DOI: https://doi.org/10.26089/NumMet.v20r432

Application of asymptotic analysis methods for solving a coefficient inverse problem for a system of nonlinear singularly perturbed reaction-diffusion equations with cubic nonlinearity

Authors

  • D.V. Lukyanenko
  • A.A. Melnikova

Keywords:

singularly perturbed problem
interior and boundary layers
reaction-diffusion equation
inverse problem with the location of moving front data

Abstract

The capabilities of asymptotic analysis methods for solving a coefficient inverse problem for a system of nonlinear singularly perturbed equations of reaction-diffusion type with cubic nonlinearity are shown. The problem considered for a system of partial differential equations is reduced to a system of algebraic equations that is much simpler for a numerical study and relates the data of the inverse problem (the information on the position of the reaction front in time) with the coefficient to be recovered. Numerical results confirm the efficiency of the proposed approach.

New computational technologies

Downloads

Published

2019-10-29

Issue

Vol. 20 (2019): Issue 4.

Section

Section 1. Numerical methods and applications

Author Biographies

D.V. Lukyanenko

Lomonosov Moscow State University,
Faculty of Physics
• Associate Professor

A.A. Melnikova

Lomonosov Moscow State University,
Faculty of Physics
• Assistant

References

  1. H. Meinhardt, Models of Biological Pattern Formation (Academic Press, London, 1982).
  2. R. FitzHugh, “Impulses and Physiological States in Theoretical Models of Nerve Membrane,” Biophys. J. 1 (6), 445-466 (1961).
  3. R. R. Aliev and A. V. Panfilov, “A Simple Two-Variable Model of Cardiac Excitation,” Chaos Soliton Fract. 7 (3), 293-301 (1996).
  4. Y. M. Ham, “Internal Layer Oscillations in FitzHugh-Nagumo Equation,” J. Comput. Appl. Math. 1999. 103 (2), 287-295 (1999).
  5. A. Hagberg and E. Meron, “Pattern Formation in Non-gradient Reaction-Diffusion Systems: The Effects of Front Bifurcations,” Nonlinearity 7 (3), 805-835 (1994).
  6. S.-L. Wu and H.-Q. Zhao, “Traveling Fronts for a Delayed Reaction-Diffusion System with a Quiescent Stage,” Commun. Nonlinear Sci. Numer. Simul. 16 (9), 3610-3621 (2011).
  7. H. Larralde, M. Araujo, S. Havlin, and H. E. Stanley, “Diffusion-Reaction Kinetics for A+B(static)->C(inert) for One-Dimensional Systems with Initially Separated Reactants,” Phys. Rev. A 46 (10), R6121-R6123 (1992).
  8. D. A. Kessler and H. Levine, “Fluctuation-Induced Diffusive Instabilities,” Nature 394 (6693), 556-558 (1998).
  9. R. O. Prum and S. Williamson, “Reaction-Diffusion Models of within-feather Pigmentation Patterning,” Proc. R. Soc. Lond. B: Biol. Sci. 269 (1493), 781-792 (2002).
  10. D. V. Lukyanenko, M. A. Shishlenin, and V. T. Volkov, “Solving of the Coefficient Inverse Problems for a Nonlinear Singularly Perturbed Reaction-Diffusion-Advection Equation with the Final Time Data,” Commun. Nonlinear Sci. Numer. Simul. 54, 233-247 (2018).
  11. D. V. Lukyanenko, V. T. Volkov, and N. N. Nefedov, “Dynamically Adapted Mesh Construction for the Efficient Numerical Solution of a Singular Perturbed Reaction-Diffusion-Advection Equation,” Model. Analiz Inform. Sist. 24 (3), 322-338 (2017).
  12. D. Lukyanenko, N. Nefedov, E. Nikulin, and V. Volkov, “Use of Asymptotics for New Dynamic Adapted Mesh Construction for Periodic Solutions with an Interior Layer of Reaction-Diffusion-Advection Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2017), Vol. 10187, pp. 107-118.
  13. A. Melnikova, N. Levashova, and D. Lukyanenko, “Front Dynamics in an Activator-Inhibitor System of Equations,” in Lecture Notes in Computer Science (Springer, Heidelberg, 2017), Vol. 10187, pp. 492-499.
