Adaptive high-order splitting schemes for large-scale differential Riccati equations

TL;DR

This paper introduces efficient high-order splitting schemes for large-scale differential Riccati equations using low-rank factorizations, overcoming previous limitations and enabling adaptive time-stepping with embedded error estimates.

Contribution

It develops a novel approach to high-order splitting schemes for large-scale Riccati equations by leveraging low-rank structures, surpassing prior constraints of order two.

Findings

Effective high-order schemes with low-rank implementation

Embedded error estimates enable adaptive time stepping

Numerical experiments show promising results

Abstract

We consider high-order splitting schemes for large-scale differential Riccati equations. Such equations arise in many different areas and are especially important within the field of optimal control. In the large-scale case, it is critical to employ structural properties of the matrix-valued solution, or the computational cost and storage requirements become infeasible. Our main contribution is therefore to formulate these high-order splitting schemes in a efficient way by utilizing a low-rank factorization. Previous results indicated that this was impossible for methods of order higher than 2, but our new approach overcomes these difficulties. In addition, we demonstrate that the proposed methods contain natural embedded error estimates. These may be used e.g. for time step adaptivity, and our numerical experiments in this direction show promising results.