On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems
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On the regularization matrix of the regularized DPSS preconditioner for non-Hermitian saddle-point problems Ju-Li Zhang1 Received: 15 January 2020 / Revised: 24 May 2020 / Accepted: 11 June 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract In this work, with respect to the regularization matrix of the new regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for non-Hermitian saddle-point problems, we provide a class of Hermitian positive semidefinite matrices, which depend on certain parameters, for practical computations. A precise description about the eigenvalue distribution of the corresponding preconditioned matrix is given. The condition number of the eigenvector matrix, which partly determines the convergence rate of the related preconditioned Krylov subspace method, is also discussed in this work. Finally, some numerical experiments are carried out to identify the effectiveness of the presented special choices for the regularization matrix to solve the non-Hermitian saddle-point problems. Keywords Non-Hermitian saddle-point problems · New RDPSS preconditioner · Regularization matrix · Matrix similar transformation · Spectral properties Mathematics Subject Classification 65M10 · 75D07
1 Introduction We consider the solution of the following saddle-point linear system A B∗ x f Au ≡ = ≡ b, −B O y g
(1.1)
where A ∈ Cn×n is non-Hermitian positive definite, B ∈ Cm×n (m ≤ n) is a rectangular matrix of full row rank, B ∗ denotes the conjugate transpose of B, f , x ∈ Cn , g, y ∈ Cm and
Communicated by Jinyun Yuan. The work is partially supported by National Natural Science Foundation of China (No. 11801362) and (No. 11601323).
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Ju-Li Zhang [email protected] School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, People’s Republic of China 0123456789().: V,-vol
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f , g are two given vectors. O is a zero matrix with suitable dimension. Then the coefficient matrix A of the system (1.1) is nonsingular, which guarantees the uniqueness and existence of the solution of the problem (1.1). Systems like (1.1) are important, and many engineering problems can be modeled as this form under some conditions, for example, computational fluid dynamics problem, constrained optimization problem, and so on. Readers can consult (Bai 2006; Benzi et al. 2005) and the references therein for a detailed discussion. In the past decades, a great deal of research has been done to develop effective methods for solving problem (1.1). When the saddle-point matrix A is large and sparse, iterative methods are preferred and have attracted many authors’ attention, for its low computational overhead per iteration. However, some classical iterative methods, such as Krylov subspace methods, often suffer from slow convergence. Then, in that case, we need a proper preconditioner to speed up the convergence rates. According to the special structure of the saddle-point matrix
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