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

doi:10.1007/s10444-022-09928-4

CSpace

	A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations
	Liu, Jun 1; Wang, Xiang-Sheng 2; Wu, Shu-Lin 3; Zhou, Tao 4
	2022-06-01
发表期刊	ADVANCES IN COMPUTATIONAL MATHEMATICS
ISSN	1019-7168
卷号	48 期号:3 页码:29
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.
关键词	Direct PinT algorithms Diagonalization technique Condition number Wave-type equations
DOI	10.1007/s10444-022-09928-4
收录类别	SCI
语种	英语
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
WOS类目	Mathematics, Applied
WOS记录号	WOS:000779137200001
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/60216
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.Southern Illinois Univ Edwardsville, Dept Math & Stat, Edwardsville, IL 62026 USA 2.Univ Louisiana Lafayette, Dept Math, Lafayette, LA 70503 USA 3.Northeast Normal Univ, Sch Math & Stat, Changchun 130024, Peoples R China 4.Chinese Acad Sci, LSEC, Inst Computat Math & Sci Engn Comp, AMSS, Beijing 100190, Peoples R China
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.