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

Abstract

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 ...

A Walk Outside Spheres for the fractional Laplacian: Fields and first eigenvalue

“…We have discussed Walk Outside Spheres for simulating the whole field rather than a point value of the solution u : D → R of Eq. (1.1), extending the algorithm of [KOS17]. By using the multilevel Monte Carlo algorithm, we improved substantially on a naive method based on independent sampling at vertices.…”

“…Walk On Spheres is a classical method for solving the Poisson problem −∆u = f on D, u = g on ∂D for a domain D ⊂ R d , boundary data g : ∂D → R and source term f : D → R. In [KOS17], the algorithm was extended to a Walk Outside Spheres (WOS) algorithm for the following problem for the fractional Laplacian: find u : D → R such that…”

“…[KOS17] provides a complete Monte Carlo algorithm for approximating u(x) for a single x ∈ D. It works by sampling the WOS process x n and the occupation density V 1 , and computing u(x) as a sample average. [KOS17] includes an implementation [OS17] and a study of the mean number of WOS steps required for the sampling of X(τ ). This paper addresses the problem of calculating the whole field u : D → R as an element of L 2 (D) and the leading eigenvalue of the fractional Laplacian.…”

“…In Section 2, we review the existence and regularity theory for Eq. (1.1) as well as the Walk Outside Spheres (WOS) algorithm from [KOS17]. Section 3 develops a more efficient algorithm for computing the solution u ∈ L 2 (D) using multilevel Monte Carlo.…”

What is the fractional Laplacian? A comparative review with new results

“…Hence, conditions prescribed merely on the boundary of Ω are not sufficient to describe the behavior of particles that are exiting the domain, and instead an exterior condition on the behavior of the process within R d \ Ω must be given to obtain a physically meaningful model. The relation to Lévy processes is more than just conceptual; we use a recent stochastic solution method, the walk-on-spheres algorithm of [22], to solve the Riesz fractional Poisson equation, and we use the resulting data in our comparisons.…”