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Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects

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Abstract

We consider an identification (inverse) problem, where the state \({\mathsf {u}}\) is governed by a fractional elliptic equation and the unknown variable corresponds to the order \(s \in (0,1)\) of the underlying operator. We study the existence of an optimal pair \(({\bar{s}}, {{\bar{{\mathsf {u}}}}})\) and provide sufficient conditions for its local uniqueness. We develop semi-discrete and fully discrete algorithms to approximate the solutions to our identification problem and provide a convergence analysis. We present numerical illustrations that confirm and extend our theory.

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Authors and Affiliations

  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, 22030, USA

    Harbir Antil

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Valparaiso, Chile

    Enrique Otárola

  3. Department of Mathematics, University of Tennessee, Knoxville, TN, 37996, USA

    Abner J. Salgado

Authors
  1. Harbir Antil
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  2. Enrique Otárola
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  3. Abner J. Salgado
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Corresponding author

Correspondence to Abner J. Salgado.

Additional information

Harbir Antil has been supported in part by NSF Grant DMS-1521590. Enrique Otárola has been supported in part by CONICYT through FONDECYT Project 3160201. Abner J. Salgado has been supported in part by NSF Grants DMS-1418784 and DMS-1720213.

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Antil, H., Otárola, E. & Salgado, A.J. Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects. J Sci Comput 77, 204–224 (2018). https://doi.org/10.1007/s10915-018-0703-0

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  • DOI: https://doi.org/10.1007/s10915-018-0703-0

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