Sensitivity Calculations for the Poisson's Equation via the Adjoint Field Method

TL;DR

This paper develops a unified, explicit method for calculating sensitivities in Poisson's equation problems using adjoint fields, explicitly considering boundary conditions and linking gradient and Gauss-Newton methods.

Contribution

It provides a detailed, adaptable derivation of adjoint sensitivity methods for Poisson's equation, including boundary condition considerations, unifying approaches for inverse problems.

Findings

Explicit derivation of adjoint sensitivities for Poisson's equation

Clarification of boundary condition roles in sensitivity calculations

Connection between gradient descent and Gauss-Newton sensitivities

Abstract

Adjoint field methods are both elegant and efficient for calculating sensitivity information required across a wide range of physics-based inverse problems. Here we provide a unified approach to the derivation of such methods for problems whose physics are provided by Poisson's equation. Unlike existing approaches in the literature, we consider in detail and explicitly the role of general boundary conditions in the derivation of the associated adjoint field-based sensitivities. We highlight the relationship between the adjoint field computations required for both gradient decent and Gauss-Newton approaches to image formation. Our derivation is based on standard results from vector calculus coupled with transparent manipulation of the underlying partial different equations thereby making the concepts employed here easily adaptable to other systems of interest.