×

zbMATH — the first resource for mathematics

A regularizing multilevel approach for nonlinear inverse problems. (English) Zbl 1403.65024
Summary: In this paper, we propose a multilevel method for solving nonlinear inverse problems \(F(x) = y\) in Banach spaces. By minimizing the discretized version of the regularized functionals at different levels, we define a sequence of regularized approximations to the sought solution, which is shown to be stable and globally convergent. The penalty term \(\Theta\) in regularized functionals is allowed to be non-smooth to include \(L^p - L^1\) or \(L^p - \mathrm{TV}\) (Total Variation) reconstructions, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. Two parameter identification examples are presented to validate the theoretical analysis and to verify the effectiveness of the method.

MSC:
65J15 Numerical solutions to equations with nonlinear operators (do not use 65Hxx)
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J06 Nonlinear ill-posed problems
Software:
TIGRA
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Beck, A.; Teboulle, M., A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sci., 2, 183-202, (2009) · Zbl 1175.94009
[2] Beck, A.; Teboulle, M., Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems, IEEE Trans. Image Process., 18, 2419-2434, (2009) · Zbl 1371.94049
[3] Bellavia, S.; Morini, B.; Riccietti, E., On an adaptive regularization for ill-posed nonlinear systems and its trust-region implementation, Comput. Optim. Appl., 64, 1-30, (2016) · Zbl 1336.93055
[4] Bunks, C.; Saleck, F. M.; Zaleski, S.; Chavent, G., Multiscale seismic waveform inversion, Geophysics, 60, 1457-1473, (1995)
[5] Burger, M.; Osher, S., Convergence rates of convex variational regularization, Inverse Probl., 20, 1411-1421, (2004) · Zbl 1068.65085
[6] Candes, E. J.; Romberg, J.; Tao, T., Stable signal recovery from incomplete and inaccurate measurements, Commun. Pure Appl. Math., 59, 1207-1223, (2006) · Zbl 1098.94009
[7] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of inverse problems, (1996), Kluwer Dordrecht · Zbl 0859.65054
[8] Español, M. I.; Kilmer, M. E., Multilevel approach for signal restoration problems with Toeplitz matrices, SIAM J. Sci. Comput., 32, 1, 299-319, (2010) · Zbl 1228.65060
[9] Español, M. I.; Kilmer, M. E., A wavelet-based multilevel approach for blind deconvolution problems, SIAM J. Sci. Comput., 36, 4, A1432-A1450, (2014) · Zbl 1303.65023
[10] Estatico, C.; Gratton, S.; Lenti, F., A conjugate gradient like method for p-norm minimization in functional spaces, Numer. Math., 137, 4, 895-922, (2017) · Zbl 1379.65029
[11] Friedman, A., Partial differential equations of parabolic type, (1983), R.E. Krieger Publishing 115 Company Malabar, Florida
[12] Groetsch, C. W., Inverse problems in mathematical sciences, (1993), Vieweg Braunschweig · Zbl 0779.45001
[13] Hanke, M., A regularization Levenberg-Marquardt scheme, with applications to inverse groundwater filtration problems, Inverse Probl., 13, 79-95, (1997) · Zbl 0873.65057
[14] Hanke, M.; Groetsch, C. W., Nonstationary iterated Tikhonov regularization, J. Optim. Theory Appl., 98, 1, 37-53, (1998) · Zbl 0910.47005
[15] Hanke, M.; Neubauer, A.; Scherzer, O., A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., 72, 21-37, (1995) · Zbl 0840.65049
[16] Hubmer, S.; Ramlau, R., Convergence analysis of a two-point gradient method for nonlinear illposed problems, Inverse Probl., 33, 9, (2017)
[17] Jin, Q., The analysis of a discrete scheme of the iteratively regularized Gauss-Newton method, Inverse Probl., 16, 1457-1476, (2000) · Zbl 1044.65510
[18] Jin, Q., Landweber-Kaczmarz method in Banach spaces with inexact inner solvers, Inverse Probl., 32, 10, (2016) · Zbl 1356.65146
