Opuscula Math. 46, no. 4 (2026), 441-474
https://doi.org/10.7494/OpMath.202606241

 
Opuscula Mathematica

Topological sensitivity analysis for the narrow escape problem

Ayadi Aya
Hassine Maatoug

Abstract. The topological sensitivity analysis method has been recognized as a promising, fast, and accurate approach for solving topology optimization and inverse problems. It is based on developing an asymptotic expansion of a design functional with respect to the creation of a small hole inside the computational domain. In this work, we extend this method to the narrow escape problem. The biological process is governed by a parabolic diffusion equation. We derive a sensitivity analysis for the parabolic problem solution with respect to the creation of a small absorbing boundary subset. We develop a rigorous mathematical framework that is valid in two- and three-dimensional space. It provides an asymptotic formula that describes the behavior of the perturbed solution with respect to the location and size of an arbitrary perturbed boundary subset. The Sobolev capacity notion has been employed to measure the smallness of the boundary subset and to describe the asymptotic behavior with respect to the perturbation size. The performed mathematical analysis is general and can be adapted for a large class of partial differential equations. The obtained asymptotic formula can serve as a useful tool to perform numerical algorithms for solving optimization and control problems.

Keywords: topological sensitivity analysis, boundary perturbation, parabolic equation, asymptotic expansion.

Mathematics Subject Classification: 65M80, 35C20, 35B40.

