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

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.

23 pages, 7 figures

Cited by in corpus (3)

• Scalable Methods for Computing Sharp Extreme Event Probabilities in Infinite-Dimensional Stochastic Systems
• A linear implicit Euler method for the finite element discretization of a controlled stochastic heat equation
• Low-rank approximated Kalman-Bucy filters using Oja's principal component flow for linear time-invariant systems