A polynomial approximation scheme for nonlinear model reduction by moment matching

TL;DR

This paper introduces a polynomial approximation scheme using moment matching and Newton iteration for nonlinear model reduction in high-dimensional dynamical systems, effectively capturing steady-state behavior.

Contribution

It presents a novel numerical method employing Galerkin residuals and polynomial basis expansion to solve invariance equations for reduced-order modeling.

Findings

Successfully approximates invariance PDEs in systems with 1000 states

Achieves effective moment matching and steady-state recovery

Applicable to systems driven by linear and nonlinear signals

Abstract

We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover the steady-state behaviour of nonlinear systems with state dimension of order 1000 driven by linear and nonlinear signal generators.