Explicit Solutions for a Riccati Equation from Transport Theory

Mehrmann, Volker; Xu, Hongguo

doi:10.1137/070708743

Cited by 27 publications

(15 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Nonsymmetric algebraic Riccati equation (7) associated with M-matrices appear in a problem in neutron transport theory; see [3,[34][35][36], where the mathematical model consists in solving an integro-differential equation. After discretization of this integro-differential equation, the problem can be expressed as the following matrix equation…”

Section: Applications To Nares From Transport Theorymentioning

confidence: 99%

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

Bentbib

Jbilou²,

Sadek

2015

Computers & Mathematics with Applications

View full text Add to dashboard Cite

a b s t r a c tIn the present paper, we consider large scale nonsymmetric matrix Riccati equations with low rank right hand sides. These matrix equations appear in many applications such as transport theory, Wiener-Hopf factorization of Markov chains, applied probability and others. We show how to apply directly Krylov methods such as the extended block Arnoldi algorithm to get low rank approximate solutions. We also combine the Newton method and block Krylov subspace methods to get approximations of the desired minimal nonnegative solution. We give some theoretical results and report some numerical experiments for the well known transport equation.

show abstract

Section: Applications To Nares From Transport Theorymentioning

confidence: 99%

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

Bentbib

Jbilou²,

Sadek

2015

Computers & Mathematics with Applications

View full text Add to dashboard Cite

show abstract

“…Symmetric algebraic Riccati equations have been the topic of extensive research, and the theory, applications, and numerical solutions of these equations are the subject of [5,6,7,8] as well as the monographs [21,29]. The minimal positive solution to the NARE (1.1), for medium size problems without the sparseness and low-ranked assumptions, has been studied recently by several authors, employing functional iterations, Newton's method, and the structure-preserving algorithm; see [1,3,4,12,13,14,15,16,17,22,23,26,27,30,31], and the references therein. Evidently, the applications associated with and the numerical solution to NAREs have attracted much attention in the past decade but this paper is the first on general large-scale NAREs.…”

Section: Introduction Consider the Nonsymmetric Algebraic Riccati Eqmentioning

confidence: 99%

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

Li¹,

Chu²,

Kuo³

et al. 2013

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

We consider the solution of the large-scale nonsymmetric algebraic Riccati equation∈ R (n 1 +n 2 )×(n 1 +n 2 ) being a nonsingular M-matrix. In addition, A and D are sparselike, with the products A −1 u, A − u, D −1 v, and D − v computable in O(n) complexity (with n = max{n 1 , n 2 }), for some vectors u and v, and B, C are low ranked. The structure-preserving doubling algorithms (SDA) by Guo, Lin, and Xu [Numer. Math., 103 (2006), pp. 392-412] is adapted, with the appropriate applications of the Sherman-MorrisonWoodbury formula and the sparse-plus-low-rank representations of various iterates. The resulting large-scale doubling algorithm has an O(n) computational complexity and memory requirement per iteration and converges essentially quadratically. A detailed error analysis, on the effects of truncation of iterates with an explicit forward error bound for the approximate solution from the SDA, and some numerical results will be presented.

show abstract

“…The Equation is a very special case of the NARE. Explicit and easy solutions to Equation are well known (see Mehrmann and Xu).…”

Section: Introductionmentioning

confidence: 99%

Fast verified computation for solutions of algebraic Riccati equations arising in transport theory

Miyajima

2017

Numerical Linear Algebra App

View full text Add to dashboard Cite

Summary A fast algorithm for enclosing the solution of the nonsymmetric algebraic Riccati equation arising in transport theory is proposed. The equation has a special structure, which is taken into account to reduce the complexity. By exploiting the structure, the enclosing process involves only quadratic complexity under a reasonable assumption. The algorithm moreover verifies the uniqueness and minimal positiveness of the enclosed solution. Numerical results show the efficiency of the algorithm.

show abstract

Explicit Solutions for a Riccati Equation from Transport Theory

Cited by 27 publications

References 27 publications

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

On some Krylov subspace based methods for large-scale nonsymmetric algebraic Riccati problems

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

Fast verified computation for solutions of algebraic Riccati equations arising in transport theory

Contact Info

Product

Resources

About