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

• The technique bypasses technically challenging derivation of stochastic representative solutions underlying Monte Carlo solution of initial boundary value problems. • In contrast to MC solutions, the technique estimates non-problem-specific impulse-response functions, i.e., Green's functions, which can be used to solve IBVP's subject to an infinite array of boundary and internal forcing conditions. • A random walker swarm is launched from a chosen response (or impulse) point, well-developed RWboundary interaction models can be implemented, and a simple estimator is used to estimate the transition density - and corresponding Green's function - between the response/impulse point and the impulse/response point. • Deleterious depletion of RW's at absorbing Dirichlet boundaries is solved using a RW respawning algorithm, while spatial variance in estimated GF's, produced by our naĩve transition density estimator, is minimized using a simple area-averaging technique. • The proposed technique is tested against three problems having known Green's functions. • The paper is presented in recipe fashion, and is aimed at non-specialists. 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. [ABSTRACT FROM AUTHOR]