Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms

Some fast elliptic solvers on parallel architectures and their complexities

The discretization of separable elliptic partial differential equations leads to linear systems with special block triangular matrices. Several methods are known to solve these systems, the most general of which is the Block Cyclic Reduction (BCR) algorithm which handles equations with nonconsistant coefficients. A method was recently proposed to parallelize and vectorize BCR. Here, the mapping of BCR on distributed memory architectures is discussed, and its complexity is compared with that of other approaches, including the Alternating-Direction method. A fast parallel solver is also described, based on an explicit formula for the solution, which has parallel computational complexity lower than that of parallel BCR

On the parallel solution of parabolic equations

Parallel algorithms for the solution of linear parabolic problems are proposed. The first of these methods is based on using polynomial approximation to the exponential. It does not require solving any linear systems and is highly parallelizable. The two other methods proposed are based on Pade and Chebyshev approximations to the matrix exponential. The parallelization of these methods is achieved by using partial fraction decomposition techniques to solve the resulting systems and thus offers the potential for increased time parallelism in time dependent problems. Experimental results from the Alliant FX/8 and the Cray Y-MP/832 vector multiprocessors are also presented

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: $10^{10}$ pages with $10^{11}$ nonzero elements and $10^{12}$ 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

Multidamping simulation framework for link-based ranking

We review methods for the approximate computation of PageRank. Standard methods are based on the eigenvector and linear system characterizations. Our starting point are recent methods based on series representation whose coefficients are damping functions, for example Linear Rank, HyperRank and TotalRank, etc. We propose a multidamping framework for interpreting PageRank and these methods. Multidamping is based on some new useful properties of Google type matrices. The approach can be generalized and could help in the exploration of new approximations for list-based ranking. This is joint work with Georgios Kollias and is supported by a Pythagoras-EPEAEK-II grant

A nested Krylov subspace method to compute the sign function of large complex matrices

We present an acceleration of the well-established Krylov-Ritz methods to compute the sign function of large complex matrices, as needed in lattice QCD simulations involving the overlap Dirac operator at both zero and nonzero baryon density. Krylov-Ritz methods approximate the sign function using a projection on a Krylov subspace. To achieve a high accuracy this subspace must be taken quite large, which makes the method too costly. The new idea is to make a further projection on an even smaller, nested Krylov subspace. If additionally an intermediate preconditioning step is applied, this projection can be performed without affecting the accuracy of the approximation, and a substantial gain in efficiency is achieved for both Hermitian and non-Hermitian matrices. The numerical efficiency of the method is demonstrated on lattice configurations of sizes ranging from 4^4 to 10^4, and the new results are compared with those obtained with rational approximation methods.Comment: 17 pages, 12 figures, minor corrections, extended analysis of the preconditioning ste

An iterative method to compute the overlap Dirac operator at nonzero chemical potential

The overlap Dirac operator at nonzero quark chemical potential involves the computation of the sign function of a non-Hermitian matrix. In this talk we present an iterative method, first proposed by us in Ref. [1], which allows for an efficient computation of the operator, even on large lattices. The starting point is a Krylov subspace approximation, based on the Arnoldi algorithm, for the evaluation of a generic matrix function. The efficiency of this method is spoiled when the matrix has eigenvalues close to a function discontinuity. To cure this, a small number of critical eigenvectors are added to the Krylov subspace, and two different deflation schemes are proposed in this augmented subspace. The ensuing method is then applied to the sign function of the overlap Dirac operator, for two different lattice sizes. The sign function has a discontinuity along the imaginary axis, and the numerical results show how deflation dramatically improves the efficiency of the method.Comment: 7 pages, talk presented at the XXV International Symposium on Lattice Field Theory, July 30 - August 4 2007, Regensburg, German

Dialkyldithiophosphate Acids (HDDPs) as Effective Lubricants of Sol–Gel Titania Coatings in Technical Dry Friction Conditions

The goal of this study was the investigation of the effectiveness of dialkyldithiophosphate acids (HDDPs) films in improving the tribological properties of thin, sol– gel derived titania coatings. Amorphous, anatase, and rutile titania coatings were obtained using sol–gel dip–coating deposition after treatment at 100, 500, and 1,000 C, respectively. Titania coatings were then modified from the liquid phase by HDDPs acids having dodecyl-(C12), tetradecyl-(C14), and hexadecyl-(C16) alkyl chains deposited by dip–coating (DC) and Langmuir–Blodgett (LB) methods. The influence of the deposition procedure, the length of the HDDPs alkyl chain and the type of titania substrate on the surface morphology and tribological properties were studied. It was found, using wetting contact angle measurements, that these modifications of titania coatings decrease the surface free energy and increase its hydrophobicity. The surface topography imaged by Atomic force microscopy (AFM), exhibit island-like or agglomerate features for the DC deposition method, while smooth topographies were observed for LB depositions. Tribological tests were conducted by means of a microtribometer operating in the normal load range 30–100 mN. An enhancement of tribological properties was observed upon modification, as compared to unmodified titania