[19] Jin, B.; Lu, X., Numerical identification of a Robin coefficient in parabolic problems, Math. Comput., 81, 1369-1398, (2012) · Zbl 1255.65170
[20] Jin, B.; Maass, P., Sparsity regularization for parameter identification problems, Inverse Probl., 28, (2012) · Zbl 1280.47063
[21] Jin, Q.; Wang, W., Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals, Inverse Probl., 29, (2013) · Zbl 1283.65056
[22] Jin, Q.; Yang, H., Levenberg-Marquardt method in Banach spaces with general convex regularization terms, Numer. Math., 133, 4, 655-684, (2016) · Zbl 1350.65051
[23] Jin, Q.; Zhong, M., Nonstationary iterated Tikhonov regularization in Banach spaces with uniformly convex penalty terms, Numer. Math., 127, 485-513, (2014) · Zbl 1297.65062
[24] Kaltenbacher, B., The problem of the convergence of the iteratively regularized Gauss-Newton method, Comput. Math. Phys., 32, 1, 1353-1359, (1992)
[25] Kaltenbacher, B., Towards global convergence for strongly nonlinear ill-posed problems via a regularizing multilevel method, Numer. Funct. Anal. Optim., 27, 637-665, (2006) · Zbl 1101.65054
[26] Kaltenbacher, B., Convergence rates of a multilevel method for the regularization of nonlinear ill-posed problems, J. Integral Equ. Appl., 20, 201-228, (2008) · Zbl 1156.65053
[27] Kaltenbacher, B., A convergence rates result for an iteratively regularized Gauss-Newton Halley method in Banach space, Inverse Probl., 31, 1, (2015) · Zbl 1314.65079
[28] Kaltenbacher, B.; Neubauer, A.; Scherzer, O., Iterative regularization methods for nonlinear ill-posed problems, (2008), Walter de Gruyter · Zbl 1145.65037
[29] Kaltenbacher, B.; Schöpfer, F.; Schuster, T., Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems, Inverse Probl., 25, (2009) · Zbl 1176.65070
[30] Kokurin, M. Y., The global search in the Tikhonov scheme, Russ. Math., 54, 17-26, (2010) · Zbl 1227.47038
[31] Marco, D., An iterative multigrid regularization method for Toeplitz discrete ill-posed problems, Numer. Math., Theory Methods Appl., 5, 1, 43-61, (2012) · Zbl 1265.65074
[32] Morigi, S.; Reichel, L.; Sgallari, F., Cascadic multiresolution methods for image deblurring, SIAM J. Imaging Sci., 1, 1, 51-74, (2008) · Zbl 1144.65092
[33] Nesterov, Y., A method of solving a convex programming problem with convergence rate O(\(1 / k^2\)), Sov. Math. Dokl., 27, 372-376, (1983) · Zbl 0535.90071
[34] Ramlau, R., A steepest descent algorithm for the global minimization of the Tikhonov functional, Inverse Probl., 18, 381-405, (2002) · Zbl 1005.65057
[35] Ramlau, R., TIGRA-an iterative algorithm for regularizing nonlinear ill-posed problems, Inverse Probl., 19, 433-465, (2003) · Zbl 1029.65059
[36] Resmerita, E.; Scherzer, O., Error estimates for non-quadratic regularization and the relation to enhancement, Inverse Probl., 22, 801-814, (2006) · Zbl 1103.65062
[37] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Phys. D, 60, 259-268, (1992) · Zbl 0780.49028
[38] Tibshirani, R., Regression shrinkage and selection via the lasso, J. R. Stat. Soc., Ser. B, 58, 267-288, (1996) · Zbl 0850.62538
[39] Wang, W.; Anzengruber, S. W.; Ramlau, R.; Han, B., A global minimization algorithm for Tikhonov functionals with sparsity constraints, Appl. Anal., 94, 580-611, (2015) · Zbl 1311.65062
[40] Wang, Y. C.; Liu, J. J., Identification of non-smooth boundary heat dissipation by partial boundary data, Appl. Math. Lett., 69, 42-48, (2017) · Zbl 1379.80007
[41] Zălinscu, C., Convex analysis in general vector spaces, (2002), World Scientific Publishing Co., Inc. River Edge, New Jersey
[42] Zhong, M.; Wang, W., A global minimization algorithm for Tikhonov functionals with p-convex (\(p \geqslant 2\)) penalty terms in Banach spaces, Inverse Probl., 32, (2016) · Zbl 1351.49038
[43] M. Zhong, W. Wang, Two Point Gradient Method with non-smooth constraints in Banach spaces, preprint.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.