A New Class of Grid-Free Monte Carlo Algorithms for Elliptic Boundary Value Problems
In this paper we consider the following mathematical model: an elliptic boundary value problem, where the partial differential equation contains advection, diffusion, and deposition parts. A Monte Carlo (MC) method to solve this equation uses a local integral representation by the Green's function and a random process called “Walks on Balls”(WOB). A new class of grid free MC algorithms for solving the above elliptic boundary value problem is suggested and studied. We prove that the integral transformation kernel can be taken as a transition density function in the Markov chain in the case when the deposition part is equal to zero. An acceptance-rejection (AR) and an inversetransformation methods are used to sample the next point in the Markov chain. An estimate for the efficiency of the AR method is obtained.
KeywordsDensity Function Markov Chain Monte Carlo Transition Density Elliptic Boundary
Unable to display preview. Download preview PDF.
- 1.Bitzadze, A.: Equations of the Mathematical Physics. Nauka, Moscow (1982)Google Scholar
- 3.Gurov, T., Withlock, P., Dimov, I.: A grid free Monte Carlo algorithm for solving elliptic boundary value problems. In: L. Vulkov, Waśniewski, J., Yalamov, P. (eds.): Numerical Analysis and its Applications. Lecture notes in Comp. Sci., Vol. 1988. Springer-Verlag (2001) 359–367.CrossRefGoogle Scholar
- 6.Ermakov, S., Nekrutkin, V., Sipin, A.: Random Processes for Solving Classical Equations of the Mathematical Physics. Nauka, Moscow (1984)Google Scholar
- 7.Mikhailov, V.: Partial Differential Equations. Nauka, Moscow (1983)Google Scholar
- 8.Miranda, C.: Equasioni alle dirivate parziali di tipo ellipttico. Springer-Verlag, Berlin (1955)Google Scholar
- 9.Sobol, I.: Monte Carlo Numerical Methods. Nauka, Moscow (1973)Google Scholar