical approximation of Navier–Stokes equations, Computational Methods in Applied Mechanical Engineering 188 (2000) 505–526]. This taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain Schur complement operators. We then present a parallel computational study that examines the performance of these methods and compares them to an additive Schwarz domain decomposition (DD) algorithm. Results are presented for two and three-dimensional steady state problems for enclosed domains and inflow/outflow systems on both structured and unstructured meshes. The numerical experiments are performed using MPSalsa, a stabilized finite element code.

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.

On domain decomposition algorithms for covolume methods for elliptic problems

In this paper, some new two-level additive Schwarz domain decomposition methods are constructed and used as preconditioners for GMRES for the nonsymmetric systems arising from the covolume methods for elliptic problems. Without any restriction on the coarse mesh size, the convergence of the preconditioned GMRES is obtained for both circumcentric and barycentric covolume methods with inexact solvers for subproblems. The rates of convergence are shown to be independent of the number of subregions and these methods are optimal for generous overlap.

On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods

AbstractIn this paper, some pathologies found for simple tests solved by means of preconditioned full iterative schemes are presented. According to these results (Sections 4 and 5), the accuracy deterioration observed should be considered as a warning for the final application given to these solutions. Even though it is well known that full iterative solvers are not the best selection for comparison, they were chosen because they are widely used by the computational fluid dynamic (CFD) community for a diversity of complex fluid dynamics applications.FEM simulated solutions are compared with analytical solutions or measured data for problems that have been considered as ‘benchmarks’ in the CFD literature. For this purpose, the study of the solution obtained via parallelized iterative methods that have been extensively used (e.g. conjugate gradients (CG), GMRes global iteration and its variants, ‘overlapping’ and ‘non‐overlapping’ additive Schwarz domain decomposition schemes) in CFD computations and those obtained with the new interface strip preconditioner (J. Comput. Meth. Sci. Engng 2003; Int. J. Numer. Meth. Engng 2005; 62(13):1873–1894) for the Schur complement method is carried out. The idea is to present the new solver as an alternative to obtain more accurate and faster solutions in the context of monolithic and non‐monolithic schemes applied to a internal/external viscous compressible/incompressible flows around bodies of complex shapes.Therefore, the target of this work is to show how the reliability of CFD codes is affected by the solver selection and why domain decomposition methods should be viewed not only as a more efficient strategy, but also to guarantee the solution quality. Copyright © 2006 John Wiley & Sons, Ltd.