Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems

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.

Nonlocal optimal design: A new perspective about the approximation of solutions in optimal design

“…To overcome some of those difficulties we propose a new model in which we replace the classical state equation and the cost functional by a model whose state equation and cost functional are written by means of nonlocal integrals. This seems to be a suitable way to approximate the solution (sometimes, relaxed solution) of the problem in the classical setting (see [19,20]). In any case, to undertake this strategy two issues are in order: existence and approximation results.…”

“…In particular, this manuscript will address the application of this formulation to a type of Optimal Design problems. [19,20] are meaningful references where this approach has been carried out. Up to our knowledge these are the only works where specific problems of optimal control have been treated under a nonlocal perspective.…”

Existence and approximation of nonlocal optimal design problems driven by parabolic equations

“…[23][24][25][26][27][28][29] However, there is not much literature concerning nonlocal optimal design models. We refer, among others, Andrés and Muñoz, 30,31 Bonder and Spedaletti, 32 D'Elia and Gunzburger, 33,34 and Zhou and Du 35,36 mainly in the context of elliptic equations. Also, Bonder et al 37 is a valuable paper, where an H-convergence study has been carried out.…”

On the Convergence of a Class of Nonlocal Elliptic Equations and Related Optimal Design Problems

“…Recent progress has been made in this direction. Some references on this field are [25][26][27][28][29].…”