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317
Preconditioning stochastic Galerkin saddle point systems
 SIAM J. Matrix Anal. Appl
"... Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDE ..."
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Cited by 110 (4 self)
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Abstract. Mixed finite element discretizations of deterministic secondorder elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic secondorder elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are blockdense and the cost of a matrix vector product is nontrivial. We implement a stochastic Galerkin discretization for the steadystate diffusion problem written as a mixed firstorder system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of blockdiagonal preconditioners of Schur complement and augmented type, for use with minres. By introducing socalled Kronecker product preconditioners we improve the robustness of cheap, meanbased preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.
Split Bregman methods and frame based image restoration, submitted
, 2009
"... Abstract. Split Bregman methods introduced in [47] have been demonstrated to be efficient tools to solve total variation (TV) norm minimization problems, which arise from partial differential equation based image restoration such as image denoising and magnetic resonance imaging (MRI) reconstruction ..."
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Cited by 103 (27 self)
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Abstract. Split Bregman methods introduced in [47] have been demonstrated to be efficient tools to solve total variation (TV) norm minimization problems, which arise from partial differential equation based image restoration such as image denoising and magnetic resonance imaging (MRI) reconstruction from sparse samples. In this paper, we prove the convergence of the split Bregman iterations, where the number of inner iterations is fixed to be one. Furthermore, we show that these split Bregman iterations can be used to solve minimization problems arising from the analysis based approach for image restoration in the literature. We apply these split Bregman iterations to the analysis based image restoration approach whose analysis operator is derived from tight framelets constructed in [59]. This gives a set of new frame based image restoration algorithms that cover several topics in image restorations, such as image denoising, deblurring, inpainting and cartoontexture image decomposition. Several numerical simulation results are provided. 1. Introduction. Image
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. SCI. COMPUT
, 2006
"... We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel mult ..."
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Cited by 68 (19 self)
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We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach.
Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations
, 1002
"... We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied ..."
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Cited by 41 (10 self)
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We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied to various finite element and MAC discretizations of the Oseen problem in two and three space dimensions. Both steady and unsteady problems are considered. Numerical experiments show the effectiveness of the proposed preconditioners for a wide range of problem parameters. Implementation on parallel architectures is also considered. The augmented Lagrangianbased approach is further generalized to deal with linear systems from stabilized finite element discretizations. Copyright c ○ 2000 John Wiley & Sons, Ltd. key words: preconditioning; saddle point problems; Oseen problem; augmented Lagrangian method; Krylov subspace methods; parallel computing 1.
An aggregationbased algebraic multigrid method
, 2008
"... An algebraic multigrid (AMG) method is presented to solve large systems of linear equations. The coarsening is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the nu ..."
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Cited by 40 (9 self)
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An algebraic multigrid (AMG) method is presented to solve large systems of linear equations. The coarsening is obtained by aggregation of the unknowns. The aggregation scheme uses two passes of a pairwise matching algorithm applied to the matrix graph, resulting in most cases in a decrease of the number of variables by a factor slightly less than four. The matching algorithm favors the strongest negative coupling(s), inducing a problem dependant coarsening. This aggregation is combined with piecewise constant (unsmoothed) prolongation, ensuring low setup cost and memory requirements. Compared with previous aggregationbased multigrid methods, the scalability is enhanced by using a socalled Kcycle multigrid scheme, providing Krylov subspace acceleration at each level. Numerical results on second order discrete scalar elliptic PDEs indicate that the proposed method may be significantly more robust than the classical AMG method as implemented in the code AMG1R5 by K. Stüben. The parallel implementation is also discussed. Satisfactory speedups are obtained on a 24 nodes processors cluster with relatively high communication latency, providing that the number of unknowns per processor is kept significant.
PRECONDITIONING DISCRETIZATIONS OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS
, 2009
"... 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 approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this approach a number of examples will be c ..."
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Cited by 25 (4 self)
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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 approach to the construction of preconditioners for unbounded saddle point problems. To illustrate this 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 examples and models which have been discussed in the literature previously. However, here each example is discussed with reference to a more unified abstract approach.
Weighted matchings for preconditioning symmetric indefinite linear systems
 SIAM J. Sci. Comput
, 2006
"... Abstract. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for inco ..."
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Cited by 24 (6 self)
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Abstract. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete LDL T factorizations. The reorderings are constructed such that the matched entries form 1 × 1or2 × 2 diagonal blocks in order to increase the diagonal dominance of the system. During the incomplete factorization only tridiagonal pivoting is used. We report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices, ranging from nonlinear elasticity to interior point optimization.
Uniform preconditioners for a parameter dependent saddle point problem with application to generalized Stokes interface equations
, 2006
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A block Newton method for nonlinear eigenvalue problems
 Numer. Math
"... We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the develo ..."
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Cited by 21 (7 self)
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We consider matrix eigenvalue problems that are nonlinear in the eigenvalue parameter. One of the most fundamental differences from the linear case is that distinct eigenvalues may have linearly dependent eigenvectors or even share the same eigenvector. This has been a severe hindrance in the development of general numerical schemes for computing several eigenvalues of a nonlinear eigenvalue problem, either simultaneously or subsequently. The purpose of this work is to show that the concept of invariant pairs offers a way of representing eigenvalues and eigenvectors that is insensitive to this phenomenon. To demonstrate the use of this concept in the development of numerical methods, we have developed a novel block Newton method for computing such invariant pairs. Algorithmic aspects of this method are considered and a few academic examples demonstrate its viability. 1