分布椭圆型最优控制问题的鲁棒有限元离散化及求解方法

IF 1 4区 数学 Q3 MATHEMATICS, APPLIED Computational Methods in Applied Mathematics Pub Date : 2023-02-18 DOI:10.1515/cmam-2022-0138
Ulrich Langer, Richard Löscher, Olaf Steinbach, Huidong Yang
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引用次数: 1

摘要

摘要考虑具有l2 L^{2}正则化的标准跟踪型分布椭圆型最优控制问题及其有限元离散化问题。我们正在研究状态u ϱ u_{\varrho}与期望状态(目标)u¯\overline{u}在正则化参数𝜚和网格尺寸方面的有限元近似u ϱ¹h u_{\varrho h}之间的l2 L^{2}误差,从而得出最优选择ϱ =h 4 \varrho=h^{4}。事实证明,对于正则化参数的选择,我们可以设计简单的类雅可比预条件MINRES或Bramble-Pasciak CG方法,这些方法允许我们在算术操作和内存需求方面以渐进最优的复杂度解决减少的离散最优性系统。通过几个具有不同规律目标(包括不连续目标)的基准问题验证了理论结果。
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Robust Finite Element Discretization and Solvers for Distributed Elliptic Optimal Control Problems
Abstract We consider standard tracking-type, distributed elliptic optimal control problems with L 2 L^{2} regularization, and their finite element discretization. We are investigating the L 2 L^{2} error between the finite element approximation u ϱ h u_{\varrho h} of the state u ϱ u_{\varrho} and the desired state (target) u ¯ \overline{u} in terms of the regularization parameter 𝜚 and the mesh size ℎ that leads to the optimal choice ϱ = h 4 \varrho=h^{4} . It turns out that, for this choice of the regularization parameter, we can devise simple Jacobi-like preconditioned MINRES or Bramble–Pasciak CG methods that allow us to solve the reduced discrete optimality system in asymptotically optimal complexity with respect to the arithmetical operations and memory demand. The theoretical results are confirmed by several benchmark problems with targets of various regularities including discontinuous targets.
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来源期刊
CiteScore
2.40
自引率
7.70%
发文量
54
期刊介绍: The highly selective international mathematical journal Computational Methods in Applied Mathematics (CMAM) considers original mathematical contributions to computational methods and numerical analysis with applications mainly related to PDEs. CMAM seeks to be interdisciplinary while retaining the common thread of numerical analysis, it is intended to be readily readable and meant for a wide circle of researchers in applied mathematics. The journal is published by De Gruyter on behalf of the Institute of Mathematics of the National Academy of Science of Belarus.
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Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability A Space-Time Finite Element Method for the Eddy Current Approximation of Rotating Electric Machines An Inverse Matrix Eigenvalue Problem for Constructing a Vibrating Rod On Error Estimates of a discontinuous Galerkin Method of the Boussinesq System of Equations Computational Methods in Applied Mathematics (CMAM 2022 Conference, Part 2)
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