Analysis of iterative        methods  for saddle point problems: a unified approach

Zulehner, Walter

doi:10.1090/s0025-5718-01-01324-2

Cited by 157 publications

(122 citation statements)

References 14 publications

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“…Then one may check that Here we just reproduce well known results (e.g, [32,36]). However, the quantitative estimates provided in Theorem 4.3 often improve significantly previous bounds.…”

supporting

confidence: 58%

“…Here we offer a complementary viewpoint, giving estimates that vary continuously in function of the main parameters (1.3), (1.4), without any restriction on these latter. Moreover, whereas we reproduce the condition µ ≥ 1 to have only real (and positive) eigenvalues with Uzawa [36] or block triangular [32] preconditioners, our analysis also reveals that scaling M A to satisfy this condition has often an adverse effect on the clustering of the eigenvalues.…”

Section: 2)mentioning

confidence: 73%

“…There are quite many works developing eigenvalue analyzes for these types of preconditioner; see [5,10] for block diagonal preconditioners, [7,32] for block triangular preconditioners, [8,15,36] for inexact Uzawa preconditioners, and [2,3,36] for block approximate factorization preconditioners -to mention just a few; we refer to [4] for many more references and historical remarks.…”

Section: 2)mentioning

confidence: 99%

“…Some previous analyzes focus on the conditions to have the preconditioned matrix positive definite in a non standard inner product, and develop related conjugate gradient like methods; see, e.g., [11,28,36]. Here we offer a complementary viewpoint, giving estimates that vary continuously in function of the main parameters (1.3), (1.4), without any restriction on these latter.…”

Section: 2)mentioning

confidence: 99%

“…Now, it is clear the the scaling of M A plays an important role for block triangular and Uzawa preconditioners. With µ ≤ 1 , we have the appealing result of Corollary 4.5, but, on the other hand, if one rescales the preconditioner for A to have µ ≥ 1 , all eigenvalues are real which may also be attractive, allowing to use conjugate gradient methods in nonstandard inner products [11,28,36]. To investigate this, we rescaled the algebraic multigrid preconditioner by a factor α , entailing that µ ≈ 0.4 α and µ = α .…”

mentioning

confidence: 99%

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A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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We consider symmetric saddle point matrices. We analyze block preconditioners based on the knowledge of a good approximation for both the top left block and the Schur complement resulting from its elimination. We obtain bounds on the eigenvalues of the preconditioned matrix that depend only of the quality of these approximations, as measured by the related condition numbers. Our analysis applies to indefinite block diagonal preconditioners, block triangular preconditioners, inexact Uzawa preconditioners, block approximate factorization preconditioners, and a further enhancement of these latter based on symmetric block Gauss-Seidel type iterations. The analysis is unified and allows the comparison of these different approaches. In particular, it reveals that block triangular and inexact Uzawa preconditioners lead to identical eigenvalue distributions. These theoretical results are illustrated on the discrete Stokes problem. It turns out that the provided bounds allow to localize accurately both real and non real eigenvalues. The relative quality of the different types of preconditioners is also as expected from the theory.

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“…Then one may check that Here we just reproduce well known results (e.g, [32,36]). However, the quantitative estimates provided in Theorem 4.3 often improve significantly previous bounds.…”

supporting

confidence: 58%

Section: 2)mentioning

confidence: 73%

Section: 2)mentioning

confidence: 99%

Section: 2)mentioning

confidence: 99%

mentioning

confidence: 99%

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A New Analysis of Block Preconditioners for Saddle Point Problems

Notay

2014

SIAM J. Matrix Anal. & Appl.

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Symmetric coupling of finite and boundary elements for exterior magnetic field problems

Kuhn

Steinbach

2002

Math Methods in App Sciences

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SUMMARYWe consider a coupled ÿnite element (fe)-boundary element (be) approach for three-dimensional magnetic ÿeld problems. The formulation is based on a vector potential in a bounded domain (fe) and a scalar potential in an unbounded domain (be). We describe a coupled variational problem yielding a unique solution where the constraints in the trial spaces are replaced by appropriate side conditions. Then we discuss a Galerkin discretization of the coupled problem and prove a quasi-optimal error estimate. Finally we discuss an e cient preconditioned iterative solution strategy for the resulting linear system.

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Combination preconditioning of saddle point systems for positive definiteness

Pestana

Wathen

2012

Numerical Linear Algebra App

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Amongst recent contributions to preconditioning methods for saddle point systems, standard iterative methods in nonstandard inner products have been usefully employed. Krzyzanowski ( ˙ Numer. Linear Algebra Appl. 2011; 18:123–140) identified a two-parameter family of preconditioners in this context and Stoll and Wathen (SIAM J. Matrix Anal. Appl. 2008; 30:582–608) introduced combination preconditioning, where two preconditioners, self-adjoint with respect to different inner products, can lead to further preconditioners and associated bilinear forms or inner products. Preconditioners that render the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to a nonstandard inner product always allow a MINREStype method (W-PMINRES) to be applied in the relevant inner product. If the preconditioned matrix is also positive definite with respect to the inner product a more efficient CG-like method (W-PCG) can be reliably used. We establish eigenvalue expressions for Krzyzanowski preconditioners and show that for a specific ˙ choice of parameters, although the Krzyzanowski preconditioned saddle point matrix is self-adjoint with ˙ respect to an inner product, it is never positive definite. We provide explicit expressions for the combination of certain preconditioners and prove the rather counterintuitive result that the combination of two specific preconditioners for which only W-PMINRES can be reliably used leads to a preconditioner for which, for certain parameter choices, W-PCG is reliably applicable. That is, combining two indefinite preconditioners can lead to a positive definite preconditioner. This combination preconditioner outperforms either of the two preconditioners from which it is formed for a number of test problems

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Analysis of iterative methods for saddle point problems: a unified approach

Cited by 157 publications

References 14 publications

A New Analysis of Block Preconditioners for Saddle Point Problems

A New Analysis of Block Preconditioners for Saddle Point Problems

Symmetric coupling of finite and boundary elements for exterior magnetic field problems

Combination preconditioning of saddle point systems for positive definiteness

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