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Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems

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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.

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Notes

  1. In [20] the diffusivity is defined as a second order symmetric positive definite tensor \({\varvec{\Theta }}\) and the kernel is defined as \(\gamma :={\varvec{\alpha }}\cdot \left( {\varvec{\Theta }}{\varvec{\alpha }}\right) \). Here for simplifying the analysis of the identification problem we consider a scalar diffusivity and we do not include it in the definition of the kernel.

  2. For \(s\in (0,1)\) and for a general domain \({\widetilde{\Omega }}\in \mathbb {R}^n\), let

    $$\begin{aligned} |v|_{H^s({\widetilde{\Omega }})}^2 := \int _{\widetilde{\Omega }}\int _{\widetilde{\Omega }}\frac{\big (v(\mathbf {y})-v(\mathbf {x})\big )^2}{|\mathbf {y}-\mathbf {x}|^{n+2s}}\,d\mathbf {y}d\mathbf {x}. \end{aligned}$$

    Then, the space \(H^s({\widetilde{\Omega }})\) is defined by [32] \( H^s({\widetilde{\Omega }}) := \left\{ v\in L^2({\widetilde{\Omega }}) : \Vert v\Vert _{L^2({\widetilde{\Omega }})} + |v|_{H^s({\widetilde{\Omega }})}<\infty \right\} . \)

  3. The reason of this choice will be made clear in the following section.

  4. The non-uniqueness of the solution is not due to the nonlocality, the same result holds for the corresponding local PDE control problem.

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Acknowledgments

This work was supported in part by the US National Science Foundation Grant DMS-1315259.

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Author notes

  1. M. D’Elia

    Present address: Sandia National Laboratories, Albuquerque, NM, USA

Authors and Affiliations

  1. Florida State University, 400 Dirac Science Library, Tallahassee, FL, 32306, USA

    M. D’Elia & M. Gunzburger

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  1. M. D’Elia
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  2. M. Gunzburger
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Correspondence to M. D’Elia.

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Sandia National Laboratories is a multi program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000.

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D’Elia, M., Gunzburger, M. Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems. Appl Math Optim 73, 227–249 (2016). https://doi.org/10.1007/s00245-015-9300-x

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