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

TL;DR

This paper introduces a stochastic method for efficiently estimating Green's functions in linear advection-diffusion-reaction problems, enabling high-accuracy solutions across arbitrary geometries for various applications.

Contribution

The paper presents a novel stochastic approach to construct approximate Green's functions applicable to complex geometries, improving upon traditional Monte Carlo methods for transport problems.

Findings

Enables construction of high-accuracy Green's functions in arbitrary geometries.

Allows generation of multiple solutions for different boundary and forcing conditions.

Facilitates high-fidelity modeling for inverse problems and process optimization.

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.