Stochastic boundary collocation and spectral methods for solving PDEs

Sabelfeld Karl () and Mozartova Nadezhda ()
Additional contact information
Sabelfeld Karl: Institute of Computational Mathematics and Mathematical Geophysics, Russian Acad. Sci., Lavrentieva str., 6, 630090 Novosibirsk, Russia
Mozartova Nadezhda: Novosibirsk State University, Pirogova str., 2, 630090 Novosibirsk, Russia

Monte Carlo Methods and Applications, 2012, vol. 18, issue 3, 217-263

Abstract: We develop a stochastic boundary method (SBM) which can be considered as a randomized version of the method of fundamental solutions (MFS). We suggest solving the large system of linear equations for the weights in the expansion over the fundamental solutions by a randomized SVD method introduced by Sabelfeld and Mozartova (2011). In addition, we also deal with solving inhomogeneous problems where we use the integral representation through the Green integral formula. The relevant volume integrals are calculated by a Monte Carlo integration technique which uses the symmetry of the Green function. We construct also a stochastic boundary method based on the spectral inversion of the Poisson formula representing the solution in a disc. This is done for the Laplace equation, and the system of elasticity equations. We stress that the stochastic boundary method proposed is of high generality; it can be applied to any bounded and unbounded domain with any boundary condition provided the existence and uniqueness of the solution are proven. We present a series of numerical results which illustrate the performance of the suggested methods.

Keywords: Method of fundamental solutions; randomized SVD; low rank approximations; PCA; spectral expansions; Poisson integral formula; Lamé equation (search for similar items in EconPapers)
Date: 2012
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/mcma-2012-0008 (text/html)
For access to full text, subscription to the journal or payment for the individual article is required.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bpj:mcmeap:v:18:y:2012:i:3:p:217-263:n:2

Ordering information: This journal article can be ordered from
https://www.degruyter.com/journal/key/mcma/html

DOI: 10.1515/mcma-2012-0008

Access Statistics for this article

Monte Carlo Methods and Applications is currently edited by Karl K. Sabelfeld

More articles in Monte Carlo Methods and Applications from De Gruyter
Bibliographic data for series maintained by Peter Golla ().