A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations
A well-conditioned direct PinT algorithm for first- and second-order evolutionary equations
复制标题
用于一阶和二阶演化方程的条件良好的直接 PinT 算法
DOI：
10.1007/s10444-022-09928-4
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发表时间：
2021-08
影响因子：
1.7
通讯作者：
Tao Zhou
中科院分区：
文献类型：
--
作者：
Jun Liu;Xiang-Sheng Wang;Shu-Lin Wu;Tao Zhou
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 Cond2(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 Cond2(V)=O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$_{2}(V)=\mathcal {O}(n^{2})$\end{document}. 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.
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影响因子：
2.8
作者：
L. Brugnano;F. Mazzia;D. Trigiante
通讯作者：
L. Brugnano;F. Mazzia;D. Trigiante
影响因子：
0.9
作者：
L. Fox
通讯作者：
L. Fox
影响因子：
2.5
作者：
L. Fosdick;E. Jessup;C. C. Schauble-C.;G. Domik
通讯作者：
L. Fosdick;E. Jessup;C. C. Schauble-C.;G. Domik
DOI：
--
发表时间：
1998
期刊：
--
影响因子：
--
作者：
L. Brugnano;D. Trigiante
通讯作者：
L. Brugnano;D. Trigiante
DOI：
10.1006/jcom.1997.0442
发表时间：
1997-06
期刊：
J. Complex.
影响因子：
--
作者：
I. Gohberg;V. Olshevsky
通讯作者：
I. Gohberg;V. Olshevsky