The Adjoint Petrov-Galerkin Method for Non-Linear Model Reduction

TL;DR

The paper introduces the Adjoint Petrov-Galerkin (APG) method, a novel reduced-order modeling technique for non-linear dynamical systems that improves accuracy and efficiency by incorporating a non-linear, time-varying test basis.

Contribution

It develops a new projection-based reduced-order model using the Mori-Zwanzig formalism, combining features of adjoint stabilization and Petrov-Galerkin methods, with theoretical analysis and numerical validation.

Findings

Improves numerical accuracy over Galerkin methods.

Enhances robustness and computational efficiency.

Demonstrates effectiveness on compressible Navier-Stokes equations.

Abstract

We formulate a new projection-based reduced-ordered modeling technique for non-linear dynamical systems. The proposed technique, which we refer to as the Adjoint Petrov-Galerkin (APG) method, is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. A Markovian finite memory assumption within the Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse-scales. This procedure leads to a closed reduced-order model that displays commonalities with the adjoint stabilization method used in finite elements. The formulation is shown to be equivalent to a Petrov-Galerkin method with a non-linear, time-varying test basis, thus sharing some similarities with the least-squares Petrov-Galerkin method. Theoretical analysis examining a priori error bounds and computational cost is presented. Numerical experiments on the compressible Navier-Stokes equations demonstrate that the proposed method can lead to improvements in numerical accuracy, robustness, and computational efficiency over the Galerkin method on problems of practical interest. Improvements in numerical accuracy and computational efficiency over the least-squares Petrov-Galerkin method are observed in most cases.