A goal-oriented reduced basis method for the wave equation in inverse analysis

TL;DR

This paper develops a goal-oriented reduced basis method for wave equations with affine parameter dependence, improving efficiency and accuracy in inverse analysis applications by incorporating dual problems and a specialized sampling procedure.

Contribution

It introduces a novel goal-oriented POD-Greedy sampling method and extends reduced-basis techniques to goal-oriented wave equations with affine parameter dependence.

Findings

The method outperforms standard POD-Greedy in accuracy of output functionals.

Numerical tests on a dental implant problem validate the approach.

The procedure efficiently separates offline and online computations.

Abstract

In this paper, we extend the reduced-basis methods developed earlier for wave equations to goal-oriented wave equations with affine parameter dependence. The essential new ingredient is the dual (or adjoint) problem and the use of its solution in a sampling procedure to pick up "goal-orientedly" parameter samples. First, we introduce the reduced-basis recipe --- Galerkin projection onto a space $Y_N$ spanned by the reduced basis functions which are constructed from the solutions of the governing partial differential equation at several selected points in parameter space. Second, we propose a new "goal-oriented" Proper Orthogonal Decomposition (POD)--Greedy sampling procedure to construct these associated basis functions. Third, based on the assumption of affine parameter dependence, we use the offline-online computational procedures developed earlier to split the computational procedure into offline and online stages. We verify the proposed computational procedure by applying it to a three-dimensional simulation dental implant problem. The good numerical results show that our proposed procedure performs better than the standard POD--Greedy procedure in terms of the accuracy of output functionals.