Continuous Path Brownian Trajectories for Diffusion Monte Carlo via First- and Last-Passage Distributions

Given, James A.; Mascagni, Michael; Hwang, Chi‐Ok

doi:10.1007/3-540-45346-6_4

Cited by 7 publications

(4 citation statements)

References 22 publications

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“…His method also allows for variants of the algorithm in which the sphere sizes do not need to be optimally inscribed in D. Later, (Sabelfeld and Talay, 1995) gives an elementary proof of the |log ε| bound. Mascagni and co-authors have extensively developed the walk-on-spheres algorithm in applications; see for example (Given, Hwang, and Mascagni, 2002;Given, Mascagni, and Hwang, 2001;Hwang, Mascagni, and Given, 2003;Mackoy et al, 2013).…”

Section: The Classical Settingmentioning

confidence: 99%

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

Kyprianou

Osojnik

Shardlow

2017

IMA Journal of Numerical Analysis

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We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called 'walk-on-spheres' algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the 'walk-on-spheres' comes within ε > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic α-stable process with α ∈ (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.issued from x ∈ D, then, by the strong law of large numbers,For practical purposes, since it is impossible to take the limit, one truncates the series of estimates for large n and the central limit theorem gives O(1/n) upper bounds on the variance of the n-term sum, which serves as a numerical error estimate. Although forming the fundamental basis of most Monte Carlo methods for diffusive Dirichlettype problems, (1.3) is an inefficient numerical approach. Least of all, this is because the Monte Carlo simulation of u(x) is independent for each x ∈ D. Moreover, it is unclear how exactly to simulate the path of a Brownian motion on its first exit from D, that is to say, the quantity W τ D . This is because of the fractal properties of Brownian motion, making its path difficult to simulate. This introduces additional numerical errors over and above that of Monte Carlo simulation.A method proposed by (Muller, 1956), for the case that D is convex, sub-samples special points along the path of Brownian motion to the boundary of the domain D. The method does not require a complete simulation of its path and takes advantage of the distributional symmetry of Brownian motion. In order to describe the so-called 'walk-on-spheres', we need to first introduce some notation. We may thus set ρ 0 = x for x ∈ D and define r 1 to be the radius of the largest sphere inscribed in D that is centred at x. This sphere we will call S 1 = {y ∈ R d : |y − ρ 0 | = r 1 }. To avoid special cases, we henceforth assume that the surface area of S 1 ∩ ∂D is zero (this excludes, for example, the case that x = 0 and D is a sphere centred at the origin). Now set ρ 1 ∈ D to be a point uniformly distributed on S 1 and note that, given the assumption in the previous sentence, P x (ρ 1 ∈ ∂D) = 0. Construct the remainder of the sequence (ρ n , n ≥ 1) inductively. Given ρ n−1 , we define the radius, r n , of the largest sphere inscribed in D that is centred at ρ n−1 . Calling this sphere S n , we have that ...

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Section: The Classical Settingmentioning

confidence: 99%

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

Kyprianou

Osojnik

Shardlow

2017

IMA Journal of Numerical Analysis

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“…In many cases, simple Brownian dynamics simulations can be substantially refined, making it possible to use the walk on spheres (WOS) [8] and the Green's function first passage (GFFP) Monte Carlo methods [9]. Elimination of the WOS bias due to the boundary with GFFP, the simulation-tabulation technique [10], and last passage variants of these Monte Carlo algorithms [11] further extend the capabilities of stochastic computational methods when applied to electrostatics problems. The progress achieved so far stimulated our investigation of the "random walk on the boundary algorithm" [12] as applied to capacitance and charge density calculations.…”

Section: Introductionmentioning

confidence: 99%

The random walk on the boundary method for calculating capacitance

Mascagni

Simonov

2004

Journal of Computational Physics

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“…The Green's function first passage Monte Carlo method is the natural extension of WOS. The simulation-tabulation technique [9], and last passage variants of the algorithm [7], further extended the capabilities of stochastic computational methods when applied to solving electrostatics problems.…”

Section: Introductionmentioning

confidence: 99%

Monte Carlo Method for Calculating the Electrostatic Energy of a Molecule

Mascagni

Simonov

2003

Lecture Notes in Computer Science

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Abstract. The problem of computing the electrostatic energy of a large molecule is considered. It is reduced to solving the Poisson equation inside and the linear Poisson-Boltzmann equation in the exterior, coupled by boundary conditions. A Monte Carlo estimate for the potential point values, their derivatives, and the energy is constructed. The estimate is based on the walk on spheres and Green's function first passage algorithms; the walk in subdomains technique; and finite-difference approximations of the boundary condition. Results of some illustrative calculations are presented.

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Continuous Path Brownian Trajectories for Diffusion Monte Carlo via First- and Last-Passage Distributions

Cited by 7 publications

References 22 publications

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

Unbiased ‘walk-on-spheres’ Monte Carlo methods for the fractional Laplacian

The random walk on the boundary method for calculating capacitance

Monte Carlo Method for Calculating the Electrostatic Energy of a Molecule

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