Startseite Mathematik Convergence rates for Tikhonov regularization of a coefficient identification problem
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Convergence rates for Tikhonov regularization of a coefficient identification problem

  • Huimin Huang ORCID logo und Wensheng Zhang ORCID logo EMAIL logo
Veröffentlicht/Copyright: 19. Juni 2025
Journal of Inverse and Ill-posed Problems
Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 33 Heft 4

Abstract

This paper investigates the convergence rates for Tikhonov regularization of the problem for identifying the coefficient a ⁢ ( x ) ∈ L ∞ ⁢ ( Ω ) in the frequency-domain acoustic wave equation − div ⁢ ( a ⁢ ∇ u ) − b ⁢ u = f in general dimensional spaces; here x ∈ Ω ⊂ R d , d ≥ 1 , and b ⁢ ( x ) > 0 . We assume that we know the imprecise measurement data of 𝑢 in the subdomain Ω 1 ⊂ Ω with a measurement error of level δ > 0 , while 𝑢 satisfies the general Robin boundary condition on ∂ Ω 1 . We propose to regularize this problem by minimizing a new functional and prove that the functional attains a unique global minimum on the admissible set of a ⁢ ( x ) . Furthermore, we derive the convergence rate O ⁢ ( δ ) for the Tikhonov regularized solution with an easily satisfied source condition.

Keywords: Coefficient identification; convergence rate; functional; Tikhonov solution; wave equation
MSC 2020: 35R30; 35R25; 47A52; 49N45; 35L05

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11471328

Funding statement: This research is supported by the President Foundation of Academy of Mathematics and Systems Science, Chinese Academy of Sciences. It is partially supported by the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences. It is also partially supported by the National Natural Science Foundation of China under the grant number 11471328.

Acknowledgements

We appreciate the reviewers very much for the very valuable and constructive comments.

References

[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, Pure Appl. Math. (Amsterdam) 140, Elsevier/Academic Press, Amsterdam, 2003. Suche in Google Scholar

[2] G. Alessandrini, An identification problem for an elliptic equation in two variables, Ann. Mat. Pura Appl. (4) 145 (1986), 265–295. 10.1007/BF01790543Suche in Google Scholar

[3] H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Systems Control Found. Appl. 1, Birkhäuser, Boston, 1989. 10.1007/978-1-4612-3700-6_1Suche in Google Scholar

[4] T. F. Chan and X.-C. Tai, Identification of discontinuous coefficients in elliptic problems using total variation regularization, SIAM J. Sci. Comput. 25 (2003), no. 3, 881–904. 10.1137/S1064827599326020Suche in Google Scholar

[5] G. Chavent and K. Kunisch, The output least squares identifiability of the diffusion coefficient from an H 1 -observation in a 2-D elliptic equation, ESAIM Control Optim. Calc. Var. 8 (2002), 423–440. 10.1051/cocv:2002028Suche in Google Scholar

[6] D.-H. Chen, D. Jiang and J. Zou, Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems, Inverse Problems 36 (2020), no. 7, Article ID 075001. 10.1088/1361-6420/ab8449Suche in Google Scholar

[7] P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics Appl. Math. 40, Society for Industrial and Applied Mathematics, Philadelphia, 2002. 10.1137/1.9780898719208Suche in Google Scholar

[8] F. Colonius and K. Kunisch, Output least squares stability in elliptic systems, Appl. Math. Optim. 19 (1989), no. 1, 33–63. 10.1007/BF01448191Suche in Google Scholar

[9] Z. Ding, A proof of the trace theorem of Sobolev spaces on Lipschitz domains, Proc. Amer. Math. Soc. 124 (1996), no. 2, 591–600. 10.1090/S0002-9939-96-03132-2Suche in Google Scholar

[10] D. N. Hào and T. N. T. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Problems 26 (2010), no. 12, Article ID 125014. 10.1088/0266-5611/26/12/125014Suche in Google Scholar

[11] D. N. Hào and T. N. T. Quyen, Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I, Inverse Problems 27 (2011), Article ID 075008. 10.1088/0266-5611/27/7/075008Suche in Google Scholar

[12] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Suche in Google Scholar

[13] H. W. Engl, K. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularisation of nonlinear ill-posed problems, Inverse Problems 5 (1989), no. 4, 523–540. 10.1088/0266-5611/5/4/007Suche in Google Scholar

[14] H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems 16 (2000), no. 6, 1907–1923. 10.1088/0266-5611/16/6/319Suche in Google Scholar

