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An adaptive parareal algorithm.

Y Maday1,2, O Mula3,4

  • 1Sorbonne Université, CNRS, Université de Paris, Laboratoire Jacques-Louis Lions (LJLL), F-75005 Paris, France.

Journal of Computational and Applied Mathematics
|April 16, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces an adaptive parareal in time method to accelerate numerical simulations. By dynamically adjusting solver accuracy, it significantly improves parallel efficiency for time-dependent problems.

Keywords:
Convergence ratesDomain decompositionInexact fine solverParallel efficiencyParareal in time algorithma posteriori estimators

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Area of Science:

  • Numerical Analysis
  • Computational Science
  • Scientific Computing

Background:

  • Time domain decomposition methods accelerate numerical simulations of time-dependent problems.
  • Existing algorithms, like the parareal in time method, face efficiency limitations, hindering large-scale applications.
  • The fixed high accuracy of the fine solver in the classical parareal method impedes parallel efficiency.

Purpose of the Study:

  • To improve the parallel efficiency of the parareal in time method for time-dependent problems.
  • To develop an adaptive variant of the parareal method that dynamically adjusts fine solver accuracy.
  • To provide a theoretical framework for understanding and extending parareal-based acceleration techniques.

Main Methods:

  • Developed an adaptive variant of the parareal in time method.
  • Dynamically increased the accuracy of the fine solver across parareal iterations.
  • Theoretically analyzed the parallel efficiency and scalability of the adaptive method.
  • Illustrated performance using stiff ordinary differential equations (ODEs).

Main Results:

  • The adaptive parareal method achieves highly competitive parallel efficiency, especially when the coarse solver cost is minimal.
  • Theoretical analysis confirms that coarse solver cost and communication time are the primary remaining scalability bottlenecks.
  • The adaptive approach provides a general framework for understanding and developing improved parareal extensions.

Conclusions:

  • The adaptive parareal in time method effectively overcomes the efficiency limitations of the classical approach.
  • This advancement significantly enhances the potential for solving large-scale, high-dimensional time-dependent problems.
  • The findings pave the way for more scalable and efficient numerical simulations in computational science.