Projection-Based Model Reduction with Dynamically Transformed Modes

TL;DR

This paper introduces a novel model reduction framework for transport problems using dynamic transformations, generalizing MFEM and enabling efficient low-dimensional approximations with error control.

Contribution

It generalizes the moving finite element method to arbitrary basis functions and develops a nonlinear manifold projection approach for improved model reduction.

Findings

Effective in reducing model complexity for transport phenomena

Provides an a-posteriori error bound for the reduced model

Demonstrates success through analytical and numerical examples

Abstract

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.