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

We consider an identification (inverse) problem, where the state u is governed by a fractional elliptic equation and the unknown variable corresponds to the order s ∈ ( 0 , 1 ) of the underlying operator. We study the existence of an optimal pair ( s ¯ , u ¯ ) and provide sufficient conditions for i...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 77; no. 1; pp. 204 - 224
Main Authors: Antil, Harbir, Otárola, Enrique, Salgado, Abner J.
Format: Journal Article
Language:English
Published: New York Springer US 01.10.2018
Springer Nature B.V
Subjects:
Algorithms
Computational Mathematics and Numerical Analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Optimization
Partial differential equations
Theoretical
Fully-discrete methods
65M15
Finite elements
Stability
Optimal control problems
35J70
26A33
Identification (inverse) problems
Convergence
Bisection algorithm
49K21
Fractional diffusion
49J20
65M60
49M25
65M12
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:We consider an identification (inverse) problem, where the state u is governed by a fractional elliptic equation and the unknown variable corresponds to the order s ∈ ( 0 , 1 ) of the underlying operator. We study the existence of an optimal pair ( s ¯ , 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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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-018-0703-0