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Probing Methods for Saddle-Point Problems
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Probing methods for saddle-point problems

Profile image of Eric de SturlerEric de Sturler

2006

Abstract

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddle-point problems. We consider matrices where the Schur complement has rapid decay over some graph known a priori. This occurs for many matrices arising from the discretization of systems of partial differential equations, and this graph is then related to the mesh. We propose the use of probing methods to approximate these Schur complements in preconditioners for saddle-point problems. We demonstrate these techniques for the block-diagonal and constraint preconditioners proposed by [Murphy, Golub and Wathen '00], [de Sturler and Liesen '04] and [Siefert and de Sturler '05]. However, these techniques are applicable to many other preconditioners as well. We discuss the implementation of probing methods, and we consider the application of those approximations in preconditioners for Navier-Stokes problems and metal deformation problems. Finally, we study eigenvalue clustering for the preconditioned matrices, and we present convergence and timing results for various problem sizes. These results demonstrate the effectiveness of the proposed preconditioners with probing-based approximate Schur complements.

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Related papers

Preconditioners for Generalized Saddle-Point Problems
Eric de Sturler

SIAM Journal on Numerical Analysis, 2006

We examine block-diagonal preconditioners and efficient variants of indefinite preconditioners for block two-by-two generalized saddle-point problems. We consider the general, nonsymmetric, nonsingular case. In particular, the (1,2) block need not equal the transposed (2,1) block. Our preconditioners arise from computationally efficient splittings of the (1,1) block. We provide analyses for the eigenvalue distributions and other properties of the preconditioned matrices. We extend the results of [de Sturler and Liesen 2003] to matrices with non-zero (2,2) block and to allow for the use of inexact Schur complements. To illustrate our eigenvalue bounds, we apply our analysis to a model Navier-Stokes problem, computing the bounds, comparing them to actual eigenvalue perturbations and examining the convergence behavior.

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Probing Methods for Generalized Saddle-Point Problems
Eric de Sturler

2005

Several Schur complement-based preconditioners have been proposed for solving (generalized) saddlepoint problems. We consider probing-based methods for approximating those Schur complements in the preconditioners of the type proposed by [Murphy, Golub and Wathen '00], [de Sturler and Liesen '03] and [Siefert and de Sturler '04]. This approach can be applied in similar preconditioners as well. We discuss the implementation of probing-based approximations to Schur complements. We consider the application of those approximations in preconditioners for Navier-Stokes problems and metal deformation problems. Finally, we present eigenvalue clustering for the preconditioned matrices, and convergence and timing results. These demonstrate the effectiveness of the proposed preconditioners with probing-based approximate Schur complements.

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An analysis of low-rank modifications of preconditioners for saddle point systems
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ℋ︁-Matrix Preconditioners for Symmetric Saddle-Point Systems from Meshfree Discretization
Sabine Le Borne

Numerical Linear Algebra with Applications, 2008

Meshfree methods are suitable for solving problems on irregular domains, avoiding the use of a mesh. To deal with the boundary conditions, we can use Lagrange multipliers and obtain a sparse, symmetric and indefinite system of saddle-point type. Many methods have been developed to solve the indefinite system. Previously, we presented an algebraic method to construct an LU-based preconditioner for the saddle-point system obtained by meshfree methods, which combines the multilevel clustering method with the H-matrix arithmetic. The corresponding preconditioner has both H-matrix and sparse matrix subblocks. In this paper we refine the above method and propose a way to construct a pure H-matrix preconditioner. We compare the new method with the old method, JOR and smoothed algebraic multigrid methods. The numerical results show that the new preconditioner outperforms the preconditioners based on the other methods.

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