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

  • Harbir Antil
  • Enrique Otárola
  • Abner J. Salgado
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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.

Keywords

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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References

  1. 1.
    Adams, R.: Sobolev Spaces, Pure and Applied Mathematics, vol. 65. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)Google Scholar
  2. 2.
    Antil, H., Otárola, E.: A FEM for an optimal control problem of fractional powers of elliptic operators. SIAM J. Control Optim. 53(6), 3432–3456 (2015).  https://doi.org/10.1137/140975061 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Cabré, X., Tan, J.: Positive solutions of nonlinear problems involving the square root of the Laplacian. Adv. Math. 224(5), 2052–2093 (2010).  https://doi.org/10.1016/j.aim.2010.01.025 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007).  https://doi.org/10.1080/03605300600987306 MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Caffarelli, L., Stinga, P.: Fractional elliptic equations, Caccioppoli estimates and regularity. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(3), 767–807 (2016).  https://doi.org/10.1016/j.anihpc.2015.01.004 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Capella, A., Dávila, J., Dupaigne, L., Sire, Y.: Regularity of radial extremal solutions for some non-local semilinear equations. Commun. Partial Differ. Equ. 36(8), 1353–1384 (2011).  https://doi.org/10.1080/03605302.2011.562954 MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, L.: iFEM: an integrated finite element methods package in MATLAB. Technical Report, University of California at Irvine, Tech. rep. (2009)Google Scholar
  8. 8.
    Deckelnick, K., Hinze, M.: Convergence and error analysis of a numerical method for the identification of matrix parameters in elliptic PDEs. Inverse Probl. 28(11), 115015 (2012).  https://doi.org/10.1088/0266-5611/28/11/115015 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nochetto, R., Otárola, E., Salgado, A.: A PDE approach to fractional diffusion in general domains: a priori error analysis. Found. Comput. Math. 15(3), 733–791 (2015).  https://doi.org/10.1007/s10208-014-9208-x MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Nochetto, R., Otárola, E., Salgado, A.: A PDE approach to space–time fractional parabolic problems. SIAM J. Numer. Anal. 54(2), 848–873 (2016).  https://doi.org/10.1137/14096308X MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Otárola, E.: A piecewise linear FEM for an optimal control problem of fractional operators: error analysis on curved domains. ESAIM Math. Model. Numer. Anal. 51(4), 1473–1500 (2017).  https://doi.org/10.1051/m2an/2016065 MathSciNetMATHGoogle Scholar
  12. 12.
    Sprekels, J., Valdinoci, E.: A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation. SIAM J. Control Optim. 55(1), 70–93 (2017).  https://doi.org/10.1137/16M105575X MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Stinga, P., Torrea, J.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Partial Differ. Equ. 35(11), 2092–2122 (2010).  https://doi.org/10.1080/03605301003735680 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations, Graduate Studies in Mathematics, Theory, Methods and Applications, vol. 112. American Mathematical Society, Providence, RI (2010).  https://doi.org/10.1090/gsm/112. (Translated from the 2005 German original by Jürgen Sprekels)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

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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