我们分析了单层重叠区域分解预条件子SORAS（对称化优化限制加性Schwarz）应用于一般线性系统的收敛性，该系统的矩阵不一定是对称/自伴的，也不一定是正定的。通过推广[I.G. Graham, E.A. Spence, 和 J. Zou, SIAM J.Numer.Anal., 2020]中为Helmholtz方程发展的理论，我们确定了一组假设和估计，这些假设和估计足以获得预条件矩阵范数的上界，以及其值域到原点距离的下界。我们强调，我们的理论在以下意义上是通用的：它不特定于某一个边值问题。此外，它不依赖于单元足够小的粗网格。作为该框架的一个说明，我们为具有Robin型传输条件的非均匀反应-对流-扩散方程的重叠区域分解方法证明了新的估计（为了证明该方程的稳定性假设，我们考虑了强制双线性形式的情形，尽管它是非对称的）。
英文摘要：
We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in [I.G. Graham, E.A. Spence, and J. Zou, SIAM J.Numer.Anal., 2020], we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction-convection-diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).
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…
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 …
A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；Department of Mathematics, The Chinese University of Hong Kong（香港中文大学数学系）
A two-level domain-decomposition preconditioner for the time-harmonic Maxwell's equations
Université Côte d’Azur, CNRS, LJAD（蔚蓝海岸大学，法国国家科学研究中心，拉格朗日分析几何实验室）；University of Strathclyde（思克莱德大学）；University of Bath（巴斯大学）；UPMC Univ Paris 06, LJLL（巴黎第六大学皮埃尔和玛丽·居里大学，拉格朗日分析实验室）
Solving the indefinite Helmholtz equation is not only crucial for the understanding of many physical phenomena but also represents an outstandingly-difficult benchmark problem for the successful application of numerical methods. Here we introduce a new approach for evolving efficient preconditioned iterative solvers for Helmholtz problems with multi-objective grammar-guided genetic programming. Our approach is based on a novel context-free grammar, which enables the construction of multigrid preconditioners that employ a tailored sequence of operations on each discretization level. To find sol…
Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized wave equation are sought, e.g. by simulating for a long time with a time-harmonic forcing term. Disadvantages of the time-domain method are that the solutions are affected by temporal discretization errors and that the spatial discretization cannot be freely chosen, since it is inherited from the time-domain scheme. In this work we address these issues. Given a…
Spectral coarse spaces based on indefinite operators: the $H_k$-GenEO method
Inria（法国国家信息与自动化研究所）；Univ. Lille（里尔大学）；Laboratoire Paul Painlevé（保罗·潘勒韦实验室）；Eindhoven University of Technology（埃因霍温理工大学）；University of Strathclyde（斯特拉斯克莱德大学）；University of Bath（巴斯大学）
In this paper, we consider low-rank approximations for the solutions to the stochastic Helmholtz equation with random coefficients. A Stochastic Galerkin finite element method is used for the discretization of the Helmholtz problem. Existence theory for the low-rank approximation is established when the system matrix is indefinite. The low-rank algorithm does not require the construction of a large system matrix which results in an advantage in terms of CPU time and storage. Numerical results show that, when the operations in a low-rank method are performed efficiently, it is possible to obtai…
McMg: A Learned Phase-Space Multi-channel Multigrid Preconditioner for Helmholtz Equation
School of Mathematics, Jilin University（吉林大学数学学院）；Ocean University of China（中国海洋大学）；Shenzhen Loop Area Institute（深圳河套学院）；King Abdullah University of Science and Technology（卡布斯科学技术大学）
Incomplete factorization is a widely used preconditioning technique for Krylov subspace methods for solving large-scale sparse linear systems. Its multilevel variants, such as ILUPACK, are more robust for many symmetric or unsymmetric linear systems than the traditional, single-level incomplete LU (or ILU) techniques. However, the previous multilevel ILU techniques still lacked robustness and efficiency for some large-scale saddle-point problems, which often arise from systems of partial differential equations (PDEs). We introduce HILUCSI, or Hierarchical Incomplete LU-Crout with Scalability-o…
