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

Christian Bertram, Heike Faßbender

TL;DR

This paper develops a Petrov-Galerkin projection framework for solving large-scale continuous-time algebraic Riccati equations by projecting onto a block rational Krylov subspace via a generalized BRAD. It derives a small, Hermitian Riccati equation for the reduced unknown $Y_j$ and forms a low-rank approximation $X_j = Z_jY_jZ_j^H$, with $Z_j$ spanning the Krylov subspace. A key contribution is the efficient, non-orthogonal residual evaluation that reduces to small $2p\times 2p$ matrices, plus a truncation strategy that yields even lower-rank approximations while preserving a projected residual. The method is extended to generalized Riccati equations and validated through numerical experiments against RADI and RKSM, showing competitive convergence and well-controlled storage, especially when the orthogonal projection $\underline{L}_j=\underline{K}_j$ is used. The work offers a flexible framework that leverages BRAD structure and shift strategies to tackle large-scale CAREs in practical applications.

Abstract

A class of (block) rational Krylov subspace based projection method for solving 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 to reduce 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.

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

TL;DR

This paper develops a Petrov-Galerkin projection framework for solving large-scale continuous-time algebraic Riccati equations by projecting onto a block rational Krylov subspace via a generalized BRAD. It derives a small, Hermitian Riccati equation for the reduced unknown and forms a low-rank approximation , with spanning the Krylov subspace. A key contribution is the efficient, non-orthogonal residual evaluation that reduces to small matrices, plus a truncation strategy that yields even lower-rank approximations while preserving a projected residual. The method is extended to generalized Riccati equations and validated through numerical experiments against RADI and RKSM, showing competitive convergence and well-controlled storage, especially when the orthogonal projection is used. The work offers a flexible framework that leverages BRAD structure and shift strategies to tackle large-scale CAREs in practical applications.

Abstract

A class of (block) rational Krylov subspace based projection method for solving large-scale continuous-time algebraic Riccati equation (CARE) with a large, sparse and and of full low rank is proposed. The CARE is projected onto a block rational Krylov subspace spanned by blocks of the form for some shifts The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to The resulting projected Riccati equation is solved for the small square Hermitian Then the Hermitian low-rank approximation to is set up where the columns of span The residual norm can be computed efficiently via the norm of a readily available matrix. We suggest to reduce the rank of the approximate solution even further by truncating small eigenvalues from This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.
Paper Structure (14 sections, 2 theorems, 75 equations, 6 figures, 2 tables, 2 algorithms)

This paper contains 14 sections, 2 theorems, 75 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. General Projection Method
  4. Efficient Implementation
  5. Shift Selection
  6. Efficient Residual Norm Evaluation
  7. Truncation of the approximate solution
  8. Generalized Riccati Equations
  9. Numerical Experiments
  10. Example 1
  11. Example 2
  12. Example 3
  13. Summary of Findings
  14. Concluding Remarks

Key Result

Proposition 4

\newlabeltheo411 Let $\Upsilon \coloneqq \underline{K}_jY_j \underline{H}_j^H + (I-0.5\tilde{\pi}_j)\tilde{C}\tilde{C}^H \in \mathbb{C}^{(j+1)p\times (j+1)p}$ and $T\coloneqq \Upsilon W (U^HW)^{-1} \in \mathbb{C}^{(j+1)p\times p}.$ Let $[U~T] = QR,$ with $Q \in \mathbb{C}^{(j+1)p\times 2p}, R

Figures (6)

  • Figure 1: Example 1: Relative residual norms for shifts generated by mess_lrradi with option 'gen-ham-opti'. \newlabelfig11
  • Figure 2: Example 1: Relative residual norms for different shifts. \newlabelfig21
  • Figure 3: Example 1: Relative residual norms for shifts generated by mess_lrradi and the option 'heur'. \newlabelfig31
  • Figure 4: Example 2: Relative residual norms for different set of shifts. \newlabelfig41
  • Figure 5: Example 2: Relative residual norms for 'heur' shifts generated with mess_lrradi. \newlabelfig51
  • ...and 1 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Proposition 4
  • Proof 1
  • Remark 5
  • Theorem 6
  • Proof 2