Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Simoncini, Valeria

doi:10.1137/16m1059382

Cited by 52 publications

(51 citation statements)

References 33 publications

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“…To monitor the progress and to stop the iteration, the Euclidean or Frobenius norm of the residual R j := R(Q j Y j Q * j ) can also be computed efficiently without explicitly forming this large, dense n × n matrix [47,59,95,96,98]. Following [47], the CARE residual in RKSM has the form (for M = I)…”

Section: Projection Methodsmentioning

confidence: 99%

See 1 more Smart Citation

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

Benner¹,

Bujanović²,

Kürschner³

et al. 2020

SIAM J. Sci. Comput.

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In this paper, we discuss numerical methods for solving large-scale continuous-time algebraic Riccati equations. These methods have been the focus of intensive research in recent years, and significant progress has been made in both the theoretical understanding and efficient implementation of various competing algorithms. There are several goals of this manuscript: first, to gather in one place an overview of different approaches for solving large-scale Riccati equations, and to point to the recent advances in each of them. Second, to analyze and compare the main computational ingredients of these algorithms, to detect their strong points and their potential bottlenecks. And finally, to compare the effective implementations of all methods on a set of relevant benchmark examples, giving an indication of their relative performance. *

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Section: Projection Methodsmentioning

confidence: 99%

“…Initially, these parameters were computed in advance, prior to any iteration of the Riccati solver. In the last years, dynamic shift generation strategies [16,23,47,48,69,96] have attracted increasing attention.…”

Section: Shiftsmentioning

confidence: 99%

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

Benner¹,

Bujanović²,

Kürschner³

et al. 2020

SIAM J. Sci. Comput.

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show abstract

“…(10) computationally infeasible for practical fluid flow applications. To remedy this, we employ a Galerkin projection based method [44,45] that yields a low-rank approximation of the solution to Eq. (10).…”

Section: Controller Designmentioning

confidence: 99%

Reduced-order control using low-rank dynamic mode decomposition

Sashittal

Bodony

2019

Theor. Comput. Fluid Dyn.

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In this work we perform full-state LQR feedback control of fluid flows using non-intrusive data-driven reduced-order models. We propose a model reduction method called low-rank Dynamic Mode Decomposition (lrDMD) that solves for a rank-constrained linear representation of the dynamical system. lrDMD is shown to have lower data reconstruction error compared to standard Optimal Mode Decomposition (OMD) and Dynamic Mode Decomposition (DMD), but with an increased computational cost arising from solving a non-convex matrix optimization problem. We demonstrate model order reduction on the complex linearized Ginzburg-Landau equation in the globally unstable regime and on the unsteady flow over a flat plate at a high angle of attack. In both cases, lowdimensional full-state feedback controller is constructed using reduced-order models constructed using DMD, OMD and lrDMD. It is shown that lrDMD stabilizes the Ginzburg-Landau system with a lower order controller and is able to suppress vortex shedding from an inclined flat plate at a cost lower than either DMD or OMD. It is further shown that lrDMD yields an improved estimate of the adjoint system, for a given rank, relative to DMD and OMD.

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“…Algorithm 1 gives a basic illustration of this method. We refer to the relevant literature [22,46,47] for implementation details and only comment on some critical steps. For…”

Section: Low-rank Solution Of Correction Equation For Carementioning

confidence: 99%

“…end for 11: end procedure selecting the shift parameters in line 7 we employ the adaptive procedure from [22,46,47]. This may result in complex shifts or, more precisely, in complex conjugate pairs of shifts.…”

Section: Low-rank Solution Of Correction Equation For Carementioning

confidence: 99%

Low-rank updates and divide-and-conquer methods for quadratic matrix equations

2019

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In this work, we consider two types of large-scale quadratic matrix equations: Continuoustime algebraic Riccati equations, which play a central role in optimal and robust control, and unilateral quadratic matrix equations, which arise from stochastic processes on 2D lattices and vibrating systems. We propose a simple and fast way to update the solution to such matrix equations under low-rank modifications of the coefficients. Based on this procedure, we develop a divide-and-conquer method for quadratic matrix equations with coefficients that feature a specific type of hierarchical low-rank structure, which includes banded matrices. This generalizes earlier work on linear matrix equations. Numerical experiments indicate the advantages of our newly proposed method versus iterative schemes combined with hierarchical holds for an induced norm · then ϕ J (z) has exactly k eigenvalues (counting multiplicities) in the open unit disc and k eigenvalues with modulus greater than 1, where k denotes the cardinality of J. This implies the result of the lemma.Setting ψ(λ) := −B −1 J (λA J + λ −1 C J ), the condition (30) clearly holds if we can show that the spectral radius ρ(ψ(λ)) is less than 1 for every λ on the unit circle. Note that |λA J + λ −1 C J | ≤ A J +C J because A J , C J are non-negative. Combined with the fact that −B J is an M-matrix, which implies −B −1 J 0, and the monotonicity of the spectral radius, we obtain (1)).Using −B −1 J 0 we also have (A J + B J + C J + I)e e =⇒ (A J + C J )e −B J e =⇒ ψ(1)e e.

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Analysis of the Rational Krylov Subspace Projection Method for Large-Scale Algebraic Riccati Equations

Cited by 52 publications

References 33 publications

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

A Numerical Comparison of Different Solvers for Large-Scale, Continuous-Time Algebraic Riccati Equations and LQR Problems

Reduced-order control using low-rank dynamic mode decomposition

Low-rank updates and divide-and-conquer methods for quadratic matrix equations

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