Volume 62, pp. 138-162, 2024.

A class of Petrov-Galerkin Krylov methods for algebraic Riccati equations

Christian Bertram and Heike Faßbender

Abstract

A class of (block) rational Krylov-subspace-based projection methods for solving the large-scale continuous-time algebraic Riccati equation (CARE) 0=R(X):=AHX+XA+CHC−XBBHX with a large, sparse A, and B and C of full low rank is proposed. The CARE is projected onto a block rational Krylov subspace Kj spanned by blocks of the form (AH−skI)−1CH for some shifts sk, k=1,…,j. The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to Kj. The resulting projected Riccati equation is solved for the small square Hermitian Yj. Then the Hermitian low-rank approximation Xj=ZjYjZjH to X is set up where the columns of Zj span Kj. The residual norm ‖R(Xj)‖F can be computed efficiently via the norm of a readily available 2p×2p matrix. We suggest reducing the rank of the approximate solution Xj even further by truncating small eigenvalues from Xj. This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of Kj. This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.

Full Text (PDF) [1.1 MB], BibTeX , DOI: 10.1553/etna_vol62s138

Key words

algebraic Riccati equation, large-scale matrix equation, (block) rational Krylov subspace, projection method

AMS subject classifications

15A24, 65F15

Links to the cited ETNA articles

[2] Vol. 42 (2014), pp. 147-164 Peter Benner and Tobias Breiten: Rational interpolation methods for symmetric Sylvester equations
[18] Vol. 33 (2008-2009), pp. 53-62 M. Heyouni and K. Jbilou: An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation