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Greedy Kernel Methods for Accelerating Implicit Integrators for Parametric ODEs

  • Tim Brünnette
  • Gabriele SantinEmail author
  • Bernard Haasdonk
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

We present a novel acceleration method for the solution of parametric ODEs by single-step implicit solvers by means of greedy kernel-based surrogate models. In an offline phase, a set of trajectories is precomputed with a high-accuracy ODE solver for a selected set of parameter samples, and used to train a kernel model which predicts the next point in the trajectory as a function of the last one. This model is cheap to evaluate, and it is used in an online phase for new parameter samples to provide a good initialization point for the nonlinear solver of the implicit integrator. The accuracy of the surrogate reflects into a reduction of the number of iterations until convergence of the solver, thus providing an overall speedup of the full simulation. Interestingly, in addition to providing an acceleration, the accuracy of the solution is maintained, since the ODE solver is still used to guarantee the required precision. Although the method can be applied to a large variety of solvers and different ODEs, we will present in details its use with the Implicit Euler method for the solution of the Burgers equation, which results to be a meaningful test case to demonstrate the method’s features.

Notes

Acknowledgements

The authors would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/2) at the University of Stuttgart.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Tim Brünnette
    • 1
  • Gabriele Santin
    • 1
    Email author
  • Bernard Haasdonk
    • 1
  1. 1.University of Stuttgart, Institute of Applied Analysis and Numerical SimulationStuttgartGermany

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