Related papers: A New Subspace Iteration method for the Algebraic …
The approximate solution of large-scale algebraic Riccati equations is considered. We are interested in approximate solutions which yield a Riccati residual matrix of a particular small rank. It is assumed that such approximate solutions…
In the numerical solution of the algebraic Riccati equation $A^* X + X A - X BB^* X + C^* C =0$, where $A$ is large, sparse and stable, and $B$, $C$ have low rank, projection methods have recently emerged as a possible alternative to the…
This work is concerned with the numerical solution of large-scale symmetric positive definite matrix equations of the form $A_1XB_1^\top + A_2XB_2^\top + \dots + A_\ell X B_\ell^\top = F$, as they arise from discretized partial differential…
We propose a Riemannian optimization approach for computing low-rank solutions of the algebraic Riccati equation. The scheme alternates between fixed-rank optimization and rank-one updates. The fixed-rank optimization is on the set of…
Algebraic Riccati equations with indefinite quadratic terms play an important role in applications related to robust controller design. While there are many established approaches to solve these in case of small-scale dense coefficients,…
In the present paper, we consider large-scale continuous-time differential matrix Riccati equations having low rank right-hand sides. These equations are generally solved by Backward Differentiation Formula (BDF) or Rosenbrock methods…
This paper presents an effective low-rank generalized alternating direction implicit iteration (R-GADI) method for solving large-scale sparse and stable Lyapunov matrix equations and continuous-time algebraic Riccati matrix equations. The…
In [3] it was shown that four seemingly different algorithms for computing low-rank approximate solutions $X_j$ to the solution $X$ of large-scale continuous-time algebraic Riccati equations (CAREs) $0 = \mathcal{R}(X) :=…
This paper introduces a new algorithm for solving large-scale continuous-time algebraic Riccati equations (CARE). The advantage of the new algorithm is in its immediate and efficient low-rank formulation, which is a generalization of the…
In this paper, we propose an RADI-type method for large-scale stochastic continuous-time algebraic Riccati equations with sparse and low-rank matrices. This new variant of RADI-type methods is developed by integrating the core concept of…
This paper proposes an effective low-rank alternating direction doubling algorithm (R-ADDA) for computing numerical low-rank solutions to large-scale sparse continuous-time algebraic Riccati matrix equations. The method is based on the…
In this work, we consider two types of large-scale quadratic matrix equations: Continuous-time algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from…
The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M-matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive…
Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati…
The low-rank alternating direction implicit (ADI) method is an efficient and effective solver for large-scale standard continuous-time algebraic Riccati equations that admit low-rank solutions. However, the existing low-rank ADI algorithm…
This paper considers large-scale nonsymmetric continuous-time algebraic Riccati equations (NAREs) that admit low-rank solutions. Low-rank alternating direction implicit (ADI) methods have proven to be an efficient approach for solving…
The numerical solution of the algebraic Riccati equation is a challenging task especially for very large problem dimensions. In this paper we present a new algorithm that combines the very appealing computational features of projection…
We propose a new class of short matrix recurrences for the solution of nonsymmetric linear equations of the type $\mathbf{A}_1\mathbf{X}\mathbf{B}_1+\ldots+\mathbf{A}_p\mathbf{X}\mathbf{B}_p=CD^T$. These iterative methods combine local…
We consider a Krylov subspace approximation method for the symmetric differential Riccati equation $\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The method we consider is based on projecting the large scale equation onto a Krylov subspace…
We propose a computational framework for computing low-rank approximations to the ensemble of solutions of a parametrized system of the form $A(\xi)x(\xi)+g(x(\xi))=b(\xi)$ for multiple parameter values. The central idea is to reinterpret…