Scalable Computation of Energy Functions for Nonlinear Balanced Truncation

TL;DR

This paper introduces a scalable Taylor-series-based method for computing energy functions in nonlinear balanced truncation, enabling model reduction of large-scale nonlinear systems by solving high-dimensional Hamilton-Jacobi-Bellman equations efficiently.

Contribution

It proposes a unifying, scalable approach using Taylor series to approximate energy functions, facilitating nonlinear model reduction on large systems.

Findings

Successfully applied to Burgers and Kuramoto-Sivashinsky equations.

Demonstrates ability to handle structured linear systems with billions of unknowns.

Provides a unified framework for open- and closed-loop balancing.

Abstract

Nonlinear balanced truncation is a model order reduction technique that reduces the dimension of nonlinear systems in a manner that accounts for either open- or closed-loop observability and controllability aspects of the system. A computational challenges that has so far prevented its deployment on large-scale systems is that the energy functions required for characterization of controllability and observability are solutions of various high-dimensional Hamilton-Jacobi-(Bellman) equations, which are computationally intractable in high dimensions. This work proposes a unifying and scalable approach to this challenge by considering a Taylor-series-based approximation to solve a class of parametrized Hamilton-Jacobi-Bellman equations that are at the core of nonlinear balancing. The value of a formulation parameter provides either open-loop balancing or a variety of closed-loop balancing options. To solve for the coefficients of Taylor-series approximations to the energy functions, the presented method derives a linear tensor system and heavily utilizes it to numerically solve structured linear systems with billions of unknowns. The strength and scalability of the algorithm is demonstrated on two semi-discretized partial differential equations, namely the Burgers and the Kuramoto-Sivashinsky equations.