Showing papers in "Numerical Linear Algebra With Applications in 2003"
TL;DR: A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts and the main result is a bound on the condition number based on inequalities involving the matrices of the preconditionser.
Abstract: A convergence theory is presented for a substructuring preconditioner based on constrained energy minimization concepts. The substructure spaces consist of local functions with zero values of the constraints, while the coarse space consists of minimal energy functions with the constraint values continuous across substructure interfaces. In applications, the constraints include values at corners and optionally averages on edges and faces. The preconditioner is reformulated as an additive Schwarz method and analysed by building on existing results for balancing domain decomposition. The main result is a bound on the condition number based on inequalities involving the matrices of the preconditioner. Estimates of the form C(1+log2(H/h)) are obtained under the standard assumptions of substructuring theory. Computational results demonstrating the performance of method are included. Published in 2003 by John Wiley & Sons, Ltd.
220 citations
TL;DR: A parallel version of the algebraic recursive multilevel solver (ARMS) is developed for distributed computing environments that adopts the general framework of distributed sparse matrices and relies on solving the resulting distributed Schur complement system.
Abstract: A parallel version of the algebraic recursive multilevel solver (ARMS) is developed for distributed computing environments. The method adopts the general framework of distributed sparse matrices and relies on solving the resulting distributed Schur complement system. Numerical experiments are presented which compare these approaches on regularly and irregularly structured problems. Copyright © 2003 John Wiley & Sons, Ltd.
139 citations
TL;DR: A novel technique for computing a sparse incomplete factorization of a general symmetric positive definite matrix A based on A‐orthogonalization, which results in a reliable solver for highly ill‐conditioned linear systems.
Abstract: We describe a novel technique for computing a sparse incomplete factorization of a general symmetric positive definite matrix A. The factorization is not based on the Cholesky algorithm (or Gaussian elimination), but on A-orthogonalization. Thus, the incomplete factorization always exists and can be computed without any diagonal modification. When used in conjunction with the conjugate gradient algorithm, the new preconditioner results in a reliable solver for highly ill-conditioned linear systems. Comparisons with other incomplete factorization techniques using challenging linear systems from structural analysis and solid mechanics problems are presented. Copyright © 2003 John Wiley & Sons, Ltd.
103 citations
TL;DR: Preconditioning methods for matrices on saddle point form, as typically arising in equality constrained optimization problems, are surveyed and special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and a preconditionser on the same saddle pointform as the given matrix.
Abstract: Preconditioning methods for matrices on saddle point form, as typically arising in equality constrained optimization problems, are surveyed. Special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and a preconditioner on the same saddle point form as the given matrix. Both methods result in eigenvalues with positive real parts and small or zero imaginary parts. The behaviour of the methods are illustrated by solving a regularized Stokes problem. Copyright © 2002 John Wiley & Sons, Ltd.
75 citations
TL;DR: The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function to be confirmed.
65 citations
TL;DR: This paper proves that the coarsening technique preserves the M matrix property, and gives several numerical examples illustrating the robustness of the method with respect to the variations in both the diffusion and convection coefficients.
Abstract: In this paper we propose a practical and robust multigrid method for convection–diffusion problems based on a new coarsening techniques for unstructured grids. The idea is to use a graph matching technique to define proper coarse subspaces. Such an approach is based on the graph corresponding to the stiffness matrix, and is purely algebraic. We prove that our coarsening technique preserves the M matrix property. We also give several numerical examples illustrating the robustness of the method with respect to the variations in both the diffusion and convection coefficients. Copyright © 2002 John Wiley & Sons, Ltd.
58 citations
TL;DR: The balancing domain decomposition method for mixed finite elements by Cowsar, Mandel, and Wheeler is extended to the case of mortar Mixed finite elements on non-matching multiblock grids, and quasi-optimal condition number bounds are derived.
Abstract: The balancing domain decomposition method for mixed finite elements by Cowsar, Mandel, and Wheeler is extended to the case of mortar mixed finite elements on non-matching multiblock grids. The algorithm involves an iterative solution of a mortar interface problem with one local Dirichlet solve and one local Neumann solve per subdomain on each iteration. A coarse solve is used to guarantee that the Neumann problems are consistent and to provide global exchange of information across subdomains. Quasi-optimal condition number bounds are derived, which are independent of the jump in coefficients between subdomains. Numerical experiments confirm the theoretical results. Copyright © 2002 John Wiley & Sons, Ltd.
