Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems

Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Identification of the Diffusion Parameter in Nonlocal Steady Diffusion Problems

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-015-9300-x
Publisher site
See Article on Publisher Site

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.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2016

References

  • Continuous and discontinuous finite element methods for a peridynamics model of mechanics
    Chen, X; Gunzburger, MD
  • Reformulation of elasticity theory for discontinuities and long-range forces
    Silling, SA
  • Limits of dense graph sequences
    Lovasz, L; Szegedy, B
  • The effect of long-range forces on the dynamics of a bar
    Weckner, O; Abeyaratne, R
  • Image recovery via nonlocal operators
    Lou, Y; Zhang, X; Osher, S; Bertozzi, A

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