TY - JOUR

T1 - Random walk methods for Monte Carlo simulations of Brownian diffusion on a sphere

AU - Novikov, A.

AU - Kuzmin, D.

AU - Ahmadi, O.

N1 - Funding Information:
The work of Omid Ahmadi and Dmitri Kuzmin was supported by the German Research Association ( DFG ) under grant KU 1530/13-1 . The work of Alexei Novikov was supported by the US NSF grants DMS-1515187 and DMS-1813943 .

PY - 2020/1/1

Y1 - 2020/1/1

N2 - This paper is focused on efficient Monte Carlo simulations of Brownian diffusion effects in particle-based numerical methods for solving transport equations on a sphere (or a circle). Using the heat equation as a model problem, random walks are designed to emulate the action of the Laplace–Beltrami operator without evolving or reconstructing the probability density function. The intensity of perturbations is fitted to the value of the rotary diffusion coefficient in the deterministic model. Simplified forms of Brownian motion generators are derived for rotated reference frames, and several practical approaches to generating random walks on a sphere are discussed. The alternatives considered in this work include projections of Cartesian random walks, as well as polar random walks on the tangential plane. In addition, we explore the possibility of using look-up tables for the exact cumulative probability of perturbations. Numerical studies are performed to assess the practical utility of the methods under investigation.

AB - This paper is focused on efficient Monte Carlo simulations of Brownian diffusion effects in particle-based numerical methods for solving transport equations on a sphere (or a circle). Using the heat equation as a model problem, random walks are designed to emulate the action of the Laplace–Beltrami operator without evolving or reconstructing the probability density function. The intensity of perturbations is fitted to the value of the rotary diffusion coefficient in the deterministic model. Simplified forms of Brownian motion generators are derived for rotated reference frames, and several practical approaches to generating random walks on a sphere are discussed. The alternatives considered in this work include projections of Cartesian random walks, as well as polar random walks on the tangential plane. In addition, we explore the possibility of using look-up tables for the exact cumulative probability of perturbations. Numerical studies are performed to assess the practical utility of the methods under investigation.

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U2 - 10.1016/j.amc.2019.124670

DO - 10.1016/j.amc.2019.124670

M3 - Article

AN - SCOPUS:85071247839

VL - 364

JO - Applied Mathematics and Computation

JF - Applied Mathematics and Computation

SN - 0096-3003

M1 - 124670

ER -