52 citations
TL;DR: The Newton–Kantorovich method for the solution of the non-linear problem is studied, including the role of the best linear approximation as a starting point for this method.
Abstract: This paper is concerned with the development of a ‘best’ rank one transitive approximation to a general paired comparison matrix in a least-squares sense. A direct attack on the non-linear problem is frequently replaced by a sub-optimal linear problem and, here, the best procedure of this kind is obtained. The Newton–Kantorovich method for the solution of the non-linear problem is also studied, including the role of the best linear approximation as a starting point for this method. Numerical examples are included. Copyright © 2002 John Wiley & Sons, Ltd.
42 citations
TL;DR: Experimental results upon matrices with large size, arising from space discretization of 2D advection–diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques.
Abstract: In this paper we compare Krylov subspace methods with Faber series expansion for approximating the matrix exponential operator on large, sparse, non-symmetric matrices. We consider in particular the case of Chebyshev series, corresponding to an initial estimate of the spectrum of the matrix by a suitable ellipse. Experimental results upon matrices with large size, arising from space discretization of 2D advection–diffusion problems, demonstrate that the Chebyshev method can be an effective alternative to Krylov techniques. Copyright © 2002 John Wiley & Sons, Ltd.
36 citations
32 citations
TL;DR: The problem of generating a matrix A with specified eigen-pair, where A is a symmetric and anti-persymmetric matrix, is presented and an existence theorem is given and proved and an expression is provided for this nearest matrix.
Abstract: The problem of generating a matrix A with specified eigen-pair, where A is a symmetric and anti-persymmetric matrix, is presented. An existence theorem is given and proved. A general expression of such a matrix is provided. We denote the set of such matrices by En. The optimal approximation problem associated with En is discussed, that is: to find the nearest matrix to a given matrix A* by A∈En. The existence and uniqueness of the optimal approximation problem is proved and the expression is provided for this nearest matrix. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: The Laplace equation under mixed boundary conditions on a polygonal domain Ω is considered and regularity estimates in terms of Sobolev norms of fractional order for this type of problem are proved.
Abstract: We consider the Laplace equation under mixed boundary conditions on a polygonal domain Ω. Regularity estimates in terms of Sobolev norms of fractional order for this type of problem are proved. The analysis is based on new interpolation results and multilevel representation of norms on the Sobolev spaces Hα(Ω). The Fourier transform and the construction of extension operators to Sobolev spaces on ℝ2 are avoided in the proofs of the interpolation theorems. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: Optimal-order error estimates in some H1-equivalence norms are established for the proposed discontinuous finite element methods and an optimal- order error estimate is derived in the L2 norm for the symmetric formulation.
Abstract: A new finite element method is proposed and analysed for second order elliptic equations using discontinuous piecewise polynomials on a finite element partition consisting of general polygons. The new method is based on a stabilization of the well-known primal hybrid formulation by using some least-squares forms imposed on the boundary of each element. Two finite element schemes are presented. The first one is a non-symmetric formulation and is absolutely stable in the sense that no parameter selection is necessary for the scheme to converge. The second one is a symmetric formulation, but is conditionally stable in that a parameter has to be selected in order to have an optimal order of convergence.
Optimal-order error estimates in some H1-equivalence norms are established for the proposed discontinuous finite element methods. For the symmetric formulation, an optimal-order error estimate is also derived in the L2 norm. The new method features a finite element partition consisting of general polygons as opposed to triangles or quadrilaterals in the standard finite element Galerkin method. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: The results show greatly improved convergence rate when the transformation is applied for solving sample diffusion and elasticity problems and the cost grows and can get very high with the number of of nonzeros per row.
Abstract: SUMMARY This paper studies the use of a generalized hierarchical basis transformation at each level of a multilevel block factorization. The factorization may be used as a preconditioner to the conjugate gradient method, or the structure it sets up may be used to define a multigrid method. The basis transformation is performed with an averaged piecewise constant interpolant and is applicable to unstructured elliptic problems. The results show greatly improved convergence rate when the transformation is applied for solving sample diffusion and elasticity problems. The cost of the method, however, grows and can get very high with the number of of nonzeros per row. Copyright c ∞ 2002 John Wiley & Sons, Ltd.
