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) $0 = \mathcal{R}(X) := A^HX + XA + C^HC - XBB^HX$ 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 $\mathcal{K}_j$ spanned by blocks of the form $(A^H - s_kI)^{-1}C^H$ for some shifts $s_k,\ k = 1, \ldots, j.$ The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to $\mathcal{K}_j.$ The resulting projected Riccati equation is solved for the small square Hermitian $Y_j.$ Then the Hermitian low-rank approximation $X_j = Z_jY_jZ_j^H$ to $X$ is set up where the columns of $Z_j$ span $\mathcal{K}_j.$ The residual norm $\|R(X_j )\|_F$ can be computed efficiently via the norm of a readily available $2p \times 2p$ matrix. We suggest reducing the rank of the approximate solution $X_j$ even further by truncating small eigenvalues from $X_j.$ This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of $\mathcal{K}_j.$ This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.

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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

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