[15] C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Res. Notes Math. 105, Pitman, Boston, 1984. Suche in Google Scholar

[16] M. Hanke, A regularizing Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Problems 13 (1997), no. 1, 79–95. 10.1088/0266-5611/13/1/007Suche in Google Scholar

[17] K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems, J. Math. Anal. Appl. 188 (1994), no. 3, 1040–1066. 10.1006/jmaa.1994.1479Suche in Google Scholar

[18] D. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Problems 28 (2012), no. 10, Article ID 104002. 10.1088/0266-5611/28/10/104002Suche in Google Scholar

[19] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Appl. Math. Sci. 120, Springer Science+Business Media, LLC, 2011. Suche in Google Scholar

[20] I. Knowles, Parameter identification for elliptic problems, J. Comput. Appl. Math. 131 (2001), no. 1–2, 175–194. 10.1016/S0377-0427(00)00275-2Suche in Google Scholar

[21] R. V. Kohn and B. D. Lowe, A variational method for parameter identification, RAIRO Modél. Math. Anal. Numér. 22 (1988), no. 1, 119–158. 10.1051/m2an/1988220101191Suche in Google Scholar

[22] J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972. 10.1007/978-3-642-65217-2Suche in Google Scholar

[23] V. A. Morozov, Methods for Solving Incorrectly Posed Problems, Springer, New York, 1984. 10.1007/978-1-4612-5280-1Suche in Google Scholar

[24] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. Suche in Google Scholar

[25] N. Sun, Inverse Problems in Groundwater Modeling, Kluwer Academic, Dordrecht, 1994. Suche in Google Scholar

[26] A. N. Tikhonov, On the solution of ill-posed problems and the method of regularization, Dokl. Akad. Nauk SSSR 151 (1963), 501–504. Suche in Google Scholar

[27] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems, Scripta Ser. Math., V. H. Winston & Sons, Washington, 1977. Suche in Google Scholar

[28] W. Yeh, Review of parameter identification procedures in ground water hydrology: The inverse problem, Water. Resour. Res. 22 (1986), no. 2, 95–108. 10.1029/WR022i002p00095Suche in Google Scholar

[29] K. Yosida, Functional Analysis, 6th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1980. Suche in Google Scholar

[30] W. Zhang, Acoustic multi-parameter full waveform inversion based on the wavelet method, Inverse Probl. Sci. Eng. 29 (2021), no. 2, 220–247. 10.1080/17415977.2020.1785444Suche in Google Scholar

[31] J. Zou, Numerical methods for elliptic inverse problems, Int. J. Comput. Math. 70 (1998), no. 2, 211–232. 10.1080/00207169808804747Suche in Google Scholar
Received: 2024-08-18
Revised: 2025-03-03
Accepted: 2025-06-05
Published Online: 2025-06-19
Published in Print: 2025-08-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. An inverse problem of finding a time-dependent parameter in a bilinear heat equation
  3. A novel method for computing core-EP inverse through elementary transformation
  4. Convergence rates for Tikhonov regularization of a coefficient identification problem
  5. Carleman estimate for stochastic degenerate wave equation with drift and its application
  6. Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions
  7. Understanding edge artifacts of the OSEM algorithm in emission tomography
  8. Unique reconstruction of the inverse spectral problem with mixed data for AKNS operator
  9. An inverse problem for nonlinear electrodynamic equations
https://doi.org/10.1515/jiip-2024-0057
Schlagwörter für diesen Artikel
Coefficient identification; convergence rate; functional; Tikhonov solution; wave equation

Artikel in diesem Heft

  1. Frontmatter
  2. An inverse problem of finding a time-dependent parameter in a bilinear heat equation
  3. A novel method for computing core-EP inverse through elementary transformation
  4. Convergence rates for Tikhonov regularization of a coefficient identification problem
  5. Carleman estimate for stochastic degenerate wave equation with drift and its application
  6. Inverse initial problem under Nash strategy for stochastic reaction-diffusion equations with dynamic boundary conditions
  7. Understanding edge artifacts of the OSEM algorithm in emission tomography
  8. Unique reconstruction of the inverse spectral problem with mixed data for AKNS operator
  9. An inverse problem for nonlinear electrodynamic equations
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern können?
Bitte schreiben Sie uns.
© 2026 De Gruyter Brill
Heruntergeladen am 30.1.2026 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2024-0057/html
Button zum nach oben scrollen