Greedy kernel methods for accelerating implicit integrators for parametric ODEs
T. Brünnette, G. Santin, and B. Haasdonk. Numerical Mathematics and Advanced Applications - ENUMATH 2017, volume Proceedings of ENUMATH 2017 of Lecture Notes in Computational Science and Engineering, Springer, (2018)
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.
%0 Conference Paper
%1 brunnette2018greedy
%A Brünnette, Tim
%A Santin, Gabriele
%A Haasdonk, Bernard
%B Numerical Mathematics and Advanced Applications - ENUMATH 2017
%D 2018
%E Radu, Florin A.
%E Kumar, Kundan
%E Berre, Inga
%E Nordbotten, Jan M.
%E Pop, Iuliu Sorin
%I Springer
%K ians imported
%N 126
%T Greedy kernel methods for accelerating implicit integrators for parametric ODEs
%V Proceedings of ENUMATH 2017
%X 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.
%@ 978-3-319-96414-0 and 978-3-319-96415-7
@inproceedings{brunnette2018greedy,
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.},
added-at = {2019-06-17T14:25:24.000+0200},
author = {Brünnette, Tim and Santin, Gabriele and Haasdonk, Bernard},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2071a5ea0ae8da316808f8076a872f2d0/britsteiner},
booktitle = {Numerical Mathematics and Advanced Applications - ENUMATH 2017},
editor = {Radu, Florin A. and Kumar, Kundan and Berre, Inga and Nordbotten, Jan M. and Pop, Iuliu Sorin},
eventdate = {2017-09-25/2017-09-29},
eventtitle = {12. European Conference on Numerical Mathematics and Advanced Applications, ENUMATH 2017},
interhash = {1bcc30dc486620f49c7013a9a7b787f1},
intrahash = {071a5ea0ae8da316808f8076a872f2d0},
isbn = {{978-3-319-96414-0} and {978-3-319-96415-7}},
keywords = {ians imported},
language = {eng},
location = {Cham},
number = 126,
publisher = {Springer},
series = {Lecture Notes in Computational Science and Engineering},
timestamp = {2019-06-17T12:34:15.000+0200},
title = {Greedy kernel methods for accelerating implicit integrators for parametric ODEs},
venue = {Bergen},
volume = {Proceedings of ENUMATH 2017},
year = 2018
}