Convergence rates for Tikhonov regularization of a coefficient identification problem

Huimin Huang; Wensheng Zhang

doi:10.1515/jiip-2024-0057

Article

Convergence rates for Tikhonov regularization of a coefficient identification problem

Huimin Huang and Wensheng Zhang

Published/Copyright: June 19, 2025

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Journal of Inverse and Ill-posed Problems Volume 33 Issue 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. Search 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/BF01790543Search 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_1Search 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/S1064827599326020Search 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:2002028Search 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/ab8449Search 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. 10.1137/1.9780898719208Search 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/BF01448191Search 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/S0002-9939-96-03132-2Search 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. 10.1088/0266-5611/26/12/125014Search 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. 10.1088/0266-5611/27/7/075008Search in Google Scholar

[12] H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems, Math. Appl. 375, Kluwer Academic, Dordrecht, 1996. 10.1007/978-94-009-1740-8Search 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. 10.1088/0266-5611/5/4/007Search 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. 10.1088/0266-5611/16/6/319Search 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. Search 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. 10.1088/0266-5611/13/1/007Search 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.1994.1479Search 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. 10.1088/0266-5611/28/10/104002Search 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. Search 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)00275-2Search 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/1988220101191Search 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. 10.1007/978-3-642-65217-2Search in Google Scholar

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

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

[25] N. Sun, Inverse Problems in Groundwater Modeling, Kluwer Academic, Dordrecht, 1994. Search 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. Search 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. Search 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/WR022i002p00095Search in Google Scholar

[29] K. Yosida, Functional Analysis, 6th ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1980. Search 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. 10.1080/17415977.2020.1785444Search in Google Scholar

[31] J. Zou, Numerical methods for elliptic inverse problems, Int. J. Comput. Math. 70 (1998), no. 2, 211–232. 10.1080/00207169808804747Search 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

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/jiip-2024-0057

Keywords for this article

Coefficient identification; convergence rate; functional; Tikhonov solution; wave equation