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The main result of this paper gives a numerically efficient method to bound the error that is made when approximating the output of a nonlinear problem depending on an unknown parameter (described by a probability distribution). The class of nonlinear problems under consideration includes high-dimensional nonlinear problems with a nonlinear output function. A goal-oriented probabilistic bound is computed by considering two phases. An offline phase dedicated to the computation of a reduced model during which the full nonlinear problem needs to be solved only a small number of times. The second phase is an online phase which approximates the output. This approach is applied to a toy model and to a nonlinear partial differential equation, more precisely the Burgers equation with unknown initial condition given by two probabilistic parameters. The savings in computational cost are evaluated and presented.
Janon, Alexandre 1 ; Nodet, Maëlle 1 ; Prieur, Christophe 1 ; Prieur, Clémentine 1
@article{M2AN_2018__52_2_705_0,
author = {Janon, Alexandre and Nodet, Ma\"elle and Prieur, Christophe and Prieur, Cl\'ementine},
title = {Goal-oriented error estimation for parameter-dependent nonlinear problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {705--728},
publisher = {EDP-Sciences},
volume = {52},
number = {2},
year = {2018},
doi = {10.1051/m2an/2018003},
zbl = {1401.49040},
mrnumber = {3834440},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018003/}
}
TY - JOUR AU - Janon, Alexandre AU - Nodet, Maëlle AU - Prieur, Christophe AU - Prieur, Clémentine TI - Goal-oriented error estimation for parameter-dependent nonlinear problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 705 EP - 728 VL - 52 IS - 2 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018003/ DO - 10.1051/m2an/2018003 LA - en ID - M2AN_2018__52_2_705_0 ER -
%0 Journal Article %A Janon, Alexandre %A Nodet, Maëlle %A Prieur, Christophe %A Prieur, Clémentine %T Goal-oriented error estimation for parameter-dependent nonlinear problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 705-728 %V 52 %N 2 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018003/ %R 10.1051/m2an/2018003 %G en %F M2AN_2018__52_2_705_0
Janon, Alexandre; Nodet, Maëlle; Prieur, Christophe; Prieur, Clémentine. Goal-oriented error estimation for parameter-dependent nonlinear problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 2, pp. 705-728. doi: 10.1051/m2an/2018003
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