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

  • Harbir Antil
  • Enrique Otárola
  • Abner J. Salgado


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.


Optimal control problems Identification (inverse) problems Fractional diffusion Bisection algorithm Finite elements Stability Fully-discrete methods Convergence 

Mathematics Subject Classification

26A33 35J70 49J20 49K21 49M25 65M12 65M15 65M60 


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Authors and Affiliations

  • Harbir Antil
    • 1
  • Enrique Otárola
    • 2
  • Abner J. Salgado
    • 3
  1. 1.Department of Mathematical SciencesGeorge Mason UniversityFairfaxUSA
  2. 2.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile
  3. 3.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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