  14. V. T. Volkov, D. V. Lukyanenko, and N. N. Nefedov, “Analytical-Numerical Approach to Describing Time-Periodic Motion of Fronts in Singularly Perturbed Reaction-Advection-Diffusion Models,” Zh. Vychisl. Mat. Mat. Fiz. 59 (1), 50-62 (2019) [Comput. Math. Math. Phys. 59 (1), 46-58 (2019)].
  15. D. V. Lukyanenko, M. A. Shishlenin, and V. T. Volkov, “Asymptotic Analysis of Solving an Inverse Boundary Value Problem for a Nonlinear Singularly Perturbed Time-Periodic Reaction-Diffusion-Advection Equation,” J. Inverse Ill-Posed Probl. 27 (2019).
    doi 10.1515/jiip-2017-0074
  16. D. V. Lukyanenko, V. B. Grigorev, V. T. Volkov, and M. A. Shishlenin, “Solving of the Coefficient Inverse Problem for a Nonlinear Singularly Perturbed Two-Dimensional Reaction-Diffusion Equation with the Location of Moving Front Data,” Comput. Math. Appl. 77 (5), 1245-1254 (2019).
  17. S. I. Kabanikhin, “Definitions and Examples of Inverse and Ill-Posed Problems,” J. Inverse Ill-Posed Probl. 16 (4), 317-357 (2008).
  18. M. I. Belishev and Ya. V. Kuryiev, “Boundary Control, Wave Field Continuation and Inverse Problems for the Wave-Equation,” Comput. Math. Appl. 1991. 22 (4-5), 27-52 (1991).
  19. M. I. Belishev and Ya. V. Kuryiev, “To the Reconstruction of a Riemannian Manifold via its Spectral Data (BC-Method),” Commun. Part. Diff. Eq. 17 (5-6), 767-804 (1992).
  20. M. I. Belishev, “Boundary Control in Reconstruction of Manifolds and Metrics (the BC Method),” Inverse Probl. 13 (5), R1-R45 (1997).
  21. L. Beilina and M. V. Klibanov, “A Globally Convergent Numerical Method for a Coefficient Inverse Problem,” SIAM J. Sci. Comput. 31 (1), 478-509 (2008).
  22. N. Pantong, J. Su, H. Shan, et al., “Globally Accelerated Reconstruction Algorithm for Diffusion Tomography with Continuous-Wave Source in an Arbitrary Convex Shape Domain,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. 2009. 26 (3), 456-472 (2009).
  23. M. V. Klibanov, M. A. Fiddy, L. Beilina, et al., “Picosecond Scale Experimental Verification of a Globally Convergent Algorithm for a Coefficient Inverse Problem,” Inverse Probl. 26 (2010).
    doi 10.1088/0266-5611/26/4/045003
  24. N. T. Thánh, L. Beilina, M. V. Klibanov, and M. A. Fiddy, “Imaging of Buried Objects from Experimental Backscattering Time-Dependent Measurements Using a Globally Convergent Inverse Algorithm,” SIAM J. Imaging Sci. 8 (1), 757-786 (2015).
  25. M. V. Klibanov, N. A. Koshev, J. Li, and A. G. Yagola, “Numerical Solution of an Ill-Posed Cauchy Problem for a Quasilinear Parabolic Equation Using a Carleman Weight Function,” J. Inverse Ill-Posed Probl. 24 (6), 761-776 (2016).
  26. M. V. Klibanov and A. G. Yagola, “Convergent Numerical Methods for Parabolic Equations with Reversed Time via a New Carleman Estimate,” Inverse Probl. (in press).
    doi 10.1088/1361-6420/ab2777
  27. S. I. Kabanikhin and M. A. Shishlenin, “Boundary Control and Gel’fand-Levitan-Krein Methods in Inverse Acoustic Problem,” J. Inverse Ill-Posed Probl. 12 (2), 125-144 (2004).
  28. S. I. Kabanikhin and M. A. Shishlenin, “Numerical Algorithm for Two-Dimensional Inverse Acoustic Problem Based on Gel’fand-Levitan-Krein Equation,” J. Inverse Ill-Posed Probl. 18 (9), 979-995 (2011).
  29. S. I. Kabanikhin, N. S. Novikov, I. V. Oseledets, and M. A. Shishlenin, “Fast Toeplitz Linear System Inversion for Solving Two-Dimensional Acoustic Inverse Problem,” J. Inverse Ill-Posed Probl. 23 (6), 687-700 (2015).
  30. S. I. Kabanikhin and M. A. Shishlenin, “Two-Dimensional Analogs of the Equations of Gelfand, Levitan, Krein, and Marchenko,” Eurasian J. Math. Comput. Appl. 3 (2), 70-99 (2015).
