Random walk on ellipsoids method for solving elliptic and parabolic equations

Shalimova Irina () and Sabelfeld Karl K. ()
Additional contact information
Shalimova Irina: Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, Russia
Sabelfeld Karl K.: Institute of Computational Mathematics and Mathematical Geophysics, Russian Academy of Sciences, Novosibirsk, Russia

Monte Carlo Methods and Applications, 2020, vol. 26, issue 4, 335-353

Abstract: A Random Walk on Ellipsoids (RWE) algorithm is developed for solving a general class of elliptic equations involving second- and zero-order derivatives. Starting with elliptic equations with constant coefficients, we derive an integral equation which relates the solution in the center of an ellipsoid with the integral of the solution over an ellipsoid defined by the structure of the coefficients of the original differential equation. This integral relation is extended to parabolic equations where a first passage time distribution and survival probability are given in explicit forms. We suggest an efficient simulation method which implements the RWE algorithm by introducing a notion of a separation sphere. We prove that the logarithmic behavior of the mean number of steps for the RWS method remains true for the RWE algorithm. Finally we show how the developed RWE algorithm can be applied to solve elliptic and parabolic equations with variable coefficients. A series of supporting computer simulations are given.

Keywords: Elliptic equations; ellipsoidal mean value relation; random walk on ellipsoids; logarithmic mean number of steps; cost estimations (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://doi.org/10.1515/mcma-2020-2078 (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:26:y:2020:i:4:p:335-353:n:8

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

DOI: 10.1515/mcma-2020-2078

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 ().