Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

TL;DR

This paper develops Krylov-based methods for efficiently approximating solutions to large nonsymmetric differential Riccati equations, with applications in transport theory and other fields, demonstrating their effectiveness through numerical experiments.

Contribution

It introduces a Krylov-type approach using the extended block Arnoldi algorithm for low rank solutions to large nonsymmetric differential Riccati equations, applicable to transport problems.

Findings

Effective low rank approximations for large-scale problems

Successful application to transport equations

Numerical results confirm efficiency and accuracy

Abstract

In the present paper, we consider large scale nonsymmetric differential matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability and others. We show how to apply Krylov-type methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as Backward Differentiation Formula (BDF) or Rosenbrok method. We also show how these technique could be easily used to solve some problems from the well known transport equation. Some numerical experiments are given to illustrate the application of the proposed methods to large-scale problems