Projection based model order reduction methods for the estimation of vector-valued variables of interest

TL;DR

This paper introduces and compares goal-oriented projection-based model order reduction techniques for efficiently estimating vector-valued functionals of solutions to parameter-dependent equations, with improved accuracy and error estimation.

Contribution

It generalizes primal-dual methods to vector-valued variables, proposes a saddle point Petrov-Galerkin approach, and develops greedy algorithms for constructing effective reduced spaces.

Findings

The proposed methods outperform standard algorithms in numerical tests.

Error estimates enable reliable assessment of approximation quality.

Goal-oriented approaches improve the accuracy of vector-valued variable estimation.

Abstract

We propose and compare goal-oriented projection based model order reduction methods for the estimation of vector-valued functionals of the solution of parameter-dependent equations. The first projection method is a generalization of the classical primal-dual method to the case of vector-valued variables of interest. We highlight the role played by three reduced spaces: the approximation space and the test space associated to the primal variable, and the approximation space associated to the dual variable. Then we propose a Petrov-Galerkin projection method based on a saddle point problem involving an approximation space for the primal variable and an approximation space for an auxiliary variable. A goal-oriented choice of the latter space, defined as the sum of two spaces, allows us to improve the approximation of the variable of interest compared to a primal-dual method using the same reduced spaces. Then, for both approaches, we derive computable error estimates for the approximations of the variable of interest and we propose greedy algorithms for the goal-oriented construction of reduced spaces. The performance of the algorithms are illustrated on numerical examples and compared to standard (non goal-oriented) algorithms.