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

Journal Article · 05 January 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]; Atzberger, P. J.  ^[1]

---

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

Here, we develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics dX_t = a(X_t)dt + b(X_t )dW_t . We formulate descriptions of Brownian motion and general drift-diffusion processes on surfaces. We consider statistics of the form u(x) = E^x [$$∫^{τ}_{0}$$ g(X_t)dt] + E^x [f (X_τ)] for a domain Ω and the exit stopping time τ = inf_t {t>0 | X_iΩ}, where f , g are general smooth functions. For computing these statistics, we develop high-order Generalized Moving Least Squares (GMLS) solvers for associated surface PDE boundary-value problems based on Backward- Kolmogorov equations. We focus particularly on the mean First Passage Times (FPTs) given by the case f = 0, g = 1 where u(x) = E^x [τ]. 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 statistics are influenced by the surface geometry, drift dynamics, and spatially dependent diffusivities.

• Research Organization: Sandia National Laboratories (SNL-NM), Albuquerque, NM (United States)
• Sponsoring Organization: USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
• Grant/Contract Number: NA0003525; SC0019246
• OSTI ID: 1882875
• Report Number(s): SAND--2022-0041J; 702770
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 453; ISSN 0021-9991
• Publisher: ElsevierCopyright Statement
• Country of Publication: United States
• Language: English