First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

Gross, B. J.; Kuberry, P.; Atzberger, P. J.

doi:10.1016/j.jcp.2021.110932

Title: First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

Journal Article · Thu Jan 06 00:00:00 EST 2022 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2021.110932· OSTI ID:1882875

Gross, B. J. ^[1]; Kuberry, P. ^[2];

^[1]

Univ. of California, Santa Barbara, CA (United States)
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $$d\mb{X}_t = a(\mb{X}_t)dt + \mb{b}(\mb{X}_t)d\mb{W}_t$$. We consider on a surface domain $$\Omega$$ the statistics $$u(\mb{x}) = \mathbb{E}^{\mb{x}}\left[\int_0^\tau g(X_t)dt \right] + \mathbb{E}^{\mb{x}}\left[f(X_\tau)\right]$$ with the exit stopping time $$\tau = \inf_t \{t > 0 \; |\; X_t \not\in \Omega\}$$. Using Dynkin's formula, we compute statistics by developing high-order Generalized Moving Least Squares (GMLS) solvers for the associated surface PDE boundary-value problems. We focus particularly on the mean First Passage Times (FPTs) given by the special case $$f = 0,\, g = 1$$ with $$u(\mb{x}) = \mathbb{E}^{\mb{x}}\left[\tau\right]$$. We perform studies for a variety of shapes showing our methods converge with high-order accuracy both in capturing the geometry and the surface PDE solutions. We then perform studies showing how FPTs are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.

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Research Organization:: Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of California, Santa Barbara, CA (United States)

Sponsoring Organization:: USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)

Grant/Contract Number:: NA0003525; SC0019246; DMS-1616353

OSTI ID:: 1882875

Alternate ID(s):: OSTI ID: 2331464

Report Number(s):: SAND-2022-0041J; 702770; TRN: US2307942

Journal Information:: Journal of Computational Physics, Vol. 453; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

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Title: First-passage time statistics on surfaces of general shape: Surface PDE solvers using Generalized Moving Least Squares (GMLS)

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