TL;DR: An unstructured multigrid method that only attempts to interpolate in the directions of geometric smoothness is proposed, which can have much lower grid and operator complexities compared to AMG, leading to lower solve timings.
Abstract: For non-M-matrices, this paper proposes an unstructured multigrid method that only attempts to interpolate in the directions of geometric smoothness. These directions are determined by analysing samples of algebraically smooth error, e. Neighbouring grid points i and j are called smoothly coupled if ei and ej are consistently nearby in value. In addition, these differences may be used to define interpolation weights. These new ideas may be incorporated into the algebraic multigrid method. Test results show that the new method can have much lower grid and operator complexities compared to AMG, leading to lower solve timings. Published in 2003 by John Wiley & Sons, Ltd.
TL;DR: Examples of preconditioners for regular elliptic systems of partial differential equations based on the Schur complement of the symbol of the operator are examined to highlight the possibilities and some of the difficulties one may encounter with this approach.
Abstract: One successful approach in the design of solution methods for saddle-point problems requires the efficient solution of the associated Schur complement problem. In the case of problems arising from partial differential equations the factorization of the symbol of the operator can often suggest useful approximations for this problem. In this work we examine examples of preconditioners for regular elliptic systems of partial differential equations based on the Schur complement of the symbol of the operator and highlight the possibilities and some of the difficulties one may encounter with this approach. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: The theory of Faber polynomials is used for solving N-dimensional linear initial value problems and to approximate the evolution operator creating the so-called exponential integrators.
Abstract: In this paper we use the theory of Faber polynomials for solving N-dimensional linear initial value problems. In particular, we use Faber polynomials to approximate the evolution operator creating the so-called exponential integrators. We also provide a consistence and convergence analysis. Some tests where we compare our methods with some Krylov exponential integrators are finally shown. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: Multiscale methods are applied to the coupling of finite and boundary element methods to solve an exterior two-dimensional Laplacian to corroborate the theory of the present paper.
Abstract: We apply multiscale methods to the coupling of finite and boundary element methods to solve an exterior two-dimensional Laplacian. The matrices belonging to the boundary terms of the coupled FEM–BEM system are compressed by using biorthogonal wavelet bases developed from A. Cohen, I. Daubechies and J.-C. Feauveau (Comm. Proc. Appl. Math. 1992; 45:485). The coupling yields a linear equation system which corresponds to a saddle point problem. As favourable solver, the Bramble–Pasciak–CG (Math. Comp. 1988; 50:1) is utilized. A suitable preconditioner is developed by combining the BPX (Math. Comp. 1990; 55:1) with the wavelet preconditioning (Numer. Math. 1992; 63:315). Through numerical experiments we provide results which corroborate the theory of the present paper. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: This work proposes a new approach to overcome this performance bottleneck by coupling incomplete factorization with a selective inversion scheme to replace triangular solutions by scalable matrix–vector multiplications.
Abstract: SUMMARY 11 Consider the solution of large sparse symmetric positive denite linear systems using the preconditioned conjugate gradient method. On sequential architectures, incomplete Cholesky factorizations provide ef- 13 fective preconditioning for systems from a variety of application domains, some of which may have widely diering preconditioning requirements. However, incomplete factorization based preconditioners 15 are not considered suitable for multiprocessors. This is primarily because the triangular solution step required to apply the preconditioner (at each iteration) does not scale well due to the large latency of 17 inter-processor communication. We propose a new approach to overcome this performance bottleneck by coupling incomplete factorization with a selective inversion scheme to replace triangular solutions by 19 scalable matrix-vector multiplications. We discuss our algorithm, analyze its communication latency for model sparse linear systems, and provide empirical results on its performance and scalability. Copyright 21 ? 2003 John Wiley & Sons, Ltd.
TL;DR: Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, some algebraic formulations for the approximation of the smallest non-zero eigenpairs are analysed, assuming that a sparse basis for the null space is available.
