Probabilistic potential theory applied to electrical engineering problems

Analysis of time-dependent cylindrical problems using the exodus method

“…Suppose we are interested in solving the problem defined in eqs. (1) and (2), and the potential at a specific point (ρ o ,t o ), is to be determined. We define the transition probabilities p k as the probability that a random walk starting at the point of…”

“…MCM is often used to solve Laplace's and Poisson's equations [1][2][3][4], which are elliptic partial differential equations. They are hardly applied in solving parabolic and hyperbolic partial differential equations.…”

Three-dimensional subsurface delineation via a novel method for determining the subsurface electrical profile

“…P(r + a A) + p(r -a" x A |"f) P(r" -a" A) +... + pCf -s A| ?) P(7 -j A), (3) in which [p(r *-a Ajr>], for example, is the conditional probability of jumping to O given that the particle is at r + a x A. Equation devaluates P (7) as the sum of the probabilities of reaching surface S. at some future time by the various routes through the six neighboring points around 7.…”

A unified solution of Laplace's equation in an arbitrary region

“…Other techniques based on stochastic processes have also been reported. Among these are the Monte Carlo Method (MCM) [5][6][7], the Multiple-Paths Method (MPM) [8], and the exodus method [9]. These methods are based on a random walk that may be fi xed, fl oating, on a rectangular grid, [7] or a random path [8].…”