We present a new approach to treating nonlinear operators in reduced
basis approximations of parametrized evolution equations. Our approach
is based on empirical interpolation of nonlinear differential operators
and their Frechet derivatives. Efficient offline/online decomposition
is obtained for discrete operators that allow an efficient evaluation
for a certain set of interpolation functionals. An a posteriori error
estimate for the resulting reduced basis method is derived and analyzed
numerically. We introduce a new algorithm, the PODEI-greedy algorithm,
which constructs the reduced basis spaces for the empirical interpolation
and for the numerical scheme in a synchronized way. The approach
is applied to nonlinear parabolic and hyperbolic equations based
on explicit or implicit finite volume discretizations. We show that
the resulting reduced scheme is able to capture the evolution of
both smooth and discontinuous solutions. In case of symmetries of
the problem, the approach realizes an automatic and intuitive space-compression
or even space-dimensionality reduction. We perform empirical investigations
of the error convergence and run-times. In all cases we obtain a
good run-time acceleration.
%0 Journal Article
%1 Drohmann2012
%A Drohmann, M.
%A Haasdonk, Bernard
%A Ohlberger, M.
%D 2012
%J SIAM J. Sci. Comput.
%K anm from:britsteiner ians imported
%N 2
%P A937-A969
%R 10.1137/10081157X
%T Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation
%V 34
%X We present a new approach to treating nonlinear operators in reduced
basis approximations of parametrized evolution equations. Our approach
is based on empirical interpolation of nonlinear differential operators
and their Frechet derivatives. Efficient offline/online decomposition
is obtained for discrete operators that allow an efficient evaluation
for a certain set of interpolation functionals. An a posteriori error
estimate for the resulting reduced basis method is derived and analyzed
numerically. We introduce a new algorithm, the PODEI-greedy algorithm,
which constructs the reduced basis spaces for the empirical interpolation
and for the numerical scheme in a synchronized way. The approach
is applied to nonlinear parabolic and hyperbolic equations based
on explicit or implicit finite volume discretizations. We show that
the resulting reduced scheme is able to capture the evolution of
both smooth and discontinuous solutions. In case of symmetries of
the problem, the approach realizes an automatic and intuitive space-compression
or even space-dimensionality reduction. We perform empirical investigations
of the error convergence and run-times. In all cases we obtain a
good run-time acceleration.
@article{Drohmann2012,
abstract = {We present a new approach to treating nonlinear operators in reduced
basis approximations of parametrized evolution equations. Our approach
is based on empirical interpolation of nonlinear differential operators
and their Frechet derivatives. Efficient offline/online decomposition
is obtained for discrete operators that allow an efficient evaluation
for a certain set of interpolation functionals. An a posteriori error
estimate for the resulting reduced basis method is derived and analyzed
numerically. We introduce a new algorithm, the PODEI-greedy algorithm,
which constructs the reduced basis spaces for the empirical interpolation
and for the numerical scheme in a synchronized way. The approach
is applied to nonlinear parabolic and hyperbolic equations based
on explicit or implicit finite volume discretizations. We show that
the resulting reduced scheme is able to capture the evolution of
both smooth and discontinuous solutions. In case of symmetries of
the problem, the approach realizes an automatic and intuitive space-compression
or even space-dimensionality reduction. We perform empirical investigations
of the error convergence and run-times. In all cases we obtain a
good run-time acceleration.},
added-at = {2021-09-29T14:35:09.000+0200},
author = {Drohmann, M. and Haasdonk, Bernard and Ohlberger, M.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/295bbced717a73421346d9447e5fc5bbc/mathematik},
doi = {10.1137/10081157X},
eprint = {http://epubs.siam.org/doi/pdf/10.1137/10081157X},
file = {:PDF/UNSORTED/DHO12 - RB Approx for nonlin evol.pdf:PDF},
groups = {haasdonk, haasdonk_all_papers},
interhash = {b93f6f4c10231722304edd1c8be5c5f1},
intrahash = {95bbced717a73421346d9447e5fc5bbc},
journal = {SIAM J. Sci. Comput.},
keywords = {anm from:britsteiner ians imported},
number = 2,
owner = {dwirtz},
pages = {A937-A969},
timestamp = {2021-10-15T09:10:09.000+0200},
title = {Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation},
volume = 34,
year = 2012
}