Whole field computation using Monte Carlo method

Abstract

Monte Carlo methods are generally known for solving field problems one point at a time, unlike other numerical methods such as the finite difference and finite element methods which provide the solution at all the grid nodes simultaneously. This paper provides a Monte Carlo technique for obtaining the solution everywhere at once. The technique uses absorbing Markov chains to obtain the transition probabilities for all of the grid nodes at once. The procedure is illustrated with some examples for homogeneous and inhomogeneous, rectangular and axisymmetric solution regions, and is found to be accurate. © 1997 John Wiley & Sons, Ltd.

Markov Chain Monte Carlo Solution of Laplace’s Equation in Axisymmetric Homogeneous Domain

“…The shrinking boundary and the inscribed figure methods later proposed for whole-field calculations are not significantly superior to the classical Monte Carlo methods [27] [28]. To address this gap, Markov Chains for whole-field computations was proposed by Andrey Markov [29] [30]. The applications of MCMC to rectangular and axisymmetric problems are presented in [29] [31].…”

Solution of Axisymmetric Inhomogeneous Problems with the Markov Chain Monte Carlo

“…Later, the shrinking boundary and inscribed figure methods were proposed for whole-field calculation but they still offered no significant advantage over the conventional Monte Carlo techniques [16]- [17]. Andrey Markov proposed the Markov Chains method that proved to be more efficient than shrinking boundary and inscribed figure methods for whole field computations [18]- [19]. The method is simple, accurate and robust in terms of implementation.…”

“…The method is simple, accurate and robust in terms of implementation. The Markov Chain Monte Carlo (MCMC) method involves no use of random number generator and thus not subject to randomness and the approach is potentially accurate [19]. Hence, the MCMC method is generally preferred for whole field computation.…”

Markov Chain Monte Carlo Solution of Poisson’s Equation

“…This method has been used for whole field computation for problems involving Laplace's equations [7][8][9]. This paper extends the application of MCMCM to problems involving Poisson's equation.…”