# Order reduction methods for solving large-scale differential matrix Riccati equations

## TL;DR

This paper introduces a new order reduction method using rational Krylov subspaces for efficiently solving large-scale symmetric differential matrix Riccati equations, significantly reducing computational and memory costs.

## Contribution

The paper presents a novel reduction process onto rational Krylov subspaces combined with a two-phase strategy for improved accuracy and efficiency in solving large-scale differential Riccati equations.

## Key findings

- Significant computational savings over existing methods
- Ability to solve larger problems with lower memory requirements
- Effective numerical results on benchmark problems

## Abstract

We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies, by forming low-rank approximations to the sought after solution at selected timesteps. We show that great computational and memory savings are obtained by a reduction process onto rational Krylov subspaces, as opposed to current approaches. By specifically addressing the solution of the reduced differential equation and reliable stopping criteria, we are able to obtain accurate final approximations at low memory and computational requirements. This is obtained by employing a two-phase strategy that separately enhances the accuracy of the algebraic approximation and the time integration. The new method allows us to numerically solve much larger problems than in the current literature. Numerical experiments on benchmark problems illustrate the effectiveness of the procedure with respect to existing solvers.