Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems

TL;DR

This paper introduces a residual minimizing interpolation method for efficiently approximating solutions of parameterized nonlinear dynamical systems, especially useful for quick evaluations at specific points without full time integration.

Contribution

It proposes a novel residual minimizing affine combination approach that requires only independent nonlinear evaluations, with proven properties and heuristics for improved stability and efficiency.

Findings

Comparable approximation quality to reduced basis methods

Effective in systems with stiffness and randomness

Requires only nonlinear function evaluations

Abstract

We present a method for approximating the solution of a parameterized, nonlinear dynamical system using an affine combination of solutions computed at other points in the input parameter space. The coefficients of the affine combination are computed with a nonlinear least squares procedure that minimizes the residual of the governing equations. The approximation properties of this residual minimizing scheme are comparable to existing reduced basis and POD-Galerkin model reduction methods, but its implementation requires only independent evaluations of the nonlinear forcing function. It is particularly appropriate when one wishes to approximate the states at a few points in time without time marching from the initial conditions. We prove some interesting characteristics of the scheme including an interpolatory property, and we present heuristics for mitigating the effects of the ill-conditioning and reducing the overall cost of the method. We apply the method to representative numerical examples from kinetics - a three state system with one parameter controlling the stiffness - and conductive heat transfer - a nonlinear parabolic PDE with a random field model for the thermal conductivity.