On the benefits of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi>L</mml:mi><mml:mi>D</mml:mi><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:msup></mml:math> factorization for large-scale differential matrix equation solvers

Lang, Norman; Mena, Hermann; Saak, Jens

doi:10.1016/j.laa.2015.04.006

Cited by 47 publications

(44 citation statements)

References 25 publications

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“…Other important applications lie in model order reduction [3] or in optimal control of linear time-invariant systems on finite time horizons [30]. Despite its importance, there have been but a few efforts to solve the differential Sylvester / Lyapunov or Riccati equation numerically, see [6,7,16,21,25,26,31,36]. These algorithms are usually based on applying a time discretization and solving the resulting algebraic equations.…”

Section: Introductionmentioning

confidence: 99%

Solution formulas for differential Sylvester and Lyapunov equations

Behr

Benner

Heiland

2019

Calcolo

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The differential Sylvester equation and its symmetric version, the differential Lyapunov equation, appear in different fields of applied mathematics like control theory, system theory, and model order reduction. The few available straight-forward numerical approaches if applied to largescale systems come with prohibitively large storage requirements. This shortage motivates us to summarize and explore existing solution formulas for these equations. We develop a unifying approach based on the spectral theorem for normal operators like the Sylvester operator S(X) = AX + XB and derive a formula for its norm using an induced operator norm based on the spectrum of A and B. In view of numerical approximations, we propose an algorithm that identifies a suitable Krylov subspace using Taylor series and use a projection to approximate the solution. Numerical results for large-scale differential Lyapunov equations are presented in the last sections.

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Section: Introductionmentioning

confidence: 99%

Solution formulas for differential Sylvester and Lyapunov equations

Behr

Benner

Heiland

2019

Calcolo

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“…Without such a step, each iteration of Algorithm 3.1 (for example) would add the columns in L Q to L, while the rank would likely stay similar. The compression can be performed in various ways, usually by computing either a reduced rank-revealing QR factorization or a reduced SVD [33]. Here, we employ a reduced SVD factorization, followed by a diagonalization of the small resulting system.…”

Section: Algorithm 31 Solving Dle By Strang Splittingmentioning

confidence: 99%

GPU acceleration of splitting schemes applied to differential matrix equations

2019

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We consider differential Lyapunov and Riccati equations, and generalized versions thereof. Such equations arise in many different areas and are especially important within the field of optimal control. In order to approximate their solution, one may use several different kinds of numerical methods. Of these, splitting schemes are often a very competitive choice. In this article, we investigate the use of graphical processing units (GPUs) to parallelize such schemes and thereby further increase their effectiveness. According to our numerical experiments, large speed-ups are often observed for sufficiently large matrices. We also provide a comparison between different splitting strategies, demonstrating that splitting the equations into a moderate number of subproblems is generally optimal.

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“…21 Considering the general setting described in the work of Hafizoglu et al, 21 an approximation scheme for solving the control problem and the associated Riccati equation has been proposed in the work of Levajković et al 22 The numerical solution of the SLQR relies on solving efficiently the associated Riccati equation. For the deterministic LQR problem, most of the methods for solving differential Riccati equations, for example, the previous studies [23][24][25] are based on a low-rank approximation of the solution, and their performance relies on the rapid decay of the singular values. This phenomenon is observed in singular estimate control systems in applications and has been studied in detail by, for example, the previous studies.…”

Section: Dx(t) = [A(t)x(t) + B(t)u(t) + B(t)] Dt + [C(t)x(t) + D(t)u(mentioning

confidence: 99%

Numerical solution of the finite horizon stochastic linear quadratic control problem

Damm

Mena

Stillfjord

2017

Numer. Linear Algebra Appl.

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Summary The treatment of the stochastic linear quadratic optimal control problem with finite time horizon requires the solution of stochastic differential Riccati equations. We propose efficient numerical methods, which exploit the particular structure and can be applied for large‐scale systems. They are based on numerical methods for ordinary differential equations such as Rosenbrock methods, backward differentiation formulas, and splitting methods. The performance of our approach is tested in numerical experiments.

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On the benefits of the LDLT factorization for large-scale differential matrix equation solvers

Cited by 47 publications

References 25 publications

Solution formulas for differential Sylvester and Lyapunov equations

Solution formulas for differential Sylvester and Lyapunov equations

GPU acceleration of splitting schemes applied to differential matrix equations

Numerical solution of the finite horizon stochastic linear quadratic control problem

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