Goal-oriented error estimation for parameter-dependent nonlinear problems

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 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.