### 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 language | English |
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Pages (from-to) | A2193-A2211 |

Journal | Siam Journal On Scientific Computing |

Volume | 41 |

Issue number | 4 |

DOIs | |

Publication status | Published - 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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## Cite this

*Siam Journal On Scientific Computing*,

*41*(4), A2193-A2211. https://doi.org/10.1137/18M1198922