Abstract
In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke–Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12201126
Award Identifier / Grant number: 62266002
Award Identifier / Grant number: 82060328
Funding source: Natural Science Foundation of Jiangxi Province
Award Identifier / Grant number: 20224BAB201013
Funding statement: The work of R. Zhang is partially supported by the NSF of China (No. 12201126, 62266002). The work of F. Xie is partially supported by the NSF of China (No. 82060328). This work of X. Luo is supported by the Natural Science Foundation of Jiangxi Province (No. 20224BAB201013).
References
[1] G. Bao and P. Li, Inverse medium scattering problems for electromagnetic waves, SIAM J. Appl. Math. 65 (2005), no. 6, 2049–2066. 10.1137/040607435Search in Google Scholar
[2] Z. Chen, S. Cheng, G. Nelakanti and H. Yang, A fast multiscale Galerkin method for the first kind ill-posed integral equations via Tikhonov regularization, Int. J. Comput. Math. 87 (2010), no. 1–3, 565–582. 10.1080/00207160802155302Search in Google Scholar
[3] Z. Chen, S. Ding, Y. Xu and H. Yang, Multiscale collocation methods for ill-posed integral equations via a coupled system, Inverse Problems 28 (2012), no. 2, Article ID 025006. 10.1088/0266-5611/28/2/025006Search in Google Scholar
[4] Z. Chen, C. A. Micchelli and Y. Xu, The Petrov–Galerkin method for second kind integral equations. II. Multiwavelet schemes, Adv. Comput. Math. 7 (1997), no. 3, 199–233. Search in Google Scholar
[5] Z. Chen, C. A. Micchelli and Y. Xu, Fast collocation methods for second kind integral equations, SIAM J. Numer. Anal. 40 (2002), no. 1, 344–375. 10.1137/S0036142901389372Search in Google Scholar
[6] Z. Chen, C. A. Micchelli and Y. Xu, Multiscale Methods for Fredholm Integral Equations, Cambridge Monogr. Appl. Comput. Math. 28, Cambridge University, Cambridge, 2015. 10.1017/CBO9781316216637Search in Google Scholar
[7] Z. Chen, B. Wu and Y. Xu, Fast multilevel augmentation methods for solving Hammerstein equations, SIAM J. Numer. Anal. 47 (2009), no. 3, 2321–2346. 10.1137/080734157Search in Google Scholar
[8] Z. Chen and Y. Xu, The Petrov–Galerkin and iterated Petrov–Galerkin methods for second-kind integral equations, SIAM J. Numer. Anal. 35 (1998), no. 1, 406–434. 10.1137/S0036142996297217Search in Google Scholar
[9] Z. Chen, Y. Xu and H. Yang, A multilevel augmentation method for solving ill-posed operator equations, Inverse Problems 22 (2006), no. 1, 155–174. 10.1088/0266-5611/22/1/009Search in Google Scholar
[10] J. Cheng, J. J. Liu and B. Zhang, Inverse problems for PDEs: Models, computations and applications (in Chinese), Sci. Sin. Math. 49 (2019), no. 4, 643–666. 10.1360/N012018-00076Search in Google Scholar
[11] W. Dahmen, S. Prössdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations. II. Matrix compression and fast solution, Adv. Comput. Math. 1 (1993), no. 3–4, 259–335. 10.1007/BF02072014Search in Google Scholar
[12] V. Dicken and P. Maass, Wavelet-Galerkin methods for ill-posed problems, J. Inverse Ill-Posed Probl. 4 (1996), no. 3, 203–221. 10.1515/jiip.1996.4.3.203Search in Google Scholar
[13] 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
[14] W. Erb and E. V. Semenova, On adaptive discretization schemes for the solution of ill-posed problems with semiiterative methods, Appl. Anal. 94 (2015), no. 10, 2057–2076. 10.1080/00036811.2014.964691Search in Google Scholar
[15] Z. Fu, Y. Chen, L. Li and B. Han, Analysis of a generalized regularized Gauss–Newton method under heuristic rule in Banach spaces, Inverse Problems 37 (2021), no. 12, Article ID 125003. 10.1088/1361-6420/ac2fbaSearch in Google Scholar
[16] R. Gu, Z. Fu, B. Han and H. Fu, Heuristic rule for inexact Newton–Landweber iteration with convex penalty terms of nonlinear: Ill-posed problems, Inverse Problems 39 (2023), no. 6, Article ID 065006. 10.1088/1361-6420/acccdbSearch in Google Scholar
[17] M. Hanke and T. Raus, A general heuristic for choosing the regularization parameter in ill-posed problems, SIAM J. Sci. Comput. 17 (1996), no. 4, 956–972. 10.1137/0917062Search in Google Scholar
[18] Y. Ikebe, The Galerkin method for the numerical solution of Fredholm integral equations of the second kind, SIAM Rev. 14 (1972), 465–491. 10.1137/1014071Search in Google Scholar
[19] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd ed., Appl. Math. Sci. 127, Springer, New York, 2006. Search in Google Scholar
[20] B. Jin and D. A. Lorenz, Heuristic parameter-choice rules for convex variational regularization based on error estimates, SIAM J. Numer. Anal. 48 (2010), no. 3, 1208–1229. 10.1137/100784369Search in Google Scholar
