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Boundary value problems initial condition identification by a wavelet-based Galerkin method

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Abstract

In this study, we introduce a numerical method to reconstruct, from posterior measurements of the solution, the initial condition in the one-dimensional heat conduction problem. The method uses a wavelet-based Galerkin method for the spatial discretization and spectral decomposition of the basis stiffness matrix to get the numerical solution without time discretization as in classical approaches. In fact, according to the proposed method, setting an error bound and properly selecting the eigenvalues of the discretization system, we are able to reconstruct the initial conditions without exceeding the required error. The applicability and computational efficiency of the methods are investigated by solving some numerical examples with toy solutions and the experimental results confirm its accuracy.

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References

  1. Beylkin, G.: On the representation of operator in bases of compactly supported wavelets. SIAM J. Numer. Anal. 6, 1716–1740 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  2. Beylkin, G., Keiser, J. M., Vozovoi, L.: A new class of time discretization schemes for the solution of nonlinear PDEs. J. Comput. Phys. 147, 362–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Choulli, M.: Une introduction aux problèmes inverses elliptiques et paraboliques, Mathématiques & Applications, vol. 65. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  4. Cohen, A.: Numerical Analysis of Wavelet Methods, Studies in Mathematics and Its Applications. Elsevier, Amsterdam (2003)

    Google Scholar 

  5. Cohen, A., Daubechies, I., Feauveau, J.-C.: Biorthogonal bases of compactly supported wavelets. Comm. Pure. Appli. Maths. 45, 485–560 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  6. Daubechies, I.: Ten Lectures on Wavelets, Book. SIAM, Pennsylvania (1992)

    Book  MATH  Google Scholar 

  7. DeVore, R., Zuazua, E.: Recovery of an initial temperature from discrete sampling. Math. Models. Meth. Appl. Sci. 12, 2487–2501 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Deslauriers, G., Dubuc, S.: Symmetric iterative interpolation processes. Constr. Approx. 5, 49–68 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dubuc, S.: Interpolation through an iterative scheme. J. Math. Anal. Appl. 114, 185–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numer 19, 209–286 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Imperiale, S., Moireau, P., Tonnoir, A.: Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains. ESAIM COCV 26, 45 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kadri Harouna, S., Perrier, V.: Homogeneous Dirichlet wavelets on the interval diagonalizing the derivative operator, and application to free-slip divergence-free wavelets, J. Math. Anal. Appli. 505, 125479 (2022) 

  13. Komornik, V., Loreti, P.: Fourier Series in Control Theory. Springer Science & Business Medi, Berlin (2005)

    Book  MATH  Google Scholar 

  14. Krogstad, S.: Generalized integrating factor methods for stiff PDEs. J. Comp. Phys. 203, 72–88 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Larsson, S., Thomee, V.: Partial Differential Equations with Numerical Methods. Springer Science & Business Media, Berlin (2003)

    MATH  Google Scholar 

  16. Lattès, R., Lions, J. -L.: Méthode De Quasi-Réversibilité et Applications. Dunod, Paris (1967)

    MATH  Google Scholar 

  17. Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way. Academic Press, Cambridge (2008)

    MATH  Google Scholar 

  18. Monasse, P., Perrier, V.: Orthogonal wavelet bases adapted for partial differential equations with boundary conditions. SIAM J. Math. Anal. 29, 1040–1065 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Payne, L. E.: Improperly Posed Problems in Partial Differential Equations. SIAM, Philadelphia (2017)

    MATH  Google Scholar 

  20. Phung, K. D., Wang, G., Xu, Y.: Impulse output rapid stabilization for heat equations. J. Diff. Equ. 263, 5012–5041 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  21. Qiu, C., Feng, X.: A wavelet method for solving backward heat conduction problems. Elect. J. Diff. Equ. 219, 1–9 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Seidman, T. I.: Optimal filtering for the backward heat equation. SIAM J. Num. Anal. 33, 162–170 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shukla, A., Mehra, M.: Spectral graph wavelet regularization and adaptive wavelet for the backward heat conduction problem. Inverse Problems Sci. Eng. 29, 457–488 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley (1993)

    MATH  Google Scholar 

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Authors and Affiliations

  1. LR Analysis and Control of PDEs, LR22ES03, Department of Mathematics Faculty of Sciences of Monastir, University of Monastir, 5019, Monastir, Tunisia

    Kadri Harouna Souleymane

  2. Laboratoire Mathématiques, Image et Applications (MIA), Université de La Rochelle, Avenue Michel Crépeau, 17042, La Rochelle, France

    Kaïs Ammari

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  1. Kadri Harouna Souleymane
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  2. Kaïs Ammari
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Correspondence to Kadri Harouna Souleymane.

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The data sets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. The author can also provide the numerical codes for the work evaluation if necessary.

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Souleymane, K.H., Ammari, K. Boundary value problems initial condition identification by a wavelet-based Galerkin method. Numer Algor 93, 397–414 (2023). https://doi.org/10.1007/s11075-022-01421-9

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