Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems

doi:10.1051/cocv/2020012

CSpace

	Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems
	Wu, Shu-Lin 1; Zhou, Tao 2,3
	2020-11-13
发表期刊	ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
ISSN	1292-8119
卷号	26 页码:26
摘要	Solving parabolic PDE-constrained optimization problems requires to take into account the discrete time points all-at-once, which means that the computation procedure is often time-consuming. It is thus desirable to design robust and analyzable parallel-in-time (PinT) algorithms to handle this kind of coupled PDE systems with opposite evolution directions. To this end, for two representative model problems which are, respectively, the time-periodic PDEs and the initial-value PDEs, we propose a diagonalization-based approach that can reduce dramatically the computational time. The main idea lies in carefully handling the associated time discretization matrices that are denoted by B-per and B-ini for the two problems. For the first problem, we diagonalize B-per directly and this results in a direct PinT algorithm (i.e., non-iterative). For the second problem, the main idea is to design a suitable approximation B-per B per of B-ini, which naturally results in a preconditioner of the discrete KKT system. This preconditioner can be used in a PinT pattern, and for both the Backward-Euler method and the trapezoidal rule the clustering of the eigenvalues and singular values of the preconditioned matrix is justified. Compared to existing preconditioners that are designed by approximating the Schur complement of the discrete KKT system, we show that the new preconditioner leads to much faster convergence for certain Krylov subspace solvers, e.g., the GMRES and BiCGStab methods. Numerical results are presented to illustrate the advantages of the proposed PinT algorithm.
关键词	Parabolic PDE-constrained optimization PinT algorithm diagonalization technique preconditioner GMRES BiCGStab
DOI	10.1051/cocv/2020012
收录类别	SCI
语种	英语
资助项目	NSF of China[11822111] ; NSF of China[11688101] ; NSF of China[11731006] ; NSF of China[11771313] ; NSF of Sichuan Province[2018JY0469] ; Science Challenge Project[TZ2018001] ; Science Challenge Project[TZ2016002] ; National Key Basic Research Program[2018YFB0704304] ; Youth Innovation Promotion Association (CAS)
WOS研究方向	Automation & Control Systems ; Mathematics
WOS类目	Automation & Control Systems ; Mathematics, Applied
WOS记录号	WOS:000594831500004
出版者	EDP SCIENCES S A
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/57784
专题	中国科学院数学与系统科学研究院
通讯作者	Zhou, Tao
作者单位	1.Northeast Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China 2.Chinese Acad Sci, NCMIS, Inst Computat Math & Sci Engn Comp, Acad Math & Syst Sci, Beijing 100190, Peoples R China 3.Chinese Acad Sci, LSEC, Inst Computat Math & Sci Engn Comp, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Wu, Shu-Lin,Zhou, Tao. Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems[J]. ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS,2020,26:26.
APA	Wu, Shu-Lin,&Zhou, Tao.(2020).Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems.ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS,26,26.
MLA	Wu, Shu-Lin,et al."Diagonalization-based parallel-in-time algorithms for parabolic PDE-constrained optimization problems".ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS 26(2020):26.