Low-rank approximate solutions to large-scale differential matrix Riccati equations

Abstract

In the present paper, we consider large-scale continuous-time differential matrix Riccati equations. To the authors' knowledge, the two main approaches proposed in the litterature are based on a splitting scheme or on a Rosenbrock / Backward Differentiation Formula (BDF) methods. The approach we propose is based on the reduction of the problem dimension prior to integration. We project the initial problem onto an extended block Krylov subspace and obtain a low-dimensional differential matrix Riccati equation. The latter matrix differential problem is then solved by a Backward Differentiation Formula (BDF) method and the obtained solution is used to reconstruct an approximate solution of the original problem. This process is repeated, increasing the dimension of the projection subspace until achieving a chosen accuracy. We give some theoretical results and a simple expression of the residual allowing the implementation of a stop test in order to limit the dimension of the projection space. Some numerical experiments will be given.

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

“…We first recall the EBA process applied to the pair ( A , V ), where is assumed to be nonsingular and with s ≪ n . The projection subspace that we will consider was introduced in other works and applied for solving large‐scale symmetric differential and algebraic matrix Riccati equations in other works and for solving large‐scale Lyapunov matrix equations in the work of Simoncini . This extended block Krylov subspace is given as The EBA algorithm allows the computation of an orthonormal basis of the extended Krylov subspace .…”

“…Let us see now how the obtained approximation can be expressed in a factored form. As for the algebraic case, using the singular value decomposition of Y m ( t ), and neglecting the singular values that are close to zero, the approximate solution can be given in the following factored form where Z m ,1 ( t ) and Z m ,2 ( t ) are small rank matrices.…”

A computational method for large‐scale differential symmetric Stein equation

“…Theorem 2. Let X m (t) be the approximate solution given by (5). Then the following relation is satisfieḋ…”

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

“…where  = I ⊗ A(t) + B T (t) ⊗ I, x(t) = vec(X(t)) and b(t) = vec(E(t)F(t) T ), where vec(Z ) is the long vector obtained by stacking the columns of the matrix Z, forming a sole column. For problems of moderate size, it is then possible to directly apply an integration method to solve (5). However, this approach is not suitable for large problems.…”

“…When the matrix A is nonsingular and when the computation of the products W = A −1 V is not difficult (which is the case for sparse and structured matrices), the use of the EBA is to be preferred. We notice here that such a method was used for solving large symmetric Riccati problems in the work of Guldogan et al, 5 and other new methods were also developed recently in other works 6,7 The paper is organized as follows. In Section 2, we present a first approach based on the approximation of the exponential of a matrix times a block using a Krylov projection method.…”