144 research outputs found
Scharz Preconditioners for Krylov Methods: Theory and Practice
Several numerical methods were produced and analyzed. The main thrust of the work relates to inexact Krylov subspace methods for the solution of linear systems of equations arising from the discretization of partial di#11;erential equa- tions. These are iterative methods, i.e., where an approximation is obtained and at each step. Usually, a matrix-vector product is needed at each iteration. In the inexact methods, this product (or the application of a preconditioner) can be done inexactly. Schwarz methods, based on domain decompositions, are excellent preconditioners for thise systems. We contributed towards their under- standing from an algebraic point of view, developed new ones, and studied their performance in the inexact setting. We also worked on combinatorial problems to help de#12;ne the algebraic partition of the domains, with the needed overlap, as well as PDE-constraint optimization using the above-mentioned inexact Krylov subspace methods
Asynchronous iterative computations with Web information retrieval structures: The PageRank case
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 importance of the
pages it links to. In this paper we focus on asynchronous iterative schemes to
compute PageRank over large sets of Web pages. The elimination of the
synchronizing phases is expected to be advantageous on heterogeneous platforms.
The motivation for a possible move to such large scale distributed platforms
lies in the size of matrices representing Web structure. In orders of
magnitude: pages with nonzero elements and bytes
just to store a small percentage of the Web (the already crawled); distributed
memory machines are necessary for such computations. The present research is
part of our general objective, to explore the potential of asynchronous
computational models as an underlying framework for very large scale
computations over the Grid. The area of ``internet algorithmics'' appears to
offer many occasions for computations of unprecedent dimensionality that would
be good candidates for this framework.Comment: 8 pages to appear at ParCo2005 Conference Proceeding
Preconditioned Chebyshev BiCG for parameterized linear systems
We consider the problem of approximating the solution to
for many different values of the parameter . Here we assume is
large, sparse, and nonsingular with a nonlinear dependence on . Our method
is based on a companion linearization derived from an accurate Chebyshev
interpolation of on the interval , . The
solution to the linearization is approximated in a preconditioned BiCG setting
for shifted systems, where the Krylov basis matrix is formed once. This process
leads to a short-term recurrence method, where one execution of the algorithm
produces the approximation to for many different values of the
parameter simultaneously. In particular, this work proposes
one algorithm which applies a shift-and-invert preconditioner exactly as well
as an algorithm which applies the preconditioner inexactly. The competitiveness
of the algorithms are illustrated with large-scale problems arising from a
finite element discretization of a Helmholtz equation with parameterized
material coefficient. The software used in the simulations is publicly
available online, and thus all our experiments are reproducible
07071 Abstracts Collection -- Web Information Retrieval and Linear Algebra Algorithms
From 12th to 16th February 2007, the Dagstuhl Seminar 07071 ``Web Information Retrieval and Linear Algebra Algorithms\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Newton additive and multiplicative Schwarz iterative methods
Convergence properties are presented for Newton additive and multiplicative Schwarz (AS and MS) iterative methods for the solution of nonlinear systems in several variables. These methods consist of approximate solutions of the linear Newton step using either AS or MS iterations, where overlap between subdomains can be used. Restricted versions of these methods are also considered. These Schwarz methods
can also be used to precondition a Krylov subspace method for the solution of the linear Newton
steps. Numerical experiments on parallel computers are presented, indicating the effectiveness of these
methods.The Spanish Ministry of Science and Education (TIN2005-09037-C02-02); Universidad de Alicante
(VIGROB-020); the U.S. Department of Energy (DE-FG02-05ER25672)
The effect of non-optimal bases on the convergence of Krylov subspace methods
There are many examples where non-orthogonality of a basis for Krylov subspace methods arises naturally. These methods usually require less storage or computational effort per iteration than methods using an orthonormal basis (optimal methods), but the convergence may be delayed. Truncated Krylov subspace methods and other examples of non-optimal methods have been shown to converge in many situations, often with small delay, but not in others. We explore the question of what is the effect of having a nonoptimal basis. We prove certain identities for the relative residual gap, i.e., the relative difference between the residuals of the optimal and non-optimal methods. These identities and related bounds provide insight into when the delay is small and convergence is achieved. Further understanding is gained by using a general theory of superlinear convergence recently developed. Our analysis confirms the observed fact that in exact arithmetic the orthogonality of the basis is not important, only the need to maintain linear independence is. Numerical examples illustrate our theoretical results
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