A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations

doi:10.1007/s10444-022-09928-4

	A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations
	Liu, Jun 4; Wang, Xiang-Sheng 3; Wu, Shu-Lin 2; Zhou, Tao 1
刊名	ADVANCES IN COMPUTATIONAL MATHEMATICS
	2022-06-01
卷号	48 期号:3 页码:29
关键词	Direct PinT algorithms Diagonalization technique Condition number Wave-type equations
ISSN号	1019-7168
DOI	10.1007/s10444-022-09928-4
英文摘要	In this paper, we study a direct parallel-in-time (PinT) algorithm for first- and second-order time-dependent differential equations. We use a second-order boundary value method as the time integrator. Instead of solving the corresponding all-at-once system iteratively, we diagonalize the time discretization matrix B, which yields a direct parallel implementation across all time steps. A crucial issue of this methodology is how the condition number (denoted by Cond(2)(V)) of the eigenvector matrix V of B behaves as n grows, where n is the number of time steps. A large condition number leads to large roundoff error in the diagonalization procedure, which could seriously pollute the numerical accuracy. Based on a novel connection between the characteristic equation and the Chebyshev polynomials, we present explicit formulas for V and V-1, by which we prove that Cond(2)(V) = O (n(2)). This implies that the diagonalization process is well-conditioned and the roundoff error only increases moderately as n grows, and thus, compared to other direct PinT algorithms, a much larger n can be used to yield satisfactory parallelism. A fast structure-exploiting algorithm is also designed for computing the spectral diagonalization of B. Numerical results on parallel machine are given to support our findings, where over 60 times speedup is achieved with 256 cores.
资助项目	National Natural Science Foundation of China (NSFC)[12171080] ; Natural Science Foundation of Jilin Province[JC010284408]
WOS研究方向	Mathematics
语种	英语
出版者	SPRINGER
WOS记录号	WOS:000779137200001
内容类型	期刊论文
源URL	[http://ir.amss.ac.cn/handle/2S8OKBNM/60215]
专题	中国科学院数学与系统科学研究院
通讯作者	Wu, Shu-Lin
作者单位	1.Chinese Acad Sci, LSEC, Inst Computat Math & Sci Engn Comp, AMSS, Beijing 100190, Peoples R China 2.Northeast Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China 3.Univ Louisiana Lafayette, Dept Math, Lafayette, LA 70503 USA 4.Southern Illinois Univ Edwardsville, Dept Math & Stat, Edwardsville, IL 62026 USA
推荐引用方式 GB/T 7714	Liu, Jun,Wang, Xiang-Sheng,Wu, Shu-Lin,et al. A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations[J]. ADVANCES IN COMPUTATIONAL MATHEMATICS,2022,48(3):29.
APA	Liu, Jun,Wang, Xiang-Sheng,Wu, Shu-Lin,&Zhou, Tao.(2022).A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations.ADVANCES IN COMPUTATIONAL MATHEMATICS,48(3),29.
MLA	Liu, Jun,et al."A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations".ADVANCES IN COMPUTATIONAL MATHEMATICS 48.3(2022):29.