Stochastic estimation of Green’s functions with application to diffusion and advection-diffusion-reaction problems

Abstract

A stochastic method is described for estimating Green’s functions (GF’s), appropriate to linear advection-diffusion-reaction transport problems, evolving in arbitrary geometries. By allowing straightforward construction of approximate, though high-accuracy GF’s, within any geometry, the technique solves the central challenge in obtaining Green’s function solutions. In contrast to Monte Carlo solutions of individual transport problems, subject to specific sets of conditions and forcing, the proposed technique produces approximate GF’s that can be used: a) to obtain (infinite) sets of solutions, subject to any combination of (random and deterministic) boundary, initial, and internal forcing, b) as high fidelity direct models in inverse problems, and c) as high quality process models in thermal and mass transport design, optimization, and process control problems. The technique exploits an equivalence between the adjoint problem governing the transport problem Green’s function, G(x,t|x′,t′), and the backward Kolmogorov problem governing the transition density, p(x,t|x′,t′), of the stochastic process used in Green’s function construction. We address nonspecialists and report four contributions. First, a recipe is outlined for diagnosing when stochastic Green’s function estimation can be used, and for subsequently estimating the transition density and associated Green’s function. Second, a naive estimator for the transition density is proposed and tested. Third, Green’s function estimation error produced by random walker absorption at Dirichlet boundaries is suppressed using a simple random walker splitting technique. Last, spatial discontinuity in estimated GF’s, produced by the naive estimator, is suppressed using a simple area averaging method. The paper provides guidance on choosing key numerical parameters, and the technique is tested against two simple unsteady, linear heat conduction problems, and an unsteady groundwater dispersion problem, each having known, exact GF’s.