Conditional well-posedness and error bounds for the groundwater inverse problem

Abstract

Inverse problems involving elliptic partial differential equations have many physical applications. In 1993, Knowles & Wallace published a variational method of numerical differentiation, involving minimization of an energy functional formed from the solution of an associated elliptic equation. This method has been successfully used to estimate the values of the parameters of the groundwater flow equation. The primary focus of this dissertation will be a proof of the stability of this inverse problem. Additionally, I will present theoretical error bounds on the data surface that is constructed for the parameter recovery and results of testing a new numerical algorithm.

Keywords: Inverse problems, conditional well-posedness, parameter estimation, error bounds, groundwater