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Register a new userFirst-Passage Time Statistics on Surfaces of General Shape: Surface PDE Solvers using Generalized Moving Least Squares (GMLS)
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Added by
Paul Atzberger
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We develop numerical methods for computing statistics of stochastic processes on surfaces of general shape with drift-diffusion dynamics $d{X}_t = a({X}_t)dt + {b}({X}_t)d{W}_t$. We consider on a surface domain $Omega$ the statistics $u(mathbf{x}) = mathbb{E}^{mathbf{x}}left[int_0^tau g(X_t)dt right] + mathbb{E}^{mathbf{x}}left[f(X_tau)right]$ with the exit stopping time $tau = inf_t {t > 0 ; |; X_t otin Omega}$. Using Dynkins 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 $u(mathbf{x}) = mathbb{E}^{mathbf{x}}left[tauright]$. 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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We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.
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