Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

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• 作者：Owe Axelsson ; Shiraz Farouq ; Maya Neytcheva
• 关键词：PDE ; constrained optimization problems ; Finite elements ; Iterative solution methods ; Preconditioning
• 刊名：Numerical Algorithms
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：74
• 期：1
• 页码：19-37
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Numeric Computing; Algorithms; Algebra; Theory of Computation; Numerical Analysis;
• 出版者：Springer US
• ISSN：1572-9265
• 卷排序：74

文摘

The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.