A Monte Carlo method for poisson's equation

DeLaurentis, John M.; Romero, Louis A.

doi:10.1016/0021-9991(90)90199-b

Cited by 39 publications

(37 citation statements)

References 6 publications

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“…This method is what we refer to as the simulation-tabulation method [1]. It should be noted that the cumulative radial distribution found here is the same as that used previously by other researchers [9], and the analytic form of the conditional cumulative angular distribution is the two-dimensional Poisson kernel [11]. The ith estimate of the solution to Poisson's equation is given by Here, x i j is the center of the circle of radius r i j .…”

Section: Numerical Experimentsmentioning

confidence: 70%

“…Thus, we use the standard WOS approach of "fattening" the boundary by to create a capture region that is used to terminate the walk [3]. The error associated with this approximation has been theoretically estimated in previous WOS studies [7,9] and with our GFFP method no -absorption layer is needed [1].…”

Section: Modified "Walk On Spheres"mentioning

confidence: 99%

“…We use as our domain, D, the unit disk minus the first quadrant, which was used in previous research by DeLaurentis and Romero [9] (See Fig. 1).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…In addition, this random-walk based approach has been extended to solve other, more complicated, partial differential equations including Poisson's equation, and the linearized Poisson-Boltzmann equation [3][4][5][6][7][8][9]. In WOS and our Green's function first-passage (GFFP) method [1], instead of using detailed Brownian trajectories inside the domain, discrete jumps are made using the first-passage probability distributions.…”

Section: Introductionmentioning

confidence: 99%

“…Instead of simulating the detailed Brownian trajectory, we use discrete WOS jumps together with an h-conditioned Green's function [11] to incorporate the source term in the Poisson equation. While different Green's functions have been used in previous work of others [3,5,9] for the source term in the Poisson's equation, the h-conditioned Green's function has never been used in this way. The h-conditioned Green's function gives the probability density distribution of a Brownian trajectory inside a ball during its passage from the ball's center to an exit point on the ball's boundary.…”

Section: Introductionmentioning

confidence: 99%

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A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Hwang

Mascagni

Given

2003

Mathematics and Computers in Simulation

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This study presents a Feynman-Kac path-integral implementation for solving the Dirichlet problem for Poisson's equation. The algorithm is a modified "walk on spheres" (WOS) that includes the Feynman-Kac path-integral contribution for the source term. In our approach, we use an h-conditioned Green's function instead of simulating Brownian trajectories in detail to implement this path-integral computation. The h-conditioned Green's function allows us to represent the integral of the right-hand-side function from the Poisson equation along Brownian paths as a volume integral with respect to a residence time density function: the h-conditioned Green's function. The h-conditioned Green's function allows us to solve the Poisson equation by simulating Brownian trajectories involving only large jumps, which is consistent with both WOS and our Green's function first-passage (GFFP) method [J. Comput. Phys. 174 (2001) 946]. As verification of the method, we tabulate the h-conditioned Green's function for Brownian motion starting at the center of the unit circle and making first-passage on the boundary of the circle, find an analytic expression fitting the h-conditioned Green's function, and provide results from a numerical experiment on a two-dimensional Poisson problem.

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Section: Numerical Experimentsmentioning

confidence: 70%

Section: Modified "Walk On Spheres"mentioning

confidence: 99%

“…We use as our domain, D, the unit disk minus the first quadrant, which was used in previous research by DeLaurentis and Romero [9] (See Fig. 1).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Hwang

Mascagni

Given

2003

Mathematics and Computers in Simulation

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Whole field computation using Monte Carlo method

Sadiku

Garcia

1997

Int. J. Numer. Model.

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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.

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Applications, Algorithms, and Software for Massively Parallel Computing

Benner

Harris

1991

At&T Technical Journal

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Sandia National Laboratories has developed massively parallel (MP) software that provides high absolute performance and nearly perfect parallel efficiencies. We have used this software on first-and second-generation 1K-processor hypercubes with more than thousand-fold parallel speedups and supercomputer performance. As we developed MP software, we made significant research advances in parallel numerical methods, such as parallel time stepping. The hardware diagnostic tools and parallel graphics algorithms that we have developed and provided to vendors can pinpoint hardware problems and visualize performance in thousand-processor ensembles. Our distributed-computing software will provide supercomputer performance from workstation networks. In collaboration with university and industry researchers in parallel computing, we have proven the concept of distributed parallel supercomputing with parallel computations on a network of VAX@ computers. (VAX is a registered trademark of Digital Equipment Corporation.)

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A Monte Carlo method for poisson's equation

Cited by 39 publications

References 6 publications

A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

A Feynman–Kac path-integral implementation for Poisson’s equation using an h-conditioned Green’s function

Whole field computation using Monte Carlo method

Applications, Algorithms, and Software for Massively Parallel Computing

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