[21] Q. Jin, Hanke–Raus heuristic rule for variational regularization in Banach spaces, Inverse Problems 32 (2016), no. 8, Article ID 085008. 10.1088/0266-5611/32/8/085008Search in Google Scholar
[22] Q. Jin, On a heuristic stopping rule for the regularization of inverse problems by the augmented Lagrangian method, Numer. Math. 136 (2017), no. 4, 973–992. 10.1007/s00211-016-0860-8Search in Google Scholar
[23] Q. Jin and W. Wang, Analysis of the iteratively regularized Gauss–Newton method under a heuristic rule, Inverse Problems 34 (2018), no. 3, Article ID 035001. 10.1088/1361-6420/aaa0fbSearch in Google Scholar
[24] S. I. Kabanikhin, Inverse and Ill-Posed Problems: Theory and Applications, Inverse Ill-posed Probl. Ser. 55, Walter de Gruyter, Berlin, 2011. 10.1515/9783110224016Search in Google Scholar
[25] B. Kaltenbacher, A. Neubauer and O. Scherzer, Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Ser. Comput. Appl. Math. 6, Walter de Gruyter, Berlin, 2008. 10.1515/9783110208276Search in Google Scholar
[26] S. Kindermann and K. Raik, Heuristic parameter choice rules for Tikhonov regularization with weakly bounded noise, Numer. Funct. Anal. Optim. 40 (2019), no. 12, 1373–1394. 10.1080/01630563.2019.1604546Search in Google Scholar
[27] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, 2nd ed., Appl. Math. Sci. 120, Springer, New York, 2011. Search in Google Scholar
[28] J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications, Comput. Sci. Eng. 10, Society for Industrial and Applied Mathematics, Philadelphia, 2012. 10.1137/1.9781611972344Search in Google Scholar
[29] M. T. Nair, Linear Operator Equations: Approximation and Regularization, World Scientific, Hackensack, 2009. 10.1142/7055Search in Google Scholar
[30] S. Pereverzev and E. Schock, On the adaptive selection of the parameter in regularization of ill-posed problems, SIAM J. Numer. Anal. 43 (2005), no. 5, 2060–2076. 10.1137/S0036142903433819Search in Google Scholar
[31] D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9 (1962), 84–97. 10.1145/321105.321114Search in Google Scholar
[32] R. Plato, The Galerkin scheme for Lavrentiev’s m-times iterated method to solve linear accretive Volterra integral equations of the first kind, BIT 37 (1997), no. 2, 404–423. 10.1007/BF02510220Search in Google Scholar
[33] R. Plato and G. Vainikko, On the regularization of projection methods for solving ill-posed problems, Numer. Math. 57 (1990), no. 1, 63–79. 10.1007/BF01386397Search in Google Scholar
[34] A. Rieder, A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization, Numer. Math. 75 (1997), no. 4, 501–522. 10.1007/s002110050250Search in Google Scholar
[35] Y. F. Wang, Y. Zhang, D. V. Lukyanenko and A. G. Yagola, Recovering aerosol particle size distribution function on the set of bounded piecewise-convex functions, Inverse Probl. Sci. Eng. 21 (2013), no. 2, 339–354. 10.1080/17415977.2012.700711Search in Google Scholar
[36] H. Yang and R. Zhang, A fast multilevel iteration method for solving linear ill-posed integral equations, J. Inverse Ill-Posed Probl. 30 (2022), no. 3, 409–423. 10.1515/jiip-2020-0127Search in Google Scholar
[37] Y. Zhang and B. Hofmann, On fractional asymptotical regularization of linear ill-posed problems in Hilbert spaces, Fract. Calc. Appl. Anal. 22 (2019), no. 3, 699–721. 10.1515/fca-2019-0039Search in Google Scholar
[38] Y. Zhang and B. Hofmann, On the second-order asymptotical regularization of linear ill-posed inverse problems, Appl. Anal. 99 (2020), no. 6, 1000–1025. 10.1080/00036811.2018.1517412Search in Google Scholar
[39] Y. Zhang, D. V. Lukyanenko and A. G. Yagola, An optimal regularization method for convolution equations on the sourcewise represented set, J. Inverse Ill-Posed Probl. 23 (2015), no. 5, 465–475. 10.1515/jiip-2014-0047Search in Google Scholar
[40] Y. Zhang, D. V. Lukyanenko and A. G. Yagola, Using Lagrange principle for solving two-dimensional integral equation with a positive kernel, Inverse Probl. Sci. Eng. 24 (2016), no. 5, 811–831. 10.1080/17415977.2015.1077445Search in Google Scholar
[41] R. Zhang, X. Luo, W. Hu and D. Zhou, Adaptive multilevel iteration methods for solving ill-posed integral equations via a coupled system, Inverse Problems 37 (2021), no. 9, Article ID 095002. 10.1088/1361-6420/ac1361Search in Google Scholar
[42] R. Zhang and B. Zhou, Heuristic parameter choice rule for solving linear ill-posed integral equations in finite dimensional space, J. Comput. Appl. Math. 400 (2022), Article ID 113741. 10.1016/j.cam.2021.113741Search in Google Scholar
[43] Z. Zhang and Q. Jin, Heuristic rule for non-stationary iterated Tikhonov regularization in Banach spaces, Inverse Problems 34 (2018), no. 11, Article ID 115002. 10.1088/1361-6420/aad918Search in Google Scholar
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