PinT Preconditioner for Forward-Backward Evolutionary Equations

IF 1.5 2区 数学 Q2 MATHEMATICS, APPLIED
Shu-Lin Wu, Zhiyong Wang, Tao Zhou
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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 44, Issue 4, Page 1771-1798, December 2023.
Abstract. Solving the linear system [math] is often the major computational burden when a forward-backward evolutionary equation must be solved in a problem, where [math] is the so-called all-at-once matrix of the forward subproblem after space-time discretization. An efficient solver requires a good preconditioner for [math]. Inspired by the structure of [math], we precondition [math] by [math] with [math] being a block [math]-circulant matrix constructed by replacing the Toeplitz matrices in [math] by the [math]-circulant matrices. By a block Fourier diagonalization of [math], the computation of the preconditioning step [math] is parallelizable for all the time steps. We give a spectral analysis for the preconditioned matrix [math] and prove that for any one-step stable time-integrator the eigenvalues of [math] spread in a mesh-independent interval [math] if the parameter [math] weakly scales in terms of the number of time steps [math] as [math], where [math] is a free constant. Two applications of the proposed preconditioner are illustrated: PDE-constrained optimal control problems and parabolic source identification problems. Numerical results for both problems indicate that spectral analysis predicts the convergence rate of the preconditioned conjugate gradient method very well.
正反向进化方程的PinT预条件
矩阵分析与应用,第44卷,第4期,第1771-1798页,2023年12月。摘要。求解线性系统[数学]往往是主要的计算负担,当必须在一个问题中求解一个向前向后的进化方程时,其中[数学]是所谓的时空离散化后的前向子问题的一次性矩阵。一个有效的解算器需要一个好的[数学]前提条件。受[math]结构的启发,我们将[math]作为[math]的先决条件,其中[math]是一个块[math]-循环矩阵,通过将[math]中的Toeplitz矩阵替换为[math]-循环矩阵来构建。通过[math]的块傅立叶对角化,预处理步骤[math]的计算可以对所有时间步骤并行化。我们给出了预条件矩阵[math]的谱分析,并证明了对于任何一步稳定时间积分器[math],如果参数[math]在[math]的时间步数[math]方面弱缩放[math],则[math]的特征值在网格无关区间[math]中传播,其中[math]是一个自由常数。给出了该预调节器的两种应用:pde约束最优控制问题和抛物型源识别问题。数值结果表明,谱分析能很好地预测预条件共轭梯度法的收敛速度。
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来源期刊
CiteScore
2.90
自引率
6.70%
发文量
61
审稿时长
6-12 weeks
期刊介绍: The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.
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