Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

4 Issues per year


CiteScore 2016: 0.70

SCImago Journal Rank (SJR) 2016: 0.647
Source Normalized Impact per Paper (SNIP) 2016: 0.908

Mathematical Citation Quotient (MCQ) 2016: 0.33

Online
ISSN
1569-3961
See all formats and pricing
More options …
Volume 16, Issue 3-4

Issues

Stochastic iterative projection methods for large linear systems

Karl Sabelfeld
  • Institute of Computational Mathematics and Mathem. Geophysics, Russian Acad. Sci., Lavrentieva str., 6, 630090 Novosibirsk, Russia. E-mail:
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Nadja Loshchina
Published Online: 2010-11-18 | DOI: https://doi.org/10.1515/mcma.2010.020

Abstract

We suggest a randomized version of the projection methods belonging to the class of a “row-action” methods which work well both for systems with quadratic nonsingular matrices and for overdetermined systems. These methods belong to a type known as Projection on Convex Sets methods. Here we present a method beyond the conventional Markov chain based Neumann–Ulam scheme. The main idea is in a random choice of blocks of rows in the projection method so that in average, the convergence is improved compared to the conventional periodic choice of the rows. We suggest an acceleration of the row projection method by using the Johnson–Lindenstrauss (J–L) theorem to find, among the randomly chosen rows, in a sense an optimal row. We extend this randomized method for solving linear systems coupled with systems of linear inequalities. Applied to finite-difference approximations of boundary value problems, the method appears to be an extremely efficient Random Walk algorithm whose convergence is exponential, and the cost does not depend on the dimension of the matrix. In addition, the algorithm calculates the solution in all grid points, and is easily parallelizable.

Keywords.: Random projections; random row-action; Kaczmarz's method; random sparsification; overdetermined systems; Johnson–Lindenstrauss theorem

About the article

Received: 2009-09-15

Revised: 2009-12-10

Published Online: 2010-11-18

Published in Print: 2010-12-01


Citation Information: Monte Carlo Methods and Applications, Volume 16, Issue 3-4, Pages 343–359, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma.2010.020.

Export Citation

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[3]
K.K. Sabelfeld
Engineering Analysis with Boundary Elements, 2012, Volume 36, Number 7, Page 1092

Comments (0)

Please log in or register to comment.
Log in