Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

Schöberl, Joachim; Zulehner, Walter

doi:10.1137/060660977

Cited by 151 publications

(173 citation statements)

References 18 publications

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“…This matrix is typically very large and sparse, and the system (4.2) is generally solved iteratively. It was shown in [33] that two preconditioners that are optimal in terms of the mesh size taken are We note that other preconditioners have been developed which may be better suited for small β, for example a block triangular preconditioner [3], or preconditioners that are β independent [30,43]. Here our aim is not to argue that this is the way one should solve such control problems-justifying such a claim would require more exhaustive tests and domain-specific theory, and as such is beyond the scope of the work.…”

Section: Pde-constrained Optimizationmentioning

confidence: 99%

“…Note that the norm of the control appears in the cost functional, along with a Tikhonov regularization parameter, β, to ensure that the problem is well-posed. If we discretize the problem using finite elements, then the minimum of the discretized cost functional is found by solving the linear system of equations [33], [43] …”

Section: Pde-constrained Optimizationmentioning

confidence: 99%

“…It is common to have two or more different candidate preconditioners for the same linear system, each possessing different properties. For example, in the case of saddle-point problems, block diagonal preconditioners and constraint preconditioners provide such a choice; see, e.g., [4], [24], [26], [33], [43]. What we propose in this paper is a way to use more than one preconditioner simultaneously.…”

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confidence: 99%

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GMRES with multiple preconditioners

Greif

Rees

Szyld

2016

SeMA

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Abstract. We propose a variant of GMRES, where multiple (two or more) preconditioners are applied simultaneously, while maintaining minimal residual optimality properties. To accomplish this, a block version of Flexible GMRES is used, but instead of considering blocks associated with multiple right hand sides, we consider a single right-hand side and grow the space by applying each of the preconditioners to all current search directions, minimizing the residual norm over the resulting larger subspace. To alleviate the difficulty of rapidly increasing storage requirements, we present a heuristic limited-memory selective algorithm, and demonstrate the effectiveness of this approach. Numerical results for problems in PDE-constrained optimization and fluid flow problems are presented, illustrating the viability and the potential of the proposed method.

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Section: Pde-constrained Optimizationmentioning

confidence: 99%

Section: Pde-constrained Optimizationmentioning

confidence: 99%

mentioning

confidence: 99%

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GMRES with multiple preconditioners

Greif

Rees

Szyld

2016

SeMA

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“…This is in agreement with the theory, as this problem has more unknowns and its matrix is indefinite. A better approach for the solution of this problem may reduce the time involved in solving this problem (see the work of Schöberl and Zulehner [33] and Zulehner [36] for instance). Figure 1.…”

Section: Computational Timementioning

confidence: 99%

Finite element discretization of Darcy's equations with pressure dependent porosity

Girault¹,

Murat²,

Salgado³

2010

ESAIM: M2AN

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Abstract.We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.Mathematics Subject Classification. 76S05, 65N30.

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“…If neither direct factorization nor the Schur complement reduction are applicable, iterative solvers have to be used. For a survey of saddle point solvers we refer to [2] and the references summarized in [32]. As the reduced problem on ker C is positive definite, we focus on conjugate gradient methods.…”

Section: Computing (Simplified) Normal Steps and Adjoint Updatesmentioning

confidence: 99%

An affine covariant composite step method for optimization with PDEs as equality constraints

Lubkoll

Schiela

Weiser

2016

Optimization Methods and Software

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AMS MSC 2000: 49M37, 90C55, 90C06We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.

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Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

Cited by 151 publications

References 18 publications

GMRES with multiple preconditioners

GMRES with multiple preconditioners

Finite element discretization of Darcy's equations with pressure dependent porosity

An affine covariant composite step method for optimization with PDEs as equality constraints

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