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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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Correspondence to Abner J. Salgado.

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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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