Hyper-reduction for Petrov-Galerkin reduced order models

TL;DR

This paper introduces a Petrov-Galerkin based hyper-reduction method for reduced order models that effectively minimizes the residual for both SPD and non-SPD Jacobians, simplifying implementation and avoiding the need for a complementary mesh.

Contribution

It proposes a novel Petrov-Galerkin minimization approach that enhances hyper-reduction for reduced order models, applicable to nonlinear problems and eliminating the need for a complementary mesh.

Findings

Applicable to both SPD and non-SPD Jacobians

Enables element-by-element assembly

Simplifies hyper-reduction process

Abstract

Projection-based Reduced Order Models minimize the discrete residual of a "full order model" (FOM) while constraining the unknowns to a reduced dimension space. For problems with symmetric positive definite (SPD) Jacobians, this is optimally achieved by projecting the full order residual onto the approximation basis (Galerkin Projection). This is sub-optimal for non-SPD Jacobians as it only minimizes the projection of the residual, not the residual itself. An alternative is to directly minimize the 2-norm of the residual, achievable using QR factorization or the method of the normal equations (LSPG). The first approach involves constructing and factorizing a large matrix, while LSPG avoids this but requires constructing a product element by element, necessitating a complementary mesh and adding complexity to the hyper-reduction process. This work proposes an alternative based on Petrov-Galerkin minimization. We choose a left basis for a least-squares minimization on a reduced problem, ensuring the discrete full order residual is minimized. This is applicable to both SPD and non-SPD Jacobians, allowing element-by-element assembly, avoiding the use of a complementary mesh, and simplifying finite element implementation. The technique is suitable for hyper-reduction using the Empirical Cubature Method and is applicable in nonlinear reduction procedures.