Symplectic Model-Reduction with a Weighted Inner Product

TL;DR

This paper extends symplectic model reduction for Hamiltonian systems to incorporate general inner products, improving stability and applicability, and demonstrates its effectiveness on elastic beam and sine-Gordon models.

Contribution

It introduces a generalized symplectic reduction method with a greedy basis construction that preserves structure under non-Euclidean inner products.

Findings

The method maintains symplectic structure with general inner products.

The greedy basis is norm-bounded, ensuring stability.

Effective reduced models for elastic beam and sine-Gordon equation.

Abstract

In the recent years, considerable attention has been paid to preserving structures and invariants in reduced basis methods, in order to enhance the stability and robustness of the reduced system. In the context of Hamiltonian systems, symplectic model reduction seeks to construct a reduced system that preserves the symplectic symmetry of Hamiltonian systems. However, symplectic methods are based on the standard Euclidean inner products and are not suitable for problems equipped with a more general inner product. In this paper, we generalize symplectic model reduction to allow for the norms and inner products that are most appropriate to the problem while preserving the symplectic symmetry of the Hamiltonian systems. To construct a reduced basis and accelerate the evaluation of nonlinear terms, a greedy generation of a symplectic basis is proposed. Furthermore, it is shown that the greedy approach yields a norm-bounded reduced basis. The accuracy and the stability of this model reduction technique are illustrated through the development of reduced models for a vibrating elastic beam and the sine-Gordon equation.