Goal-oriented error estimation for parameter-dependent nonlinear problems

Alexandre Janon 1, 2, 3 Maëlle Nodet 4 Christophe Prieur 5 Clémentine Prieur 4
4 AIRSEA - Mathematics and computing applied to oceanic and atmospheric flows
Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology, UGA - Université Grenoble Alpes, LJK - Laboratoire Jean Kuntzmann, Inria Grenoble - Rhône-Alpes
5 GIPSA-SYSCO - SYSCO
GIPSA-DA - Département Automatique
Abstract : 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 a 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.
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Alexandre Janon, Maëlle Nodet, Christophe Prieur, Clémentine Prieur. Goal-oriented error estimation for parameter-dependent nonlinear problems . ESAIM: Mathematical Modelling and Numerical Analysis, EDP Sciences, 2018, 52 (2), pp.705-728. ⟨10.1051/m2an/2018003⟩. ⟨hal-01290887⟩

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