ANALYSIS OF MEAN-SQUARE STABILITY OF THE PARAREAL ALGORITHM
Wu Shulin1, Wang Zhiyong2, Huang Chengming3
1. School of Science, Sichuan University of Science and Engineering, Zigong 643000, Sichuan, China;
2. School of Applied Mathematics, University of Electronic Science and Technology of China, Chengdu 610054, China;
3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Parareal algorithm is a very efficient parallel in time computation methods. Compared with traditional parallel methods, this algorithm has the advantages of faster convergence, higher parallel performance and easy coding. This algorithm was first proposed by Lions, Maday and Turinici in 2001 and has attracted many researchers over the past few years. Recently, the application and theoretical analysis of this algorithm for stochastic computation have been investigated by some researchers. In this paper, we analyze the Mean-square stability of the Parareal algorithm in stochastic computation. The sufficient conditions under which the Parareal algorithm is stable are obtained and it is shown that: a) the algorithm converges superlinearly on any bounded time interval and b) the convergence speed is only linear on unbounded time intervals. Finally, numerical results are given to validate our theoretical conclusions.
. ANALYSIS OF MEAN-SQUARE STABILITY OF THE PARAREAL ALGORITHM[J]. Mathematica Numerica Sinica, 2011, 33(2): 113-124.
Bal G. On the convergence and the stability of the parareal algorithm to solve partial differential equations [J]. In Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng., 2003, 40: 426-432.
Bal G and Maday Y. A “parareal” time discretization for non-linear pde’s with application to the pricing of an American put. In Recent Developments in Domain Decomposition Methods[J]. Lect. Notes Comput. Sci. Eng., 2002, 23: 189-202.
Bal G. Parallelization in time of (stochastic) ordinary differential equations[J]. Submitted (a PDF file is available at http://www.columbia.edu/ gb2030).
Cortial J and Farhat C. A time-parallel implicit method for accelerating the solution of non-linear structural dynamics problems[J]. Internat. J. Numer. Methods Engrg., 2008, 77: 451-470.
Engblom S. Parallel in time simulation of multiscale stochastic chemical kinetics[J]. 2008, to appear in the IT technical report series (a PDF file is available at http://user.it.uu.se/stefane/ssa parareal.pdf.
Engblom S. Time-parallel Simulation of Stochastic Chemical Kinetics[J]. Numerical Analysis and Applied Mathematics: International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, 2008, 1048: 174-177.
Farhat C and Chandesris M. Time-decomposed parallel time-integrators: Theory and feasibility studies for fluid, structure, and fluid-structure applications[J]. Internat. J. Numer. Methods Engrg., 2003, 58: 1397-1434.
Farhat C, Cortial J, Dastillung C. and Bavestrello H. Time-parallel implicit integrators for the near-real-time prediction of linear structural dynamic responses[J]. Internat. J. Numer. Methods Engrg., 2006, 67: 697-724.
Gander M J and Petcu M. Analysis of a Krylov Subspace Enhanced Parareal Algorithm[J]. ESAIM Proc., 2008, in press.
Gander M J, Vandewalle S. Analysis of the parareal time-parallel time-integration method[J]. SIAM J. Sci. Comput., 2007, 29: 556-578.
Higham D J. Mean-square and asymptotic stability of the stochastic theta method[J]. SIAM J. Numer. Anal., 2001, 38: 753-769.
Koskodan R and Allen E. Extrapolation of the Stochastic Theta Numerical Method for Stochastic Differential Equations[J]. Stochastic Analysis and Applications, 2006, 24: 475-487.
Lions J L, Maday Y and Turinici G. A “parareal” in time discretization of PDE’s[J]. C. R. Acad. Sci. Paris S′er. I Math., 2001, 332: 661-668.
SubberWand Sarkar A. Performance of A Parallel Time Integrator For Noisy Nonlinear System[J]. Journal of Probabilistic Engineering Mechanics, 2008, to appear.
Staff G A and R?nquist E M. Stability of the parareal algorithm[J]. In Proceedings of the 15th International Domain Decomposition Conference, Lect. Notes Comput. Sci. Eng., 2003, 40: 449-56.
Wu S L, Shi B C and Huang C M. Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs[J]. Communication in Computational Physics, 2009, 6: 883-902.
Wu S L, Shi B C and Huang C M. Convergence and Stability analysis of the Parareal-Richardson Algorithm[J]. Manuscript.