Block Preconditioners for KKT Systems in PDE—Governed Optimal Control Problems

Battermann, Astrid; Sachs, E. W.

doi:10.1007/978-3-0348-8233-0_1

Cited by 37 publications

(36 citation statements)

References 22 publications

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“…This preconditioner in combination with the iterative solution of (1) by GMRES was analyzed in [1]. However, since P 2 is symmetric, one may also employ the symmetric QMR method.…”

Section: Solving the Kkt Systemmentioning

confidence: 99%

“…One preconditioning approach that is compatible with the design requirement that the preconditioner should be constructed automatically with as little user interaction as possible is the geometric multigrid method [6,7]. Hence the method proposed here differs from the examples in [3] in that we do not require problem-dependent preconditioners for e y and e y , and from [1] in allowing inexact inverses for e y and e y . How we use this approach in our context is explained in more detail in section 3.…”

Section: Solving the Kkt Systemmentioning

confidence: 99%

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Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization

Griesse

Walther

2005

Proc Appl Math and Mech

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The presented approach aims at solving an equality constrained, finite-dimensional optimization problem, where the constraints arise from the discretization of some partial differential equation (PDE) on a given space grid. For this purpose, a stationary point of the Lagrangian is computed using Newton's method, which requires the repeated solution of KKT systems. The proposed algorithm focuses on two topics: Firstly, Algorithmic Differentiation (AD) will be used to evaluate the necessary computations of gradients, Jacobian-vector products, and Hessian-vector products, so that only the objective f (y, u) and the PDE constraint e(y, u) = 0 have to be specified by the user. Secondly, we solve the KKT system iteratively using the QMR algorithm, with preconditioning provided by a multigrid approach. We wish to explore whether the Jacobian-vector products provided by AD are sufficient to construct suitable multigrid preconditioners. Our approach is then embedded into a globalized optimization routine. Numerical results for optimization problems involving a nonlinear reaction-diffusion model will be given.

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“…This preconditioner in combination with the iterative solution of (1) by GMRES was analyzed in [1]. However, since P 2 is symmetric, one may also employ the symmetric QMR method.…”

Section: Solving the Kkt Systemmentioning

confidence: 99%

Section: Solving the Kkt Systemmentioning

confidence: 99%

Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization

Griesse

Walther

2005

Proc Appl Math and Mech

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“…For research papers on reliable preconditioning in (unconstrained) optimal control of PDEs we refer to, e.g., [2][3][4]. The literature on preconditioning techniques in the case of additional inequality constraints and, in particular, in connection with active set solvers is relatively scarce.…”

Section: Introductionmentioning

confidence: 99%

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

Hintermüller¹,

Kopacka²,

Volkwein³

2008

ESAIM: COCV

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Abstract.Optimal control problems for the heat equation with pointwise bilateral control-state constraints are considered. A locally superlinearly convergent numerical solution algorithm is proposed and its mesh independence is established. Further, for the efficient numerical solution reduced space and Schur complement based preconditioners are proposed which take into account the active and inactive set structure of the problem. The paper ends by numerical tests illustrating our theoretical findings and comparing the efficiency of the proposed preconditioners.Mathematics Subject Classification. 49K20, 65K05.

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“…This gives a preconditioned system whose spectrum depends on S −1 * S with K * = K I . Inexact versions of this methodology have been suggested in the other contexts, specifically magnetostatics [14] and optimal control in ground-water flow modelling [3], but have not been considered in the context of biharmonic problems until now. In Section 4 we present some numerical experiments illustrating the computational efficiency This means that the L 2 norm of ω h is approximated so that of our inexact constraint preconditioning approach.…”

Section: Introductionmentioning

confidence: 99%

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Silvester

Mihajlović

2004

BIT Numerical Mathematics

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Abstract.We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. (2000): 65F10, 65N12, 65N22, 65N55. AMS subject classification

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Block Preconditioners for KKT Systems in PDE—Governed Optimal Control Problems

Cited by 37 publications

References 22 publications

Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization

Towards Matrix‐Free AD‐Based Preconditioning of KKT Systems in PDE‐Constrained Optimization

Mesh-independence and preconditioning for solving parabolic control problems with mixed control-state constraints

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

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