Finding the smallest eigenvalue by the inverse Monte Carlo method with refinement

Finding the smallest eigenvalue of a given square matrix A of order n is computationally very intensive problem. The most popular method for this problem is the Inverse Power Method which uses LU-decomposition and forward and backward solving of the factored system at every iteration step. An alternative to this method is the Resolvent Monte Carlo method which uses representation of the resolvent matrix [I -qA](-m) as a series and then performs Monte Carlo iterations (random walks) on the elements of the matrix. This leads to great savings in computations, but the method has many restrictions and a very slow convergence. In this paper we propose a method that includes fast Monte Carlo procedure for finding the inverse matrix, refinement procedure to improve approximation of the inverse if necessary, and Monte Carlo power iterations to compute the smallest eigenvalue. We provide not only theoretical estimations about accuracy and convergence but also results from numerical tests performed on a number of test matrices.

Robustness and applicability of Markov chain Monte Carlo algorithms for eigenvalue problems

In this paper we analyse applicability and robustness of Markov chain Monte Carlo algorithms for eigenvalue problems. We restrict our consideration to real symmetric matrices. Almost Optimal Monte Carlo (MAO) algorithms for solving eigenvalue problems are formulated. Results for the structure of both - systematic and probability error are presented. It is shown that the values of both errors can be controlled independently by different algorithmic parameters. The results present how the systematic error depends on the matrix spectrum. The analysis of the probability error is presented. It shows that the close (in some sense) the matrix under consideration is to the stochastic matrix the smaller is this error. Sufficient conditions for constructing robust and interpolation Monte Carlo algorithms are obtained. For stochastic matrices an interpolation Monte Carlo algorithm is constructed. A number of numerical tests for large symmetric dense matrices are performed in order to study experimentally the dependence of the systematic error from the structure of matrix spectrum. We also study how the probability error depends on the balancing of the matrix. (c) 2007 Elsevier Inc. All rights reserved.

Durability assessment of PbTe/II-IV infrared filters (Space Shuttle 1st LDEF)

Residual stress having been further reduced, selected infrared coatings composed of thin films of (PbTe/ ZnS (or ZnSe) can now be made which comply with the durability requirements of MIL-48616 whilst retaining transparency. Such improved durability is due to the sequence:- i) controlled deposition, followed by ii) immediate exposure to air, followed by iii) annealing in vacuo to relieve stress. (At the time of writing we assume the empiric procedure "exposure to air/annealing in vacuo" acts to relieve the inherent stresses of deposition). As part of their testing, representative sample filters prepared by the procedure are being assembled for the shuttle's 1st Long Duration Exposure Facility (to be placed in earth orbit for a considerable period and then recovered for analysis). The sample filters comprise various narrowband-designs to permit deduction of the constituent thin film optical properties. The Reading assembly also contains representative sample of the infrared crystals, glasses, thin-film absorbers and bulk absorbers, and samples of shorter-wavelength filters prepared similarly but made with Ge/SiO. Findings on durability and transparency after exposure will be reported.

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two-sided estimates for this term in a variety of situations.

Proceedings of the 1st European conference on disability, virtual reality and associated technologies (ECDVRAT 1996)

The proceedings of the conference