Results 1  10
of
182
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. Sci. Comput
, 2006
"... Abstract. 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 no ..."
Abstract

Cited by 52 (22 self)
 Add to MetaCart
Abstract. 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. Key words. Navier–Stokes equations, finite element, iterative methods, multigrid, preconditioning AMS subject classifications. 65F10, 65N22, 65F50 DOI. 10.1137/050646421 1. Introduction. We consider the numerical solution of the steady Navier– Stokes equations governing the flow of a Newtonian, incompressible viscous fluid. Let Ω ⊂ R d (d =2,3) be a bounded, connected domain with a piecewise smooth
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 ..."
Abstract

Cited by 36 (9 self)
 Add to MetaCart
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
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 ..."
Abstract

Cited by 31 (12 self)
 Add to MetaCart
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.
A comparison of preconditioners for incompressible Navier–Stokes solvers
 International Journal for Numerical Methods in Fluids 2008; 57:1731–1751. DOI: 10.1002/fld.1684
"... We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve th ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
We consider solution methods for large systems of linear equations that arise from the finite element discretization of the incompressible Navier–Stokes equations. These systems are of the socalled saddle point type, which means that there is a large block of zeros on the main diagonal. To solve these types of systems efficiently, several block preconditioners have been published. These types of preconditioners require adaptation of standard finite element packages. The alternative is to apply a standard ILU preconditioner in combination with a suitable renumbering of unknowns. We introduce a reordering technique for the degrees of freedom that makes the application of ILU relatively fast. We compare the performance of this technique with some block preconditioners. The performance appears to depend on grid size, Reynolds number and quality of the mesh. For mediumsized problems, which are of practical interest, we show that the reordering technique is competitive with the block preconditioners. Its simple implementation makes it worthwhile to implement it in the standard finite element method software. Copyright q 2007
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 ..."
Abstract

Cited by 17 (4 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
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.
Inexact Constraint Preconditioners for Linear Systems Arising in Interior Point Methods
, 2005
"... Abstract. Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jaco ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
Abstract. Issues of indefinite preconditioning of reduced Newton systems arising in optimization with interior point methods are addressed in this paper. Constraint preconditioners have shown much promise in this context. However, there are situations in which an unfavorable sparsity pattern of Jacobian matrix may adversely affect the preconditioner and make its inverse representation unacceptably dense hence too expensive to be used in practice. A remedy to such situations is proposed in this paper. An approximate constraint preconditioner is considered in which sparse approximation of the Jacobian is used instead of the complete matrix. Spectral analysis of the preconditioned matrix is performed and bounds on its nonunit eigenvalues are provided. Preliminary computational results are encouraging. Keywords Interiorpoint methods, Iterative solvers, Preconditioners, Approximate Jacobian.
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 ..."
Abstract

Cited by 13 (6 self)
 Add to MetaCart
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
PRESSURE SCHUR COMPLEMENT PRECONDITIONERS FOR THE DISCRETE OSEEN PROBLEM ∗
"... Abstract. We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these app ..."
Abstract

Cited by 12 (1 self)
 Add to MetaCart
Abstract. We consider several preconditioners for the pressure Schur complement of the discrete steady Oseen problem. Two of the preconditioners are well known from the literature and the other is new. Supplemented with an appropriate approximate solve for an auxiliary velocity subproblem, these approaches give rise to a family of the block preconditioners for the matrix of the discrete Oseen system. In the paper we critically review possible advantages and difficulties of using various Schur complement preconditioners. We recall existing eigenvalue bounds for the preconditioned Schur complement and prove such for the newly proposed preconditioner. These bounds hold both for LBB stable and stabilized finite elements. Results of numerical experiments for several model twodimensional and threedimensional problems are presented. In the experiments we use LBB stable finite element methods on uniform triangular and tetrahedral meshes. One particular conclusion is that in spite of essential improvement in comparison with “simple ” scaled massmatrix preconditioners in certain cases, none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex (e.g., circulating) advection vector field.