An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifte... more An implementation of GMRES with multiple preconditioners (MPGMRES) is proposed for solving shifted linear systems with shift-and-invert preconditioners. With this type of preconditioner, the Krylov subspace can be built without requiring the matrix-vector product with the shifted matrix. Furthermore, the multipreconditioned search space is shown to grow only linearly with the number of preconditioners. This allows for a more efficient implementation of the algorithm. The proposed implementation is tested on shifted systems that arise in computational hydrology and the evaluation of different matrix functions. The numerical results indicate the effectiveness of the proposed approach.
Abstract: There are several ideas being used today for Web information retrieval, and specificall... more Abstract: There are several ideas being used today for Web information retrieval, and specifically in Web search engines. The PageRank algorithm is one of those that introduce a content-neutral ranking function over Web pages. This ranking is applied to the set of pages returned by the Google search engine in response to posting a search query. PageRank is based in part on two simple common sense concepts:(i) A page is important if many important pages include links to it.(ii) A page containing many links has reduced impact on ...
The e ect of a threshold variant TPABLO of the permutation (and partitioning) algorithm PABLO on ... more The e ect of a threshold variant TPABLO of the permutation (and partitioning) algorithm PABLO on the performance of certain preconditionings is explored. The goal of these permutations is to produce matrices with dense diagonal blocks, and in the threshold variant, with large entries in the diagonal blocks. Experiments are reported using matrices arising from the discretization of elliptic partial di erential equations. The iterative solvers used are GMRES, QMR, BiCGStab and CGNR. The preconditioners are di erent incomplete factorizations. It is shown that preprocessing the matrices with TPABLO has a positive e ect on the overall performance, resulting in better convergence rates for highly nonsymmetric problems.
The Schwarz method can be used for the iterative solution of elliptic boundary value problems on ... more The Schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain Ω. One subdivides Ω into smaller, more manageable, subdomains and solves the differential equation in these subdomains using appropriate boundary conditions. Schwarz-Robin methods use Robin conditions on the artificial interfaces for information exchange at each iteration. Optimized Schwarz Methods (OSM) are those in which one optimizes the Robin parameters. While the convergence theory of classical Schwarz methods (with Dirichlet conditions on the artificial interface) is well understood, the overlapping Schwarz-Robin methods still lack a complete theory. In this paper, an abstract Hilbert space version of the OSM is presented, together with an analysis of conditions for its convergence. It is also shown that if the overlap is relatively uniform, these convergence conditions are met for Schwarz-Robin methods for two-dimensional elliptic problems, for any positive Robin parameter. In the discrete setting, we obtain that the convergence rate ω(h) varies like a polylogarithm of h. Numerical experiments show that the methods work well and that the convergence rate does not appear to depend on h.
A number of conjugate gradient methods are considered for a class of linear systems of real algeb... more A number of conjugate gradient methods are considered for a class of linear systems of real algebraic equations. This class includes all symmetric and certain special nonsymmetric problems, which give rise to three-term recursions. All the algorithms are characterized variationally. This makes it possible to derive error estimates systematically in terms of certain polynomial approximation problems. Bounds are obtained, which are functions of the extreme eigenvalues of the basic iteration operator.
Comparison Theorems Of General Stationary Iterative Methods For Singular Matrices
... ARTICLE{Marek00comparisontheorems, author = {Ivo Marek and ... 13, Numerical solutions of spa... more ... ARTICLE{Marek00comparisontheorems, author = {Ivo Marek and ... 13, Numerical solutions of sparse singular systems of equations arising from ergodic Markov chains - Barker. 13, Frobenius theory of positive operators; comparison theorems and applications - Marek - 1970. ...
Nous formulons plusieurs modèles algorithmiques et mathématiques pour des itérations asynchrones,... more Nous formulons plusieurs modèles algorithmiques et mathématiques pour des itérations asynchrones, y inclus le modèle des communications flexibles de Miellou, El Baz et Spitéri. Nouś etablissons la convergence de ces itérations dans le cadre d'une décomposition "deuxétapes" pour des systèmes linéaires. ABSTRACT. We formulate different computational and mathematical models for asynchronous parallel iterations, including the recent flexible communication model by Miellou, El Baz and Spitéri. Within this framework we establish the convergence of asynchronoustwo-stage schemes for solving linear systems.
