Monte Carlo approximations of the Neumann problem

Maire, Sylvain; Tanré, Étienne

doi:10.1515/mcma-2013-0010

Cited by 18 publications

(20 citation statements)

References 29 publications

(50 reference statements)

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“…We did not attempt to optimize the integration method (using a simple trapezoid rule), but there is clearly no advantage to be gained by using the direct representation of local time. Another disadvantage, more fully described in Reference , is that the choice of the delta function width is a bit delicate. We experimented with several and picked the best ones(Table ).…”

Section: Test Cases and Numerical Resultsmentioning

confidence: 99%

Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Raghavan¹,

Brady²

2020

Engineering Reports

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The Neumann problem, particularly for problems exterior to a bounding surface, is very useful in a number of situations. Those that are described by elliptic and parabolic partial differential equations are immediately amenable to probabilistic representations for the solutions, the numerical implementation of which has several advantages including parallelism and for obtaining solutions at specific desired points or subregions. Although these representations have been known for a long time to mathematicians, the numerical implementation of these seems to have been little used. The purpose of this note is to point out that a suitable characterization of the mathematical representation results in an efficient scheme and to give some numerical examples, including from our own work involving detailed modeling of drug transport in tissue.

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Section: Test Cases and Numerical Resultsmentioning

confidence: 99%

Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Raghavan¹,

Brady²

2020

Engineering Reports

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“…For the Poisson equation, the contribution of the source term might be computed as a conditional integral [19]. Moreover, the proper truncation of time period is unknown, though it is proven that the variance of the approximation increases linearly of T [13].…”

Section: Discussionmentioning

confidence: 99%

“…Other literatures [9][10] [13] [14] have also explored similar problems. Especially, in [13] schemes based on the WOS, Euler schemes and kinetic approximations are proposed to treat inhomogeneous Neumann problems. It turns out that the pointwise resolution is much harder due to the choice of the truncation of time.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

Zhou

Cai

2016

J Sci Comput

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In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman-Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorohod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres (WOS) method, approximation of the boundary local time is obtained and the Feynman-Kac formula is calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.

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“…Boundary conditions, in particular of Neumann and Robin types, pose another challenge to particle-based methods in general [e.g., 17, and the references therein], and to their use in hybrid simulations. (Even such a well-studied discrete model as Brownian motion appears to be rigorously analyzed and numerically implemented only on domains with reflecting boundaries [11] or, at steady state, for Neumann boundaries [15]; we are not aware of similar analyses of other discrete methods on bounded domains with general boundary conditions. )…”

Section: Introductionmentioning

confidence: 99%

On the use of reverse Brownian motion to accelerate hybrid simulations

Bakarji

Tartakovsky

2017

Journal of Computational Physics

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Multiscale and multiphysics simulations are two rapidly developing fields of scientific computing. Efficient coupling of continuum (deterministic or stochastic) constitutive solvers with their discrete (stochastic, particle-based) counterparts is a common challenge in both kinds of simulations. We focus on interfacial, tightly coupled simulations of diffusion that combine continuum and particle-based solvers. The latter employs the reverse Brownian motion (rBm), a Monte Carlo approach that allows one to enforce inhomogeneous Dirichlet, Neumann, or Robin boundary conditions and is trivially parallelizable. We discuss numerical approaches for improving the accuracy of rBm in the presence of inhomogeneous Neumann boundary conditions and alternative strategies for coupling the rBm solver with its continuum counterpart. Numerical experiments are used to investigate the convergence, stability, and computational efficiency of the proposed hybrid algorithm.

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Monte Carlo approximations of the Neumann problem

Cited by 18 publications

References 29 publications

Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

On the use of reverse Brownian motion to accelerate hybrid simulations

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