Volume 52, Number 2, March–April 2018
|Page(s)||705 - 728|
|Published online||11 July 2018|
Goal-oriented error estimation for parameter-dependent nonlinear problems
Laboratoire de Mathématiques d’Orsay, Université Paris-Sud,
2 Grenoble INP, Institute of Engineering Univ. Grenoble Alpes, LJK, 38000 Grenoble, France
3 Grenoble INP, Institute of Engineering Univ. Grenoble Alpes, GIPSA-lab, 38000 Grenoble, France
* Corresponding author: e-mail: firstname.lastname@example.org
Accepted: 12 January 2018
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.
Mathematics Subject Classification: 49Q12 / 62F12 / 65C20 / 82C80
Key words: Goal-oriented / probabilistic error estimation / nonlinear problems / uncertainty quantification
© The authors. Published by EDP Sciences, SMAI 2018
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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