Solving differential Riccati equations: A nonlinear space-time method using tensor trains

TL;DR

This paper introduces a novel all-at-once space-time approach using tensor trains and Newton-Kleinman iteration to efficiently solve nonlinear differential Riccati equations, reducing computational effort compared to traditional methods.

Contribution

It proposes a new space-time method with low-rank tensor train approximation for solving nonlinear differential Riccati equations globally in time.

Findings

Reduces the number of operator applications needed

Provides low-rank solutions efficiently

Outperforms traditional time-stepping methods

Abstract

Differential algebraic Riccati equations are at the heart of many applications in control theory. They are time-depent, matrix-valued, and in particular nonlinear equations that require special methods for their solution. Low-rank methods have been used heavily computing a low-rank solution at every step of a time-discretization. We propose the use of an all-at-once space-time solution leading to a large nonlinear space-time problem for which we propose the use of a Newton-Kleinman iteration. Approximating the space-time problem in low-rank form requires fewer applications of the discretized differential operator and gives a low-rank approximation to the overall solution.