Identification of the diffusion parameter in nonlocal steady diffusion problems

TL;DR

This paper addresses the inverse problem of identifying diffusion parameters in nonlocal steady diffusion equations using an optimal control framework, with theoretical analysis and numerical validation.

Contribution

It introduces a variational formulation for the inverse problem using nonlocal calculus and provides finite element discretization with error estimates.

Findings

Existence of at least one optimal solution for certain kernels.

Finite element method achieves reliable approximation with error bounds.

Nonlocal models can estimate discontinuous diffusion parameters.

Abstract

The problem of identifying the diffusion parameter appearing in a nonlocal steady diffusion equation is considered. The identification problem is formulated as an optimal control problem having a matching functional as the objective of the control and the parameter function as the control variable. The analysis makes use of a nonlocal vector calculus that allows one to define a variational formulation of the nonlocal problem. In a manner analogous to the local partial differential equations counterpart, we demonstrate, for certain kernel functions, the existence of at least one optimal solution in the space of admissible parameters. We introduce a Galerkin finite element discretization of the optimal control problem and derive a priori error estimates for the approximate state and control variables. Using one-dimensional numerical experiments, we illustrate the theoretical results and show that by using nonlocal models it is possible to estimate non-smooth and discontinuous diffusion parameters.