KMS Of Academy of mathematics and systems sciences, CAS
A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations | |
Liu, Jun1; Wang, Xiang-Sheng2; Wu, Shu-Lin3; Zhou, Tao4 | |
2022-06-01 | |
Source Publication | ADVANCES IN COMPUTATIONAL MATHEMATICS |
ISSN | 1019-7168 |
Volume | 48Issue:3Pages:29 |
Abstract | 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. |
Keyword | Direct PinT algorithms Diagonalization technique Condition number Wave-type equations |
DOI | 10.1007/s10444-022-09928-4 |
Indexed By | SCI |
Language | 英语 |
Funding Project | National Natural Science Foundation of China (NSFC)[12171080] ; Natural Science Foundation of Jilin Province[JC010284408] |
WOS Research Area | Mathematics |
WOS Subject | Mathematics, Applied |
WOS ID | WOS:000779137200001 |
Publisher | SPRINGER |
Citation statistics | |
Document Type | 期刊论文 |
Identifier | http://ir.amss.ac.cn/handle/2S8OKBNM/60216 |
Collection | 中国科学院数学与系统科学研究院 |
Corresponding Author | Wu, Shu-Lin |
Affiliation | 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 |
Recommended Citation 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. |
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