A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its ... more A matrix is said to have the Perron-Frobenius property (strong Perron-Frobenius property) if its spectral radius is an eigenvalue (a simple positive and strictly dominant eigenvalue) with a corresponding semipositive (positive) eigenvector. It is known that a matrix A with the Perron-Frobenius property can always be the limit of a sequence of matrices A(ε) with the strong Perron-Frobenius property such that A − A(ε) ≤ ε. In this note we study the form that the parameterized matrices A(ε) and their spectral characteristics can take. We show that it is possible to have A(ε) cubic, its spectral radius quadratic and the corresponding positive eigenvector linear (all as functions of ε); further, if the spectral radius of A is simple, positive and strictly dominant, then A(ε) can be taken to be quadratic and its spectral radius linear (in ε). Two other cases are discussed: when A is normal it is shown that the sequence of approximating matrices A(ε) can be written as a quadratic polynomial in trigonometric functions, and when A has semipositive left and right Perron-Frobenius eigenvectors and ρ(A) is simple, the sequence A(ε) can be represented as a polynomial in trigonometric functions of degree at most six.
A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue wi... more A matrix is said to have the Perron-Frobenius property if its spectral radius is an eigenvalue with a corresponding nonnegative eigenvector. Matrices having this and similar properties are studied in this paper as generalizations of nonnegative matrices. Sets consisting of such generalized nonnegative matrices are studied and certain topological aspects such as connectedness and closure are proved. Similarity transformations leaving such sets invariant are completely described, and it is shown that a nonnilpotent matrix eventually capturing the Perron-Frobenius property is in fact a matrix that already has it.
Numerical experiments are presented whereby the e ect of reorderings on the convergence of precon... more Numerical experiments are presented whereby the e ect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are di erent variants of incomplete factorizations. It is shown that reorderings for direct methods, such as Reverse Cuthill-McKee and Minimum Degree, can be very bene cial. The bene t can be seen in the reduction of the number of iterations and also in measuring the appropriate deviation of the preconditioned operator from the identity.
Numerical experiments are presented whereby the effect of reorderings on the convergence of preco... more Numerical experiments are presented whereby the effect of reorderings on the convergence of preconditioned Krylov subspace methods for the solution of nonsymmetric linear systems is shown. The preconditioners used in this study are different variants of incomplete factorizations. It is shown that certain reorderings for direct methods, such as Reverse Cuthill-McKee and Minimum Degree, can be very beneficial. The benefit can be seen in the reduction of the number of iterations and also in measuring the appropriate deviation of the preconditioned operator from the identity.
The e ect of a threshold variant TPABLO of the permutation (and partitioning) algorithm PABLO on ... more The e ect of a threshold variant TPABLO of the permutation (and partitioning) algorithm PABLO on the performance of certain preconditionings is explored. The goal of these permutations is to produce matrices with dense diagonal blocks, and in the threshold variant, with large entries in the diagonal blocks. Experiments are reported using matrices arising from the discretization of elliptic partial di erential equations. The iterative solvers used are GMRES, QMR, BiCGStab and CGNR. The preconditioners are di erent incomplete factorizations. It is shown that preprocessing the matrices with TPABLO has a positive e ect on the overall performance, resulting in better convergence rates for highly nonsymmetric problems.
A matrix is said to have the Perron-Frobenius property if it has a positive dominant eigenvalue t... more A matrix is said to have the Perron-Frobenius property if it has a positive dominant eigenvalue that corresponds to a nonnegative eigenvector. Matrices having this and similar properties are studied in this paper. Characterizations of collections of such matrices are given in terms of the spectral projector. Some combinatorial, spectral, and topological properties of such matrices are presented, and the similarity transformations preserving the Perron-Frobenius property are completely described. In addition, certain results associated with nonnegative matrices are extended to matrices having the Perron-Frobenius property.
Electronic transactions on numerical analysis ETNA
A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems o... more A class of asynchronous Schwarz methods for the parallel solution of nonsingular linear systems of the form Ax = f is investigated. This class includes, in particular, an asynchronous algebraic Schwarz method as well as asynchronous multisplitting. Theorems are obtained demonstrating convergence for the cases when A ,1 is nonnegative and when A is an H-matrix. The results shown are for both the situations with or without overlap between the domains in which an underlying mesh is divided, if such a mesh exists. Numerical experiments on systems of up to over ten million variables on up to 256 processors are presented. They illustrate the convergence properties of the method, as well as the fact that when the domains are not all of the same size, the asynchronous method can be up to 50% faster than the corresponding synchronous one.
Many advances in the development of Krylov subspace methods for the iterative solution of linear ... more Many advances in the development of Krylov subspace methods for the iterative solution of linear systems during the last decade and a half are reviewed. These new developments include different versions of restarted, augmented, deflated, flexible, nested, and inexact methods. Also reviewed are methods specifically tailored to systems with special properties such as special forms of symmetry and those depending on one or more parameters.
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Papers by Daniel B Szyld