Structure-Preserving Model Reduction for Port-Hamiltonian Systems Based on a Special Class of Nonlinear Approximation Ansatzes

TL;DR

This paper introduces a structure-preserving model reduction method for port-Hamiltonian systems using nonlinear ansatz functions that depend on the reduced model state, improving efficiency for transport-dominated systems.

Contribution

It proposes a residual minimization approach with a special weighted norm to obtain stable reduced models, extending classical techniques to nonlinear ansatz functions.

Findings

Reduced models maintain port-Hamiltonian structure.

Method effectively handles transport-dominated systems.

Numerical tests demonstrate stability and accuracy.

Abstract

We discuss structure-preserving model order reduction for port-Hamiltonian systems based on an approximation of the full-order state by a linear combination of ansatz functions which depend themselves on the state of the reduced-order model. In recent years, such nonlinear approximation ansatzes have gained more and more attention especially due to their effectiveness in the context of model reduction for transport-dominated systems which are challenging for classical linear model reduction techniques. We demonstrate that port-Hamiltonian reduced-order models can often be obtained by a residual minimization approach where a special weighted norm is used for the residual. Moreover, we discuss sufficient conditions for the resulting reduced-order models to be stable. Finally, the methodology is illustrated by means of two transport-dominated numerical test cases, where the ansatz functions are determined based on snapshot data of the full-order state.