Nonlinear least-squares approach for large-scale algebraic Riccati equations

Khalide Jbilou, Marcos Raydan

Research output: Contribution to journalArticle

Abstract

We propose a new optimization approach for solving large-scale continuous-time algebraic Riccati equations with a low-rank right-hand side. First, we project the problem onto a Krylov-type low-dimensional subspace. Then, instead of forcing the orthogonality conditions related to the Galerkin strategy, we minimize the residual to get a low-dimensional nonlinear matrix least-squares problem that will be solved to obtain an approximate factorized solution of the initial Riccati equation. To solve the low-order minimization problems, we propose a globalized Gauss-Newton matrix approach that exhibits a smooth convergence behavior and that guarantees global convergence to stationary points. This novel procedure involves the solution of a linear symmetric matrix problem per iteration that will be solved by direct or preconditioned iterative matrix methods. To illustrate the behavior of the combined scheme, we present numerical results on some test problems.

Original languageEnglish
Pages (from-to)A2193-A2211
JournalSiam Journal On Scientific Computing
Volume41
Issue number4
DOIs
Publication statusPublished - 1 Jan 2019

Keywords

  • Extended block Arnoldi subspaces
  • Galerkin-projection methods
  • Gauss-Newton method
  • Global conjugate gradient method
  • Large-scale Riccati equations
  • Rational Krylov subspaces

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