We demonstrate that a small modification of the multiplicative, additive and restricted additive Schwarz preconditioner at the algebraic level, motivated by optimized Schwarz methods defined at the continuous level, leads to a significant reduction in the iteration count of the iterative solver. Numerical experiments using finite difference and spectral element discretizations of the positive definite Helmholtz problem and an idealized climate simulation illustrate the effectiveness of the new approach.
Optimized Schwarz preconditioning for SEM based magnetohydrodynamics
Institute for Mathematics Applied to Geosciences, National Center for Atmospheric Research（国家大气研究中心应用数学与地球科学研究所）；Department of Mathematics, Kyungpook National University（庆北大学数学系）
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 they are used both as iterative solvers and preconditioners for a Krylov method. To extend SRAS to nonlinear problems, we introduce SRASPEN (Substructured Restricted Additive Schwarz Preconditioned Exact Newton), where SRAS is used as a preconditioner f…
The computation time for reservoir simulation is dominated by the linear solver. The sets of linear equations which arise in reservoir simulation have two distinctive features: the problems are usually highly anisotropic, with a dominant vertical flow direction, and the commonly used fully implicit method requires a simultaneous solution for pressure and saturation or molar concentration variables. These variables behave quite differently, with the pressure feeling long-range effects while the saturations vary locally. In this paper we review preconditioned iterative methods used for solving t…
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell's equations
Université Côte d’Azur（蔚蓝海岸大学）；CNRS（法国国家科学研究中心）；LJAD（约瑟夫·拉格朗日实验室）；University of Strathclyde（斯特拉斯克莱德大学）；UPMC Univ Paris 6（巴黎第六大学皮埃尔和玛丽·居里大学）；LJLL（拉格朗日分析与偏微分方程实验室）
Nonlinear Preconditioning: How to use a Nonlinear Schwarz Method to Precondition Newton's Method
University of Strathclyde（斯特拉斯克莱德大学）；Université de Genève（日内瓦大学）；Université Nice Sophia Antipolis（尼斯索菲亚安蒂波利斯大学）；Inria（法国国家数字科学研究所）；Hong Kong Baptist University（香港浸会大学）
We propose domain decomposition preconditioners for the solution of an integral equation formulation of forward and inverse acoustic scattering problems with point scatterers. We study both forward and inverse problems and propose preconditioning techniques to accelerate the iterative solvers. For the forward scattering problem, we extend the domain decomposition based preconditioning techniques presented for partial differential equations in {\em "A restricted additive Schwarz preconditioner for general sparse linear systems", SIAM Journal on Scientific Computing, 21 (1999), pp. 792--797}, to…
We analyze the convergence of the (algebraic) multiplicative Schwarz method applied to linear algebraic systems with matrices having a special block structure that arises, for example, when a (partial) differential equation is posed and discretized on a domain that consists of two subdomains with an overlap. This is a basic situation in the context of domain decomposition methods. Our analysis is based on the algebraic structure of the Schwarz iteration matrices, and we derive error bounds that are based on the block diagonal dominance of the given system matrix. Our analysis does not assume t…
This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix-vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evide…
Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies
Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Euler Institute, Università della Svizzera italiana（瑞士意大利语区大学欧拉研究所）；UniDistance, Switzerland（瑞士远程大学）
We present a novel preconditioning technique for Krylov subspace algorithms to solve fluid-structure interaction (FSI) linearized systems arising from finite element discretizations. An outer Krylov subspace solver preconditioned with a geometric multigrid (GMG) algorithm is used, where for the multigrid level sub-solvers, a field-split (FS) preconditioner is proposed. The block structure of the FS preconditioner is derived using the physical variables as splitting strategy. To solve the subsystems originated by the FS preconditioning, an additive Schwarz (AS) block strategy is employed. The p…
Analysis of Block Jacobi/Gauss-Seidel and additive/multiplicative Schwarz preconditioning through the theory of GLT sequences, with applications to domain decomposition discretizations
University of Rome Tor Vergata（罗马托尔维加塔大学）；University Mohammed VI Polytechnic（穆罕默德六世polytechnic大学）；University of Insubria（因苏布里亚大学）；Uppsala University（乌普萨拉大学）
This work focuses on the development of a two-step field-split nonlinear preconditioner to accelerate the convergence of two-phase flow and transport in heterogeneous porous media. We propose a field-split algorithm named Field-Split Multiplicative Schwarz Newton (FSMSN), consisting in two steps: first, we apply a preconditioning step to update pressure and saturations nonlinearly by solving approximately two subproblems in a sequential fashion; then, we apply a global step relying on a Newton update obtained by linearizing the system at the preconditioned state. Using challenging test cases, …
Reduced basis methods build low-rank approximation spaces for the solution sets of parameterized PDEs by computing solutions of the given PDE for appropriately selected snapshot parameters. Localized reduced basis methods reduce the offline cost of computing these snapshot solutions by instead constructing a global space from spatially localized less expensive problems. In the case of online enrichment, these local problems are iteratively solved in regions of high residual and correspond to subdomain solves in domain decomposition methods. We show in this note that indeed there is a close rel…
Parallel algorithms and simulators with good scalabilities are particularly important for large-scale reservoir simulations on modern supercomputers with a large number of processors. In this paper, we introduce and study a family of highly scalable multilevel restricted additive Schwarz (RAS) methods for the fully implicit solution of subsurface flows with Peng-Robinson equation of state in two and three dimensions. With the use of a second-order fully implicit scheme, the proposed simulator is unconditionally stable with the relaxation of the time step size by the stability condition. The in…
The Poisson pressure solve resulting from the spectral element discretization of the incompressible Navier-Stokes equation requires fast, robust, and scalable preconditioning. In the current work, a parallel scaling study of Chebyshev-accelerated Schwarz and Jacobi preconditioning schemes is presented, with special focus on GPU architectures, such as OLCF's Summit. Convergence properties of the Chebyshev-accelerated schemes are compared with alternative methods, such as low-order preconditioners combined with algebraic multigrid. Performance and scalability results are presented for a variety …
Domain decomposition preconditioning for the high-frequency time-harmonic Maxwell equations with absorption
Sorbonne Université, Université Paris-Diderot SPC, CNRS, Inria, Laboratoire Jacques-Louis Lions（索邦大学，巴黎第七大学SPC，法国国家科学研究中心，法国国家信息与自动化研究所，雅克-路易·利翁斯实验室）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Université Côte d’Azur, CNRS, Laboratoire J-A Dieudonné（蔚蓝海岸大学，法国国家科学研究中心，J-A 迪厄多内实验室）；Department of Mathematical Sciences, University of Bath（巴斯大学数学科学系）
Discrete Edge Multiscale Space based Hybrid Schwarz Preconditioner for High-Frequency Helmholtz Problems
Eastern Institute for Advanced Study, Eastern Institute of Technology（宁波东方理工大学高级研究院）；School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）理工学院）；Department of Mathematics, The University of Hong Kong（香港大学数学系）；Department of Mathematics, Purdue University（普渡大学数学系）
Two-level convergence of Algebraic Multigrid with Overlapping Smoothers and Spectral Coarse Grids
Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Science and Technology Facilities Council, Scientific Computing Department, Rutherford Appleton Laboratory（英国科学与技术设施委员会，科学计算部，卢瑟福阿普尔顿实验室）
