Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.
We present robust and highly parallel multilevel non-overlapping Schwarz preconditioners, to solve an interior penalty discontinuous Galerkin finite element discretization of a system of steady state, singularly perturbed reaction-diffusion equations with a singular reaction operator, using a GMRES solver. We provide proofs of convergence for the two-level setting and the multigrid V-cycle as well as numerical results for a wide range of regimes.
Overlapping Schwarz Preconditioners for Pose-Graph SLAM in Robotics
Faculty of Mathematics and Computer Science, Technische Universität Bergakademie Freiberg（弗莱贝格矿业工业大学数学与计算机科学学院）；Universitätsrechenzentrum (URZ), Technische Universität Bergakademie Freiberg（弗莱贝格矿业大学计算中心）
Additive Schwarz preconditioner for the general finite volume element discretization of symmetric elliptic problems
University of Warsaw（华沙大学）；Bergen University College（卑尔根大学学院）；University of Bergen（卑尔根大学）；University of South Bohemia（南波希米亚大学）；Institute of Information Theory and Automation of the ASCR（捷克科学院信息论与自动化研究所）
Monolithic and Block Overlapping Schwarz Preconditioners for the Incompressible Navier-Stokes Equations
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特理工大学应用数学研究所）；Department of Mathematics and Computer Science, University of Cologne（科隆大学数学与计算机科学系）；Center for Data and Simulation Science, University of Cologne（科隆大学数据与模拟科学中心）
A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods
University of Geneva（日内瓦大学）；Zhejiang Ocean University（浙江海洋大学）
The discretization of surface intrinsic elliptic partial differential equations (PDEs) poses interesting challenges not seen in flat space. The discretization of these PDEs typically proceeds by either parametrizing the surface, triangulating the surface, or embedding the surface in a higher dimensional flat space. The closest point method (CPM) is an embedding method that represents surfaces using a function that maps points in the embedding space to their closest points on the surface. In the CPM, this mapping also serves as an extension operator that brings surface intrinsic data onto the e…
The multigroup neutron transport criticality calculations using modern supercomputers have been widely employed in a nuclear reactor analysis for studying whether or not a system is self-sustaining. However, the design and development of efficient parallel algorithms for the transport criticality calculations is challenging especially when the number of processor cores is large and an unstructured mesh is adopted. In particular, both the compute time and memory usage have to be carefully taken into consideration due to the dimensionality of the neutron transport equations. In this paper, we st…
Block Schwarz methods and preconditioning strategies using Generalized locally Toeplitz tools - part I: analysis of the preconditioners and numerical validation
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…
We introduce a domain decomposition-based nonlinear preconditioned iteration for solving nonlinear, nonsmooth elliptic optimal control problems, with a nonlinear reaction term, $L^1$ regularization and box constraints on the control function. The method is obtained by applying semismooth Newton to the fixed-point equation of the parallel optimized Schwarz iteration. As a proof of concept, numerical experiments are performed on two subdomains, as well as on a multi-subdomain test case. The results show that it is possible to obtain substantial improvements in robustness and efficiency with the …
Two-level hybrid Schwarz preconditioners with preasymptotic piecewise-polynomial coarse spaces for the high-frequency Helmholtz equation
University College London（伦敦大学学院）；University of Bath（巴斯大学）；Sorbonne Université, Université Paris Cité, CNRS, INRIA, Laboratoire Jacques-Louis Lions（索邦大学，巴黎西岱大学，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）
A nonoverlapping spectral additive Schwarz method for interior penalty discontinuous Galerkin discretizations of Anisotropic Elliptic Problems
Massachusetts Institute of Technology（麻省理工学院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；State University of New York Polytechnic Institute（纽约州立理工大学）；Worcester Polytechnic Institute（伍斯特理工学院）
We consider using the preconditioned-Krylov subspace method to solve the system of linear equations with a three-by-three block structure. By making use of the three-by-three block structure, eight inexact block factorization preconditioners, which can be put into a same theoretical analysis frame, are proposed based on a kind of inexact factorization. By generalizing Bendixson Theorem and developing a unified technique of spectral equivalence, the bounds of the real and imaginary parts of eigenvalues of the preconditioned matrices are obtained. The comparison to eleven existed exact and inexa…
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）
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）
Domain decomposition (DD) methods are widely used as preconditioner techniques. Their effectiveness relies on the choice of a locally constructed coarse space. Thus far, this construction was mostly achieved using non-assembled matrices from discretized partial differential equations (PDEs). Therefore, DD methods were mainly successful when solving systems stemming from PDEs. In this paper, we present a fully algebraic multilevel DD method where the coarse space can be constructed locally and efficiently without any information besides the coefficient matrix. The condition number of the precon…
Randomized methods are becoming increasingly popular in numerical linear algebra. However, few attempts have been made to use them in developing preconditioners. Our interest lies in solving large-scale sparse symmetric positive definite linear systems of equations where the system matrix is preordered to doubly bordered block diagonal form (for example, using a nested dissection ordering). We investigate the use of randomized methods to construct high quality preconditioners. In particular, we propose a new and efficient approach that employs Nyström's method for computing low rank approximat…
In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order ($p=1$) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, …
We propose an unconditionally robust and highly effective preconditioner for general symmetric positive definite (SPD) matrices based on structured incomplete factorization (SIF), called enhanced SIF (eSIF) preconditioner. The original SIF strategy proposed recently derives a structured preconditioner by applying block diagonal preprocessing to the matrix and then compressing appropriate scaled off-diagonal blocks. Here, we use an enhanced scaling-and-compression strategy to design the new eSIF preconditioner. Some subtle modifications are made, such as the use of two-sided block triangular pr…
An algebraic multiscale framework for two-level overlapping Schwarz preconditioners
Delft Institute of Applied Mathematics, Delft University of Technology（代尔夫特应用数学研究所，代尔夫特理工大学）；Department of Geoscience and Engineering, Delft University of Technology（地球科学与工程学院，代尔夫特理工大学）
