Related papers: A Modified Walk-on-sphere Method for High Dimensio…
In this work, we develop a probabilistic estimator for the voltage-to-current map arising in electrical impedance tomography. This novel so-called partially reflecting random walk on spheres estimator enables Monte Carlo methods to compute…
The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of L\'evy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined…
This paper presents a novel formula for the transition density of the Brownian motion on a sphere of any dimension and discusses an algorithm for the simulation of the increments of the spherical Brownian motion based on this formula. The…
A well known connection between first-passage probability of random walk and distribution of electrical potential described by Laplace equation is studied. We simulate random walk in the plane numerically as a discrete time process with…
We investigate the use of randomized quasi-Monte Carlo (RQMC) in walk on spheres algorithms to solve boundary value problems for functions with Dirichlet boundary conditions in $\mathbb{R}^d$. For harmonic functions with $d=2$, the…
A new method is presented to obtain a closed form of the generalized Green function to the Poisson and the Helmholtz equations on the $n$-dimensional unit sphere.
We give a extensive account of a recent new way of applying the Dirichlet form theory to random Poisson measures. The main application is to obtain existence of density for thelaws of random functionals of L\'evy processes or solutions of…
We consider random walks on the surface of the sphere $S_{n-1}$ ($n \geq 2$) of the $n$-dimensional Euclidean space $E_n$, in short a hypersphere. By solving the diffusion equation in $S_{n-1}$ we show that the usual law $<r^2 > \varpropto…
We consider a random walk model in a one-dimensional environment, formed by several zones of finite width with the fixed transition probabilities. It is also assumed that the transitions to the left and right neighboring points have unequal…
We study the convergence of the Method of Reflections for the Dirichlet problem of the Poisson and the Stokes equations in perforated domains which consist in the exterior of balls. We prove that the method converges if the balls are…
Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$…
Grid-free Monte Carlo methods such as walk on spheres can be used to solve elliptic partial differential equations without mesh generation or global solves. However, such methods independently estimate the solution at every point, and hence…
In this paper a method is presented for evaluating the convolution of the Green's function for the Laplace operator with a specified function $\rho(\vec x)$ at all grid points in a rectangular domain $\Omega \subset {\mathrm R}^{d}$ ($d =…
The fractional Poisson process (FPP) generalizes the standard Poisson process by replacing exponentially distributed return times with Mittag-Leffler distributed ones with an extra tail parameter, allowing for greater flexibility. The FPP…
We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time…
We propose an efficient numerical approach to simulate the boundary local time of reflected Brownian motion, as well as the time and position of the associated reaction event on a smooth boundary of a Euclidean domain. This approach…
In Boundary Element Method, Green's function with no boundary conditions is used for solving Laplace's equation with Dirichlet boundary condition. To determine the gradient of solution on the boundary, we need to solve the boundary integral…
The following document presents some novel numerical methods valid for one and several variables, which using the fractional derivative, allow to find solutions for some non-linear systems in the complex space using real initial conditions.…
We investigate the properties of some recently developed variable-order differential operators involving order transition functions of exponential type. Since the characterisation of such operators is performed in the Laplace domain it is…
A new scheme is proposed to construct an n-times differentiable function extension of an n-times differentiable function defined on a smooth domain D in d-dimensions. The extension scheme relies on an explicit formula consisting of a linear…