Analysis of block matrix preconditioners for elliptic optimal control problems

Mathew, Tarek Poonithara Abraham; Sarkis, Marcus; Schaerer, Christian E.

doi:10.1002/nla.526

Cited by 21 publications

(21 citation statements)

References 33 publications

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“…Solution algorithms for variants of the system (28) have been studied by various authors, cf. for example (Nielsen and Mardal, submitted) 42–44. The discussion here follows closely the approach taken in 44.…”

Section: Parameter‐dependent Problemsmentioning

confidence: 92%

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Preconditioning discretizations of systems of partial differential equations

Mardal

Winther

2010

Numer. Linear Algebra Appl.

257

280

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This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter-dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach.here we show how a number of different problems can be treated by the same abstract framework. Furthermore, in Section 5 we focus on the close link between the proper choice of preconditioner and the classical variational theory of Babuška [3,4].The strength of the approach taken in this paper is best illustrated by the construction of preconditioners for parameter-dependent problems. By a systematic approach we easily identify preconditioners which behave uniformly both with respect to these model parameters and the discretization parameter. The list of examples studied below includes stationary systems obtained from implicit time discretizations of the time-dependent Stokes problem, where the time step is a critical parameter. Another example we will consider is the singular perturbation problem referred to as the Reissner-Mindlin plate model, where the thickness of the plate enters as a parameter. An example of an optimal control problem will also be discussed. It is well known that the solutions of such systems depend strongly on the so-called regularization parameters, but below we show that for some problems it is possible to design preconditioners such that the conditioning of the preconditioned systems is bounded uniformly with respect to the regularization parameters.For the construction of practical preconditioners for discrete systems, the computational cost of evaluating these operators and the memory requirements of these procedures are key factors. These issues will not be discussed in full detail in this paper. The aim of the discussions here is to identify what we refer to as canonical preconditioners, which are block diagonal operators suggested by the mapping properties of the coefficient operators of the systems, i.e. the blocks will typically correspond to exact inverses of discretizations of differential operators. These canonical operators identify the basic structure of the preconditioner. However, in order to use this approach to construct practical and ...

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Section: Parameter‐dependent Problemsmentioning

confidence: 92%

“…Algorithms for other optimal control problems with elliptic PDE constraints are discussed in, e.g. Nielsen and Mardal (submitted) 42–44.…”

Section: Preconditioning Discrete Saddle Point Problemsmentioning

confidence: 99%

Preconditioning discretizations of systems of partial differential equations

Mardal

Winther

2010

Numer. Linear Algebra Appl.

257

280

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“…that the number of iterations needed does not increase significantly as α → 0. The latter type of problem has been addressed in many papers for various models [1,7,22,27,31,32,34,35,37] -i.e. for special cases of elliptic and parabolic control problems.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Minimal Residual Method Applied to Ill Posed Optimality Systems

Nielsen¹,

Mardal²

2013

SIAM J. Sci. Comput.

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We analyze the performance of the Minimal Residual Method applied to linear Karush-Kuhn-Tucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all α > 0, where α is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of α, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved with a number of iterations of order O(ln(α −1)). More precisely, in the severely ill posed case the number of iterations needed by the Minimal Residual Method cannot grow faster than O(ln(α −1)) as α → 0. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of bounded intervals. Krylov subspace methods handle such problems very well. We illuminate our theoretical findings with some numerical results for inverse problems involving partial differential equations. Our investigation is inspired by Prof. H. Egger's discussion of similar results valid for the conjugate gradient algorithm applied to the normal equations.

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“…Of course also multigrid methods have been considered for optimal control problems, see, e.g., [10]. A review of block approaches to optimal control problems can be found in [9]. A recent block approach can be found in [11].…”

Section: Introductionmentioning

confidence: 99%