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  1. D.R. Adams, L.I. Hedberg, Function Spaces and Potential Theory, Springer, Berlin, 1996. https://doi.org/10.1007/978-3-662-03282-4
  2. R.A. Adams, J.J.F. Fournier, Sobolev Spaces, Academic Press, Amsterdam, 2003.
  3. G. Allaire, Numerical Analysis and Optimization: An Introduction to Mathematical Modeling and Numerical Simulation, Oxford University Press, 2007. https://doi.org/10.1093/oso/9780199205219.001.0001
  4. S. Banach, Théorie des opérations linéaires, PWN, Warszawa, 1932.
  5. M. BenSalah, M. Hassine, Inverse source problem for a diffusion equation involving the fractional spectral Laplacian, Math. Methods Appl. Sci. 44 (2021), 917-936. https://doi.org/10.1002/mma.6799
  6. M. BenSalah, M. Hassine, Inverse source problem for a space-time fractional diffusion equation, Ricerche Mat. 73 (2024), 681-713. https://doi.org/10.1007/s11587-021-00632-x
  7. H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2010.
  8. N. Chorfi, E. Ghezaiel, M. Hassine, Topological asymptotic analysis for a tumor identification problem, Asymptot. Anal. 123 (2020), 317-333. https://doi.org/10.3233/asy-201635
  9. M. Choulli, Une introduction aux problèmes inverses elliptiques et paraboliques, Springer, Berlin, 2009.
  10. J.L. Doob, Classical Potential Theory and Its Probabilistic Counterpart, Springer, Berlin, 1984.
  11. E. Ghezaiel, M. Hassine, Topological sensitivity analysis based method for solving a geometry reconstruction problem, Appl. Math. E-Notes 19 (2019), 179-188.
  12. E. Ghezaiel, M. Hassine, Topological sensitivity analysis for the transient heat problem and applications, Math. Comput. Simulation 169 (2020), 26-50. https://doi.org/10.1016/j.matcom.2019.09.022
  13. E. Ghezaiel, M. Hassine, Topological asymptotic expansion for a thermal problem, Appl. Math. Optim. 84 (2021), 955-995. https://doi.org/10.1007/s00245-020-09667-2
  14. D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, Berlin, 2015. https://doi.org/10.1007/978-3-642-61798-0
  15. P. Grisvard, Elliptic Problems in Nonsmooth Domains, SIAM, Philadelphia, 2011. https://doi.org/10.1137/1.9781611972030
  16. M. Hassine, S. Chaouch, Topological asymptotic expansion for the full Navier-Stokes equations, Asymptot. Anal. 133 (2022), 91-121. https://doi.org/10.3233/ASY-221807
  17. M. Hassine, M. Ouni, Topological sensitivity analysis for the 3D nonlinear Navier-Stokes equations, Asymptot. Anal. 135 (2023), 277-304. https://doi.org/10.3233/ASY-231855
  18. D. Holcman, Z. Schuss, Diffusion laws in dendritic spines, J. Math. Neurosci. 1 (2011), Art. 10. https://doi.org/10.1186/2190-8567-1-10
  19. H.L.F. von Helmholtz, Theorie der Luftschwingungen in Röhren mit offenen Enden, J. Reine Angew. Math. 57 (1860), 1-72. https://doi.org/10.1515/crll.1860.57.1
  20. D. Holcman, Z. Schuss, Escape through a small opening: receptor trafficking in a synaptic membrane, J. Stat. Phys. 117 (2004), 191-230. https://doi.org/10.1007/s10955-004-5712-8
  21. D. Holcman, A. Triller, Modeling synaptic dynamics and receptor trafficking, Biophys. J. 91 (2006), 2405-2415. https://doi.org/10.1529/biophysj.106.081935
  22. M. Hrizi, M. BenSalah, M. Hassine, Determination of the initial density in nonlocal diffusion from final time measurements, Discrete Contin. Dyn. Syst. Ser. S 15 (2022), 1469-1498. https://doi.org/10.3934/dcdss.2022029
  23. M. Hrizi, M. Hassine, Reconstruction of contact regions in semiconductor transistors using Dirichlet-Neumann cost functional approach, Appl. Anal. 100 (2019), 893-922. https://doi.org/10.1080/00036811.2019.1623393
  24. M. Hrizi, M. Hassine, A.A. Novotny, Imaging of mass distributions from partial domain measurement, J. Inverse Ill-Posed Probl. 30 (2022), 713-727. https://doi.org/10.1515/jiip-2020-0143
  25. M. Hrizi, M. Hassine, A.A. Novotny, Reconstruction of pointwise sources in a time-fractional diffusion equation, Fract. Calc. Appl. Anal. 26 (2023), 193-219. https://doi.org/10.1007/s13540-022-00127-y
  26. J.D. Jackson, Classical Electrodynamics, John Wiley & Sons, New York, 1998.
  27. N.S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, Berlin, 1972. https://doi.org/10.1007/978-3-642-65183-0
  28. J.-L. Lions, E. Magenes, Problèmes aux limites non homogènes et applications, Dunod, Paris, 1968. https://doi.org/10.5802/aif.111
  29. R. Malek, M. Hassine, M. Hrizi, Singular geometry perturbation based method for shape-topology optimization in unsteady Stokes flow, J. Math. Anal. Appl. 517 (2023), 126648. https://doi.org/10.1016/j.jmaa.2022.126648
  30. J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Springer, Berlin, 2001.
  31. J.W.S. Rayleigh, The Theory of Sound, vol. 2, 2nd ed., Dover, New York, 1945.
  32. R. Roux, Méthodes numériques déterministes, Master Thesis, Université Pierre et Marie Curie, 2014. https://doi.org/10.1007/s00446-016-0271-1
  33. Z. Schuss, A. Singer, D. Holcman, The narrow escape problem for diffusion in cellular microdomains, Proc. Natl. Acad. Sci. USA 104 (2007), 16098-16103. https://doi.org/10.1073/pnas.0706599104
  • Ayadi Aya
  • University of Monastir, Department of Mathematics, Avenue of the Environment, 5019 Monastir, Tunisia
  • Communicated by Vicenţiu D. Rădulescu.
  • Received: 2026-04-06.
  • Revised: 2026-06-23.
  • Accepted: 2026-06-24.
  • Published online: 2026-08-04.
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Cite this article as:
Ayadi Aya, Hassine Maatoug, Topological sensitivity analysis for the narrow escape problem, Opuscula Math. 46, no. 4 (2026), 441-474, https://doi.org/10.7494/OpMath.202606241

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