Inexact Newton regularization in Banach spaces based on two-point gradient method with uniformly convex penalty terms

Abstract In this paper, we propose and analyze a new inexact Newton regularization method for solving nonlinear inverse problems in Banach spaces. The method consists of an outer Newton iteration and an inner scheme which provides increments by applying a regularization technique to the local linearized equation around the current iterate. In order to accelerate the convergence, we employ a two-point gradient method as inner regularization scheme, which is based on the Landweber iteration and an extrapolation strategy. The penalty term is allowed to be non-smooth, including L 1 -like and total variation-like penalty functionals, to detect special features of solutions such as sparsity and piecewise constancy. We present, under certain assumptions, the detailed analysis of convergence and regularization properties of the method. Finally, some numerical experiments on elliptic parameter identification and Robin coefficient reconstruction problems are provided to illustrate the effectiveness of reconstructing the properties of sought solutions and the acceleration effect of the method.

[1]  Andreas Rieder,et al.  On convergence rates of inexact Newton regularizations , 2001, Numerische Mathematik.

[2]  Y. Nesterov.  A method for solving the convex programming problem with convergence rate O(1/k^2) , 1983 .

[3]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[4]  Qinian Jin,et al.  Inexact Newton–Landweber iteration for solving nonlinear inverse problems in Banach spaces , 2012 .

[5]  Ronny Ramlau,et al.  Nesterov’s accelerated gradient method for nonlinear ill-posed problems with a locally convex residual functional , 2018, Inverse Problems.

[6]  M. Hanke.  Conjugate gradient type methods for ill-posed problems , 1995 .

[7]  Kamil S. Kazimierski,et al.  Accelerated Landweber iteration in Banach spaces , 2010 .

[8]  F. Schöpfer,et al.  Fast regularizing sequential subspace optimization in Banach spaces , 2008 .

[9]  Marc Teboulle,et al.  Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems , 2009, IEEE Transactions on Image Processing.

[10]  Bo Han,et al.  An accelerated sequential subspace optimization method based on homotopy perturbation iteration for nonlinear ill-posed problems , 2019, Inverse Problems.

[11]  Wei Wang,et al.  Landweber iteration of Kaczmarz type with general non-smooth convex penalty functionals , 2013, 1307.4311.

[12]  Tal Schuster,et al.  Nonlinear iterative methods for linear ill-posed problems in Banach spaces , 2006 .

[13]  Andreas Rieder,et al.  Inexact Newton Regularization Using Conjugate Gradients as Inner Iteration , 2005, SIAM J. Numer. Anal..

[14]  Qinian Jin,et al.  Landweber-Kaczmarz method in Banach spaces with inexact inner solvers , 2016, 1603.07490.

[15]  B. Kaltenbacher,et al.  Iterative methods for nonlinear ill-posed problems in Banach spaces: convergence and applications to parameter identification problems , 2009 .

[16]  Bo Han,et al.  Fast subspace optimization method for nonlinear inverse problems in Banach spaces with uniformly convex penalty terms , 2019, Inverse Problems.

[17]  Qinian Jin.  On the order optimality of the regularization via inexact Newton iterations , 2012, Numerische Mathematik.

[18]  M. Hanke,et al.  A convergence analysis of the Landweber iteration for nonlinear ill-posed problems , 1995 .

[19]  Andreas Rieder,et al.  A Kaczmarz Version of the REGINN-Landweber Iteration for Ill-Posed Problems in Banach Spaces , 2014, SIAM J. Numer. Anal..

[20]  Bo Han,et al.  A new Kaczmarz-type method and its acceleration for nonlinear ill-posed problems , 2019, Inverse Problems.

[21]  M. Hegland,et al.  Accelerated Landweber iteration with convex penalty for linear inverse problems in Banach spaces , 2015 .

[22]  Wei Wang,et al.  Regularization of inverse problems by two-point gradient methods in Banach spaces , 2019, Numerische Mathematik.

[23]  Andreas Rieder,et al.  On the regularization of nonlinear ill-posed problems via inexact Newton iterations , 1999 .

[24]  Min Zhong,et al.  Nonstationary iterated Tikhonov regularization in Banach spaces with uniformly convex penalty terms , 2014, Numerische Mathematik.

[25]  Qinian Jin,et al.  Inexact Newton-Landweber Iteration in Banach Spaces with NonSmooth Convex Penalty Terms , 2015, SIAM J. Numer. Anal..

[26]  Ronny Ramlau,et al.  Convergence analysis of a two-point gradient method for nonlinear ill-posed problems , 2017 .

[27]  B. Han,et al.  A proximal regularized Gauss-Newton-Kaczmarz method and its acceleration for nonlinear ill-posed problems , 2020 .

[28]  M. Hanke.  Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems , 1997 .

[29]  L. Rudin,et al.  Nonlinear total variation based noise removal algorithms , 1992 .

[30]  I. Ciorǎnescu.  Geometry of banach spaces, duality mappings, and nonlinear problems , 1990 .

[31]  Jijun Liu,et al.  Identification of non-smooth boundary heat dissipation by partial boundary data , 2017, Appl. Math. Lett..

[32]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[33]  R. Boţ,et al.  Iterative regularization with a general penalty term—theory and application to L1 and TV regularization , 2012 .

[34]  Barbara Kaltenbacher,et al.  Regularization Methods in Banach Spaces , 2012, Radon Series on Computational and Applied Mathematics.

[35]  Andreas Rieder,et al.  Towards a general convergence theory for inexact Newton regularizations , 2009, Numerische Mathematik.

[36]  C. Zălinescu.  Convex analysis in general vector spaces , 2002 .