Abstract: Given the generalized symmetric eigenvalue problem Ax=λMx, with A semidefinite and M definite, we analyse some algebraic formulations for the approximation of the smallest non-zero eigenpairs, assuming that a sparse basis for the null space is available. In particular, we consider the inexact version of the Shift-and-Invert Lanczos method, and we show that apparently different algebraic formulations provide the same approximation iterates, under some natural hypotheses. Our results suggest that alternative strategies need to be explored to really take advantage of the special problem setting, other than reformulating the algebraic problem. Experiments on a real application problem corroborate our theoretical findings. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: All preconditioners considered are very robust for the cases with the discontinuity ratio of 1000 across the interface, and numerical comparison of their performance on non‐matching grids is presented.
Abstract: We consider elliptic problems with discontinuous coefficients defined on a union of two polygonal subdomains. The problems are discretized by the finite element method on non-matching triangulation across the interface. The discrete problems are described by the mortar technique in the space with constraints (the mortar condition) and in the space without constraints using Lagrange multipliers. To solve the discrete problems Preconditioned conjugate gradient iterations are used with Neumann–Dirichlet and Neumann–Neumann preconditioners in the first case, and dual Neumann–Dirichlet and dual Neumann–Neumann (or FETI, the finite element tearing and interconnecting) in the second case. An analysis of convergence of all four of these preconditioners is given. Numerical comparison of their performance on non-matching grids is presented. The general observation is that all preconditioners considered are very robust for the cases with the discontinuity ratio of 1000 across the interface. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: The paper deals with nested two-level decompositions of 3D domains into tetrahedra and the corresponding spaces of continuous piecewise linear finite element (FE) functions.
Abstract: The paper deals with nested two-level decompositions of 3D domains into tetrahedra and the corresponding spaces of continuous piecewise linear finite element (FE) functions. The decomposition of the fine grid FE space into the coarse grid FE space and its complement can be used for the construction of various multi-level iterative methods and preconditioners, which are efficient for the finite element solution of boundary value problems (BVP) in the considered domains. The constant in the strengthened Cauchy–Bunyakowski–Schwarz (C.B.S.) inequality for the coarse grid FE space and its complement determines the efficiency of these multi-level methods and preconditioners. From this reason, this constant is investigated in this paper for the case of BVP with anisotropic Laplacian or elasticity operators. Special emphasis is given on getting universal estimates of the C.B.S. inequality constant, which are valid for all kinds of physical or discretization anisotropy. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: A numerical method is proposed of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems and seeks possible application of this algorithm for solving the linear systems that occur in continuation problems.
Abstract: We study the Lanczos method for solving symmetric linear systems with multiple right-hand sides. First, we propose a numerical method of implementing the Lanczos method, which can provide all approximations to the solution vectors of the remaining linear systems. We also seek possible application of this algorithm for solving the linear systems that occur in continuation problems. Sample numerical results are reported. Copyright ? 2002 John Wiley & Sons, Ltd.
TL;DR: This work proposes a SPD block LDLT preconditioner whose factorized form requires a smaller amount of memory than the original matrix and can be computed adaptively and combined in a multiplicative way.
Abstract: Large linear systems arising from the discretization of partial differential equations with finite differences or finite elements on structured grids in dimension d(d ⩾ 3) require efficient preconditioners. For a symmetric and positive definite (SPD) matrix, we propose a SPD block LDLT preconditioner whose factorized form requires a smaller amount of memory than the original matrix. Moreover, the computing time for the preconditioner solves is linear with respect to the number of unknowns. The preconditioner is built in d stages: in a first stage, we use the tangential filtering decomposition of Wittum et al. and obtain a preconditioner which remains rather difficult to factorize. Then, in a second stage, we apply tangential filtering decomposition recursively to the diagonal blocks of this first preconditioner. The final stage consists of factorizing exactly the blocks corresponding to one dimensional problems. Such preconditioners can also be computed adaptively and combined in a multiplicative way. A generic programming implementation is discussed and numerical tests are presented, in particular for problems with highly heterogeneous media. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: This work presents a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix Kh,k resulting from the h‐version finite element discretization of a special degenerated problem.
Abstract: From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p-version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix Kh,k resulting from the h-version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner Ch,k for the matrix Kh,k and show that the condition number of C Kh,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: In this article, a robust interpolation scheme for non-overlapping two-level domain decomposition methods applied to two-dimensional elliptic problems with discontinuous coefficients is proposed, which maintains good scalable convergence behavior even when the jumps in the coefficients are not aligned with subdomain interfaces.
