Gu, Ruixue; Han, Bo; Tong, Shanshan; Chen, Yong An accelerated Kaczmarz type method for nonlinear inverse problems in Banach spaces with uniformly convex penalty. (English) Zbl 1456.65146 J. Comput. Appl. Math. 385, Article ID 113211, 22 p. (2021). Summary: In this paper, we propose and analyze a novel Kaczmarz type method for solving inverse problems which can be written as systems of nonlinear operator equations in Banach spaces. The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms. The penalty term is allowed to be non-smooth, including the \(L^1\) and the total variation like penalty functionals, to reconstruct special features of solutions such as sparsity and piecewise constancy. To accelerate the iteration, we introduce a sophisticated rule to determine the step sizes per iteration. Under certain conditions, we present the convergence result of the proposed method in the exact data case. When the data is given approximately, together with a suitable stopping rule, we establish the stability and regularization properties of the method. Finally, some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements are provided to validate the effectiveness of the proposed method. Cited in 4 Documents MSC: 65N21 Numerical methods for inverse problems for boundary value problems involving PDEs 65N20 Numerical methods for ill-posed problems for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65K10 Numerical optimization and variational techniques 65H10 Numerical computation of solutions to systems of equations 65H20 Global methods, including homotopy approaches to the numerical solution of nonlinear equations Keywords:nonlinear inverse problems; Kaczmarz type method; iterative regularization; Banach spaces; uniformly convex penalty Software:KELLEY PDFBibTeX XMLCite \textit{R. Gu} et al., J. Comput. Appl. Math. 385, Article ID 113211, 22 p. (2021; Zbl 1456.65146) Full Text: DOI References: [1] Engl, H. W.; Kunisch, K.; eubauer, A., Convergence rates for tikhonov regularisation of non-linear ill-posed problems, Inverse Problems, 5, 4, 523 (1989) · Zbl 0695.65037 [2] Schuster, T.; Kaltenbacher, B.; Hofmann, B.; Kazimierski, K. S., (Regularization Methods in Banach Spaces. Regularization Methods in Banach Spaces, Radon Series on Computational and Applied Mathematics, vol. 10 (2012), Walter de Gruyter: Walter de Gruyter Berlin) · Zbl 1259.65087 [3] Neubauer, A., Tikhonov regularization for non-linear ill-posed problems: optimal convergence rates and finite-dimensional approximation, Inverse Problems, 5, 541-557 (1989) · Zbl 0695.65038 [4] Hofmann, B.; Kaltenbacher, B.; Pöschl, C.; Scherzer, O., A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23, 987-1010 (2007) · Zbl 1131.65046 [5] Engl, H. W.; Hanke, M.; Neubauer, A., Regularization of Inverse Problems (1996), Kluwer: Kluwer Dordrecht · Zbl 0859.65054 [6] 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 [7] Kaltenbacher, B.; Schöpfer, F.; Schuster, T., Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems, Inverse Problems, 25, Article 065003 pp. (2009) · Zbl 1176.65070 [8] Kaltenbacher, B.; Neubauer, A.; Scherzer, O., (Iterative Regularization Methods for Nonlinear Ill-Posed Problems. Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Radon Series on Computational and Applied Mathematics, vol. 6 (2008), Walter de Gruyter: Walter de Gruyter Berlin) · Zbl 1145.65037 [9] Kowar, R.; Scherzer, O., (Convergence Analysis of A Landweber-Kaczmarz Method for Solving Nonlinear Ill-Posed Problems. Convergence Analysis of A Landweber-Kaczmarz Method for Solving Nonlinear Ill-Posed Problems, Ill-posed and Inverse Problems (Book Series), vol. 23 (2002)), 253-270 · Zbl 1048.65055 [10] Haltmeier, M.; Leitao, A.; Scherzer, O., Kaczmarz methods for regularizing nonlinear ill-posed equations: I. convergence analysis, Inverse Probl. Imaging, 1, 2, 289-298 (2007) · Zbl 1123.65051 [11] Leitao, A.; Svaiter, B. F., On projective Landweber-Kaczmarz methods for solving systems of nonlinear ill-posed equations, Inverse Problems, 32, Article 025004 pp. (2016) · Zbl 1344.65052 [12] Jin, Q.; Wang, W., Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals, Inverse Problems, 29, Article 085011 pp. (2013) · Zbl 1283.65056 [13] Jin, Q., Landweber-Kaczmarz method in Banach spaces with inexact inner solvers, Inverse Problems, 32, Article 104005 pp. (2016) · Zbl 1356.65146 [14] Cao, L.; Han, B., Convergence analysis of the homotopy perturbation method for solving nonlinear ill-posed operator equations, Comput. Math. Appl., 61, 2058-2061 (2011) · Zbl 1219.65167 [15] Cao, L.; Han, B.; Wang, W., Homotopy perturbation method for nonlinear ill-posed operator equations, Int. J. Nonlinear Sci. Numer. Simul., 10, 1319-1322 (2009) [16] Wang, J.; Wang, W.; Han, B., An iteration regularizaion method with general convex penalty for nonlinear inverse problems in banach spaces, J. Comput. Appl. Math., 361, 472-486 (2019) · Zbl 1457.65173 [17] Hein, T.; Kazimierski, K. S., Accelerated Landweber iteration in Banach spaces, Inverse Problems, 26, Article 055002 pp. (2010) · Zbl 1201.65095 [18] Hegland, M.; Jin, Q.; Wang, W., Accelerated Landweber iteration with convex penalty for linear inverse problems in Banach spaces, Appl. Anal., 94, 524-547 (2015) · Zbl 1311.65054 [19] Boţ, R.; Hein, T., Iterative regularization with a general penalty term-theory and application to \(L^1\) and TV regularization, Inverse Problems, 28, Article 104010 pp. (2012) · Zbl 1269.47054 [20] Maaß, P.; Strehlow, R., An iterative regularization method for nonlinear problems based on Bregman projections, Inverse Problems, 32, Article 115013 pp. (2016) · Zbl 1433.65104 [21] Gu, R.; Han, B.; Chen, Y., Fast subspace optimization method for nonlinear inverse problems in banach spaces with uniformly convex penalty terms, Inverse Problems, 35, 12, Article 125011 pp. (2019) · Zbl 1485.65066 [22] Cioranescu. Geometry of Banach Spaces, I., Duality Mappings and Nonlinear Problems (1990), Kluwer Academic Pub [23] Zalinescu, C., Convex Analysis in General Vector Spaces (2002), World Scientific: World Scientific Singapore · Zbl 1023.46003 [24] Schirotzek, W., Nonsmooth Analysis, Vol. 266, 1-56 (2007), Springer Berlin [25] Kelley, C. T., Iterative Methods for Optimization (1999), SIAM: SIAM Philadephia, PA · Zbl 0934.90082 [26] Rudin, L. I.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268 (1992) · Zbl 0780.49028 [27] 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 [28] 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 [29] Zhu, M.; Chan, T. F., An efficient primal-dual hybrid gradient algorithm for total variation image restoration, (UCLA CAM Report 08-34 (2008)) [30] Burger, M.; Kaltenbacher, B., Regularizing Newton-Kaczmarz methods for nonlinear ill-posed problems, SIAM J. Numer. Anal., 44, 153-182 (2006) · Zbl 1112.65049 [31] Jin, Q., Inexact Newton-Landweber iteration in Banach spaces with nonsmooth convex penalty terms, SIAM J. Numer. Anal., 53, 2389-2413 (2015) · Zbl 1326.65064 [32] Jin, B.; Maass, P., Sparsity regularization for parameter identification problems, Inverse Problems, 28, Article 123001 pp. (2012) · Zbl 1280.47063 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.