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On the Use of Reduced Basis Methods to Accelerate and Stabilize the Parareal Method

  • Feng Chen
  • Jan S. Hesthaven
  • Xueyu Zhu
Part of the MS&A - Modeling, Simulation and Applications book series (MS&A, volume 9)

Abstract

We propose a modified parallel-in-time — parareal — multi-level time integration method that, in contrast to previously proposed methods, employs a coarse solver based on a reduced model, built from the information obtained from the fine solver at each iteration. This approach is demonstrated to offer two substantial advantages: it accelerates convergence of the original parareal method for similar problems and the reduced basis stabilizes the parareal method for purely advective problems where instabilities are known to arise. When combined with empirical interpolation methods (EIM), we develop this approach to solve both linear and nonlinear problems and highlight the minimal changes required to utilize this algorithm to accelerate existing implementations. We illustrate the advantages through algorithmic design, through analysis of stability, convergence, and computational complexity, and through several numerical examples.

Keywords

Singular Value Decomposition Proper Orthogonal Decomposition Parareal Method Coarse Solver Fine Solver 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Brown UniversityProvidenceUSA
  2. 2.EPFL-SB-MATHICSEÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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