  31. S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov, and M. A. Shishlenin, “Numerical Solution of an Inverse Problem of Coefficient Recovering for a Wave Equation by a Stochastic Projection Methods,” Monte Carlo Methods Appl. 21 (3), 189-203 (2015).
  32. S. I. Kabanikhin, K. K. Sabelfeld, N. S. Novikov, and M. A. Shishlenin, “Numerical Solution of the Multidimensional Gelfand-Levitan Equation,” J. Inverse Ill-Posed Probl. 23 (5), 439-450 (2015).
  33. S. I. Kabanikhin and M. A. Shishlenin, “Comparative Analysis of Boundary Control and Gel’fand-Levitan Methods of Solving Inverse Acoustic Problem,” in Inverse Problems in Engineering Mechanics IV (Elsevier, Amsterdam, 2003), pp. 503-512.
  34. N. T. Levashova and A. A. Mel’nikova, “Step-like Contrast Structure in a Singularly Perturbed System of Parabolic Equations,” Differ. Uravn. 51 (3), 339-358 (2015) [Differ. Equ. 51 (3), 342-361 (2015)].
  35. A. A. Melnikova and M. Chen, “Existence and Asymptotic Representation of the Autowave Solution of a System of Equations,” Zh. Vychisl. Mat. Mat. Fiz. 58 (5), 705-715 (2018) [Comput. Math. Math. Phys. 58 (5), 680-690 (2018)].
  36. A. B. Vasil’eva and V. F. Butuzov, Asymptotic Methods in the Theory of Singular Perturbations (Vysshaya Shkola, Moscow, 1990) [in Russian].
  37. M. M. Lavrent’ev, “On Integral Equations of the First Kind,” Dokl. Akad. Nauk SSSR 127 (1), 31-33 (1959).
  38. A. N. Tikhonov, “Regularization of Incorrectly Posed Problems,” Dokl. Akad. Nauk SSSR 153 (1), 49-52 (1963) [Sov. Math. Dokl. 4 (6), 1624-1627 (1963)].
  39. V. V. Vasin and A. L. Ageev, Ill-Posed Problems with A Priori Information (VSP, Utrecht, 1995).
  40. A. N. Tikhonov, A. V. Goncharsky, V. V. Stepanov, and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems (Kluwer, Dordrecht, 1995).
  41. V. A. Morozov, “Regularization of Incorrectly Posed Problems and the Choice of Regularization Parameter,” Zh. Vychisl. Mat. Mat. Fiz. 6 (1), 170-175 (1966) [USSR Comput. Math. Math. Phys. 6 (1), 242-251 (1966)].
  42. A. E. Sidorova, N. T. Levashova, A. A. Melnikova, et al., “Autowave Self-Organization in Heterogeneous Natural-Anthropogenic Ecosystems,” Vestn. Mosk. Univ., Ser. 3: Fiz. Astron., No. 5, 107-113 (2016) [Moscow Univ. Phys. Bull. 71 (6), 562-568 (2016)].
  43. N. T. Levashova, A. A. Melnikova, D. V. Luk’yanenko, et al., “Modeling of Ecosystems as a Process of Self-Organization,” Mat. Model. 29 (11), 40-52 (2017).
  44. A. G. Yagola, A. S. Leonov, and V. N. Titarenko, “Data Errors and an Error Estimation for Ill-Posed Problems,” Inverse Probl. Eng. 10 (2), 117-129 (2002).
  45. K. Yu. Dorofeev, V. N. Titarenko, and A. G. Yagola, “Algorithms for Constructing a Posteriori Errors of Solutions to Ill-Posed Problems,” Zh. Vychisl. Mat. Mat. Fiz. 43 (1), 12-25 (2003) [Comput. Math. Math. Phys. 43 (1), 10-23 (2003)].
  46. V. Titarenko and A. Yagola, “Error Estimation for Ill-Posed Problems on Piecewise Convex Functions and Sourcewise Represented Sets,” J. Inverse Ill-Posed Probl. 16 (6), 625-638 (2008).
  47. A. G. Yagola and Y. M. Korolev, “Error Estimation in Ill-Posed Problems in Special Cases,” in Applied Inverse Problems. Springer Proceedings in Mathematics and Statistics (Springer, New York, 2013), Vol. 48, pp. 155-164.
  48. A. S. Leonov, “Which of Inverse Problems Can Have a Priori Approximate Solution Accuracy Estimates Comparable in Order with the Data Accuracy,” Numer. Anal. Appl. 7 (4), 284-292 (2014).
  49. A. S. Leonov, “A Posteriori Accuracy Estimations of solutions to Ill-Posed Inverse Problems and Extra-Optimal Regularizing Algorithms for Their Solution,” Numer. Anal. Appl. 5 (1), 68-83 (2012).