Solving large-scale continuous-time algebraic Riccati equations by doubling

Li, Tiexiang; Chu, Eric King-wah; Lin, Wen‐Wei; Weng, Peter Chang-Yi

doi:10.1016/j.cam.2012.06.006

Cited by 31 publications

(23 citation statements)

References 15 publications

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“…1 Not much has been done for large-scale CRREs and DRREs, except in [22]. For largescale CAREs, DAREs, and Lyapunov/Stein equations, please consult, respectively, [12,31,35,36,42], [7,43], and [34,37,44], as well as the references therein.…”

Section: Existing Methods For Rres and Aresmentioning

confidence: 99%

“…For small problems, existing methods, as in MATLAB [45, commands dare and gdare], can provide an accurate X 0 . For large-scale DRREs, when R(X, 0) = R > 0, the methods in [43] can be used. When R is singular, the shifting technique in [14] can first be applied to remove the singularity, before the application of the methods in [43].…”

Section: Computing Issues and Operation Countsmentioning

confidence: 99%

“…For large-scale DRREs, when R(X, 0) = R > 0, the methods in [43] can be used. When R is singular, the shifting technique in [14] can first be applied to remove the singularity, before the application of the methods in [43]. More efficiently, we may replace the zero eigenvalues in R, justified by [23], and then apply [43] to obtain a stabilizing X 0 .…”

Section: Computing Issues and Operation Countsmentioning

confidence: 99%

“…For small problems, the initial value X 0 is readily available, using packages like MATLAB [45, commands care and gcare]. For large-scale problems, when R(X, 0) = R > 0, the methods in [42] can be applied. Otherwise, the results in [24] imply that we can replace the zero eigenvalues in R with unity, then apply the method in [42] to obtain a stabilizing X 0 .…”

Section: Numerical Issues and Operation Countsmentioning

confidence: 99%

See 3 more Smart Citations

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Zhang

Fan²,

Chu

et al. 2015

SIAM J. Sci. Comput.

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We consider the numerical solution of the rational algebraic Riccati equations in R n , arising from stochastic optimal control in continuous and discrete time. Applying the homotopy method, we continue from the stabilizing solutions of the deterministic algebraic Riccati equations, which are readily available. The associated differential equations require the solutions of some generalized Lyapunov or Stein equations, which can be solved by the generalized Smith methods, of O(n 3 ) computational complexity and O(n 2 ) memory requirement. For large-scale problems, the sparsity and structures in the relevant matrices further improve the efficiency of our algorithms. In comparison, the alternative (modified) Newton's methods require a difficult initial stabilization step. Some illustrative numerical examples are provided. Introduction.In this paper, we consider the numerical solution of rational Riccati equations (RREs) arising in stochastic optimal control. Linear time-invariant (LTI) systems [2,46,53] with stochastic components will be considered. These can be considered to be local approximations to more general nonlinear and time-variant systems. The theory of LTI systems has been vital in control system theory and has contributed successfully to many real-life applications. In other words, the solution of the difficult RREs, which leads to the optimal control of the corresponding LTI systems with stochastic components, is of some importance to control engineers in many applications.

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Section: Existing Methods For Rres and Aresmentioning

confidence: 99%

Section: Computing Issues and Operation Countsmentioning

confidence: 99%

Section: Computing Issues and Operation Countsmentioning

confidence: 99%

Section: Numerical Issues and Operation Countsmentioning

confidence: 99%

See 2 more Smart Citations

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Zhang

Fan²,

Chu

et al. 2015

SIAM J. Sci. Comput.

Self Cite

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“…Borrowing from [24], we shall apply the Sherman-Morrison-Woodbury formula (SMWF) in order to avoid the inversion of large or unstructured matrices, and use sparse-plus-low-ranked matrices to represent iterates when appropriate. Also, some matrix operators are computed recursively, to preserve the corresponding sparsity or low-ranked structures, instead of forming them explicitly.…”

Section: Large-scale Doubling Algorithmmentioning

confidence: 99%

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

Li¹,

Chu²,

Kuo³

et al. 2013

SIAM J. Matrix Anal. & Appl.

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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.

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A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

Benner

Dufrechou

Ezzatti

et al. 2018

Concurrency and Computation

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Summary We investigate different alternatives for the solution of algebraic Riccati equations on hybrid hardware platforms (ie, CPUs+GPUs). We evaluate a mixed‐precision approach that uses single‐precision arithmetic to obtain an approximation to the solution and later improve it to the desired precision, applying some steps of an economic iterative refinement. This method exploits the higher performance of the hardware to accelerate the solver when single‐precision arithmetic is employed and simultaneously obtains a high‐accuracy solution with the iterative refinement. We extend this approach to exploit the low‐rank property of the equation, when possible, to further improve its efficiency. The experimental evaluation shows that the mixed‐precision approach reports time and energy savings and provides similar or even more accurate solutions than well‐known methods like the sign function iteration or the structure‐preserving doubling algorithm.

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Solving large-scale continuous-time algebraic Riccati equations by doubling

Cited by 31 publications

References 15 publications

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Homotopy for Rational Riccati Equations Arising in Stochastic Optimal Control

Solving Large-Scale Nonsymmetric Algebraic Riccati Equations by Doubling

A GPU‐aware mixed‐precision solver for low‐rank algebraic Riccati equations

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