A noniterative reconstruction method for the inverse potential problem for a time-fractional diffusion equation
DOI:
https://doi.org/10.12775/TMNA.2023.004Keywords
Inverse problem, topology optimization, sensitivity analysis, fractional derivativeAbstract
This paper is concerned with the reconstruction of the support of the potential term for a time-fractional diffusion equation from the final measured data. The aim of this paper is to propose an accurate approach based on the topological derivative method. The idea is to formulate the reconstruction problem as a topology optimization one minimizing a given cost function. We derive a topological asymptotic expansion for the fractional model. The unknown support is reconstructed using the level-set curve of the topological gradient. We finally make some numerical examples proving the efficiency and accuracy of the proposed algorithm.References
A.B. Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control Optim. 48 (2010), no. 5, 2871–2900.
M. Abdelwahed and M. Hassine, Topological optimization method for a geometric control problem in Stokes flow, Appl. Num. Math. 59 (2009), no. 8, 1823–1838.
O. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A Math. Theor. 40 (2007), no. 24, 6287.
A.A. Alikhanov, A priori estimates for solutions of boundary value problems for fractional-order equations, Differential Equations 46 (2010), no. 5, 660–666.
G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method, Numer. Math. 76 (1997), 27–68.
R. Almeida and D.F. Torres, Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 3, 1490–1500.
S. Amstutz and H. Andrä, A new algorithm for topology optimization using a level-set method, J. Comput. Phys. 216 (2006), no. 2, 573–588.
S. Amstutz and A.A. Novotny, Topological optimization of structures subject to von Mises stress constraints, Struct. Multidiscip. Optim. 41 (2010), no. 3, 407.
M. BenSalah, Topological sensitivity method for reconstruction of the spatial component in the source term of a time-fractional diffusion equation, Ric. Mat. (2021), 1–26.
M. BenSalah, Topological sensitivity analysis method in identifying of point sources via time-fractional diffusion equation, Acta Appl. Math. 181 (2022), no. 1, 4.
B. Berkowitz, H. Scher and S.E. Silliman, Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resources Research 36 (2000), no. 1, 149–158.
W. Chen, L. Ye and H. Sun, Fractional diffusion equations by the Kansa method, Comput. Math. Appl. 59 (2010), no. 5, 1614–1620.
M. Choulli and M. Yamamoto, An inverse parabolic problem with non-zero initial condition, Inverse Problems 13 (1997), no. 1, 19.
H.A. Eschenauer, V.V. Kobelev and A. Schumacher, Bubble method for topology and shape optimization of structures, Struct. Optim. 8 (1994), 42–51.
P. Guillaume and M. Hassine, Removing holes in topological shape optimization, ESAIM Control Optim. Calc. Var. 14 (2008), no. 1, 160–191.
M. Hassine and M. Masmoudi, The topological asymptotic expansion for the quasi-Stokes problem, ESAIM Control Optim. Calc. Var. 10 (2004), no. 4, 478–504.
F. Hecht, New development in FreeFem++, J. Numer. Math. 20 (2012), no. 3–4, 251–266.
V. Isakov, Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math. 44 (1991), no. 2, 185–209.
V. Isakov, Some inverse problems for the diffusion equation, Inverse Problems 15 (1999), no. 1, 3.
D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems 33 (2017), no. 5, 055013.
B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multiterm time-fractional diffusion equation, J. Comput. Phys. 281 (2015), 825–843.
B. Jin and W. Rundell, An inverse problem for a one-dimensional time-fractional diffusion problem, Inverse Problems 28 (2012), no. 7, 075010.
B. Jin and Z. Zhou, An inverse potential problem for subdiffusion: stability and reconstruction, Inverse Problems 37 (2020), no. 1, 015006.
B. Kaltenbacher and W. Rundell, On an inverse potential problem for a fractional reaction-diffusion equation, Inverse Problems 35 (2019), no. 6, 065004.
Y. Kian and M. Yamamoto, Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations, Inverse Problems 35 (2019) no. 11, 115006.
A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, 2006.
R. Magin, X. Feng and D. Baleanu, Solving the fractional order Bloch equation, Concepts Magn. Reson. Part A 34A (2009), no. 1, 16–23.
M. Masmoudi, The Topological Asymptotic, Computational Methods for Control Applications (H. Kawarada and J. Périaux, eds.), International Series, Gakuto, 2002.
M.M. Meerschaert, D.A. Benson, H.P. Scheffler and B. Baeumer, Stochastic solution of space-time fractional diffusion equations, Phys. Rev. E 65 (2002), no. 4, 041103.
R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Phys. Rep. 339 (2000), no. 1, 1–77.
R. Metzler and J. Klafter, Boundary value problems for fractional diffusion equations, Phys. A 278 (2000), no. 1–2, 107–25.
L. Miller and M. Yamamoto, Coefficient inverse problem for a fractional diffusion equation, Inverse Problems 29 (2013), no. 7, 075013.
J.A. Norato, M.P. Bendsøe, R.B. Haber and D.A. Tortorelli, A topological derivative method for topology optimization, Structural and Multidisciplinary Optimization. 33 (2007), 375–386.
A.A. Novotny, R.A. Feijóo, E. Taroco and C. Padra, Topological sensitivity analysis, Comput. Methods Appl. Mech. Engrg. 192 (2003), no. 7–8, 803–829.
A.A. Novotny, J. Sokolowski and A. Żochowski, Applications of the Topological Derivative Method, Springer, 2019.
A.L. Prilepko and A.B. Kostin, On certain inverse problems for parabolic equations with final and integral observation, Mat. Sb. 183 (1992), no. 4, 49–68.
A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lochpo-sitionierungskriterien, doctoral dissertation, Inst. für Mechanik und Regelungstechnik.
I.M. Sokolov and J. Klafter, From diffusion to anomalous diffusion: a century after Einstein’s Brownian motion, Chaos 15 (2005), no. 2, 026103.
J. Sokolowski and A. Żochowski, On the topological derivative in shape optimization, SIAM J. Control Optim. 37 (1999), no. 4, 1251–1272.
S.B. Yuste and K. Lindenberg, Subdiffusion-limited reactions, Chemical Physics 284 (2002), no. 1–2, 169–80.
Z. Zhang and Z. Zhou, Recovering the potential term in a fractional diffusion equation, IMA J. Appl. Math. 82 (2017), no. 3, 579–600.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Mohamed BenSalah
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
