Nested multigrid methods for time-periodic, parabolic optimal control problems

Abbeloos, Dirk; Diehl, Moritz; Hinze, Michael; Vandewalle, Stefan

doi:10.1007/s00791-011-0158-4

Cited by 20 publications

(15 citation statements)

References 18 publications

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“…As we have shown in [5] and also follows from Section 3.1, an application of the preconditioner P For completeness, we include a brief account for the analysis of the spectrum of the preconditioned matrix P −1 F S A, where A is as in (17). The following result holds true.…”

Section: The Block Matrix Preconditioner Applied To a Stokes Optimal mentioning

confidence: 81%

“…By eliminating the state and adjoint variables in the three-by-three block matrix for the optimality conditions, in [17] a Fredholm integral equation of the second kind arises. This can be seen as an infinite dimensional extension of the corresponding Schur complement matrix.…”

Section: Some Other Preconditioners For Time-periodic Problemsmentioning

confidence: 99%

“…Theorem 1. Consider the generalized eigenvalue problem λP F S v = Av, where A and P F S are defined in (17) and (18), correspondingly, Assume that M and F + F T are symmetric and positive definite and D has full rank. Then all eigenvalues λ are real and positive and…”

Section: The Block Matrix Preconditioner Applied To a Stokes Optimal mentioning

confidence: 99%

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A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control

Axelsson

Farouq

Neytcheva

2017

Journal of Computational and Applied Mathematics

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A preconditioner for optimal control problems, constrained by Stokes equation with a timeharmonic control. Journal of Computational and Applied Mathematics AbstractIn this article we construct an efficient preconditioner for solving the algebraic systems arising from discretized optimal control problems with time-periodic Stokes equations, based on a preconditioning technique for stationary Stokesconstrained optimal control problems, considered in an earlier paper by the authors. A simplified analysis of the derivation of the preconditioner and its properties is presented. The preconditioner is fully parameter-independent and the condition number of the corresponding preconditioned matrix is bounded by 2. The so-constructed preconditioner is favourably compared with another robust preconditioner for the same problem.

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Section: The Block Matrix Preconditioner Applied To a Stokes Optimal mentioning

confidence: 81%

Section: Some Other Preconditioners For Time-periodic Problemsmentioning

confidence: 99%

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A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control

Axelsson

Farouq

Neytcheva

2017

Journal of Computational and Applied Mathematics

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“…In Figure 4.7, however, we can observe numerical evidence that the two-grid Newton-Picard preconditioner is completely robust with respect to γ. Our approach is thus more robust than alternative approaches, e.g., the nested multigrid methods suggested by Abbeloos et al [1], for which the convergence rates deteriorate with decreasing γ.…”

Section: 4mentioning

confidence: 96%

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

Potschka¹,

Mommer²,

Schlöder³

et al. 2012

SIAM J. Sci. Comput.

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We develop and investigate two preconditioners for a basic linear iterative splitting method for the numerical solution of linear-quadratic optimization problems with time-periodic parabolic PDE constraints. The resulting real-valued linear system to be solved is symmetric indefinite. We propose all-at-once symmetric indefinite preconditioners based on a Newton-Picard approach which divides the variable space into slow and fast modes. The division is performed either classically with eigenspace methods or with a novel two-grid approach. We prove mesh-independent convergence for the classical Newton-Picard preconditioner, present a complexity analysis, and show numerical results for the classical and the two-grid preconditioners. Moreover, the preconditioners compare favorably with existing symmetric positive definite Schur complement preconditioners in a Krylov method context.

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“…In this approach, the optimality conditions for the constrained minimization problem (1) are solved simultaneously for the state, the adjoint and the design variable in a SQP-like fashion. Because the simulation is directly integrated in the optimization process, these methods are often called Simultaneous Analysis and Design (SAND), all-at-once approach, or one-shot approach [10,11,12,13,14,15,16]. It has been observed numerically -at least for steady state PDEs -that the full space methods can outperform the reduced space methods by about one order of magnitude measured in iteration counts and runtime [8,17].…”

mentioning

confidence: 99%

Simultaneous single-step one-shot optimization with unsteady PDEs

Günther

Gauger

Wang

2016

Journal of Computational and Applied Mathematics

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The single-step one-shot method has proven to be very efficient for PDEconstrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is nontrivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The one-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented.

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Nested multigrid methods for time-periodic, parabolic optimal control problems

Cited by 20 publications

References 18 publications

A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control

A preconditioner for optimal control problems, constrained by Stokes equation with a time-harmonic control

Newton–Picard-Based Preconditioning for Linear-Quadratic Optimization Problems with Time-Periodic Parabolic PDE Constraints

Simultaneous single-step one-shot optimization with unsteady PDEs

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