# Analysis of Krylov Subspace Approximation to Large Scale Differential Riccati Equations

## TL;DR

This paper analyzes a Krylov subspace method for large-scale symmetric differential Riccati equations, demonstrating structure preservation, superlinear convergence, and providing error estimates supported by numerical experiments.

## Contribution

It introduces a structure-preserving Krylov subspace approximation for large-scale Riccati equations with proven superlinear convergence and practical error estimation methods.

## Key findings

- The method preserves positivity and monotonicity of the Riccati flow.
- Superlinear convergence of the approximation is theoretically established.
- Numerical experiments confirm the effectiveness and accuracy of the approach.

## Abstract

We consider a Krylov subspace approximation method for the symmetric differential Riccati equation $\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The method we consider is based on projecting the large scale equation onto a Krylov subspace spanned by the matrix $A$ and the low rank factors of $X_0$ and $Q$. We prove that the method is structure preserving in the sense that it preserves two important properties of the exact flow, namely the positivity of the exact flow, and also the property of monotonicity. We also provide a theoretical a priori error analysis which shows a superlinear convergence of the method. This behavior is illustrated in the numerical experiments. Moreover, we derive an efficient a posteriori error estimate as well as discuss multiple time stepping combined with a cut of the rank of the numerical solution.