Abstract: We propose a robust interpolation scheme for non-overlapping two-level domain decomposition methods applied to two-dimensional elliptic problems with discontinuous coefficients. This interpolation is used to design a preconditioner closely related to the BPS scheme proposed in [Bramble et al. (Math. Comput. 1986; 47(175):103)]. Through numerical experiments, we show on structured and unstructured finite element problems that the new preconditioning scheme reduces to the BPS method on smooth problems but outperforms it on problems with discontinuous coefficients. In particular it maintains good scalable convergence behaviour even when the jumps in the coefficients are not aligned with subdomain interfaces. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: The relationship between the author's operator trigonometry and convergence rates and other properties of important iterative methods is extended and a new interesting trigonometric preconditioning lemma is given.
Abstract: This paper extends the relationship between the author's operator trigonometry and convergence rates and other properties of important iterative methods. A new interesting trigonometric preconditioning lemma is given. The general relationship between domain decomposition methods and the operator trigonometry is established. A new basic conceptual link between sparse approximate inverse algorithms and the operator trigonometry is observed. A new underlying fundamental inherent trigonometry of the classical successive over-relaxation scheme is exposed. Some improved trigonometric interpretations of minimum residual schemes are mentioned. Copyright © 2002 John Wiley & Sons, Ltd.
TL;DR: A numerical validation method for verifying the accuracy of approximate solutions of saddle point matrix equations is presented and it is shown that preconditioning can be used to improve the error bounds.
Abstract: A numerical validation method for verifying the accuracy of approximate solutions of saddle point matrix equations is presented and analysed. The method only requires iterative solutions of two symmetric positive definite linear systems. Moreover, it is shown that preconditioning can be used to improve the error bounds. The method is illustrated by several examples derived from mixed finite element discretization of the Stokes equations. Preliminary numerical results indicate that the method is efficient. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: It is concluded that tests of a parallel algorithm which vary the method of partitions can provide constructive information regarding the robustness of the algorithm and guidance for modifying the algorithm or the choice of partitioning algorithm to make the overall computations more robust.
Abstract: We focus on the interplay between the choice of partition (problem decomposition) and the corresponding rate of convergence of parallel numerical algorithms. Using a specific algorithm, for which the numerics depend upon the partition, we demonstrate that the rate of convergence can depend strongly on the choice of the partition. This dependence is shown to be a function of the algorithm and of the choice of problem. Information gleaned from tests using various 2-way partitions leads to new partitions for which some degree of convergence robustness is exhibited. The incorporation of a known correction for approximate Schur complements into the original algorithm yields a modified parallel algorithm which numerical experiments indicate achieves robust convergence behaviour with respect to the choice of partition. We conclude that tests of a parallel algorithm which vary the method of partitioning can provide constructive information regarding the robustness of the algorithm and guidance for modifying the algorithm or the choice of partitioning algorithm to make the overall computations more robust. Copyright © 2003 John Wiley & Sons, Ltd.
TL;DR: It is obtained that condition numbers which are related to the coefficients of ergodicity improve as the authors pass from the entire chain to the chains associated with its Perron complements.
Abstract: For an n-state ergodic homogeneous Markov chain whose transition matrix is T∈ℝn,n, it has been shown by Meyer on the one hand and by Kirkland, Neumann, and Xu on the other hand that the stationary distribution vector and that the mean first passage matrix, respectively, can be computed by a divide and conquer parallel method from the Perron complements of T. This is possible due to the facts, shown by Meyer, that the Perron complements of T are themselves transition matrices for finite ergodic homogeneous Markov chains with fewer states and that their stationary distribution vectors are multiples of the corresponding subvectors of the stationary distribution vector of the entire chain.
Here we examine various questions concerning the stability of computing the stationary distribution vectors of the Perron complements and compare them with the stability of computing the stationary distribution vector for the entire chain. In particular, we obtain that condition numbers which are related to the coefficients of ergodicity improve as we pass from the entire chain to the chains associated with its Perron complements. Copyright © 2003 John Wiley & Sons, Ltd.