Stochastic Galerkin method and port-Hamiltonian form for linear first-order ordinary differential equations

TL;DR

This paper develops a structure-preserving stochastic Galerkin method for linear port-Hamiltonian ODE systems with uncertain parameters, and demonstrates how to maintain the pH form through model order reduction, supported by numerical examples.

Contribution

It introduces a transformation approach to preserve port-Hamiltonian structure in stochastic Galerkin systems and explores structure-preserving model order reduction techniques.

Findings

Structure-preserving stochastic Galerkin projection achieved.

Reduced models retain port-Hamiltonian form.

Numerical examples validate the approach.

Abstract

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodelled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a highdimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.