Inverse Problems In Banach Spaces Research Articles

A novel two-point Landweber-type method for Regularization of non-smooth inverse problems in Banach spaces

A novel two-point Landweber-type method for Regularization of non-smooth inverse problems in Banach spaces

Generalized Inexact Newton-Landweber Iteration for Possibly Non-Smooth Inverse Problems in Banach Spaces

In this paper, we consider a generalized inexact Newton-Landweber iteration to solve nonlinear ill-posed inverse problems in Banach spaces, where the forward operator might not be Gâteaux differentiable. The method is designed with non-smooth convex penalty terms, including L1-like and total variation-like penalty functionals, to capture special features of solutions such as sparsity and piecewise constancy. Furthermore, the inaccurate inner solver is incorporated into the minimization problem in each iteration step. Under some assumptions, based on ε-subdifferential, we establish the convergence analysis of the proposed method. Finally, some numerical simulations are provided to illustrate the effectiveness of the method for solving both smooth and non-smooth nonlinear inverse problems.

A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces

Abstract In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 {L^{1}} -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method.

Convergence rates of stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces

We investigate stationary and non-stationary asymptotical regularization methods for statistical inverse problems in Banach spaces. The mean-squared errors (MSE) of both methods are estimated without the conventional assumption of the commutativity between the solution variance and the noise variance. Moreover, the a priori smoothness of solution is characterized in terms of real interpolation, which is strictly weaker than the classical settings. The sufficient and necessary conditions of optimal convergence rates of MSE are proven. Our results extend theoretical findings in Hilbert settings given in the recent paper by Lu et al. [SIAM/ASA J. Uncertain., 9 (2021), pp. 1488-1525]. Two examples of integral equations with white noise and inverse source problems in elliptic partial differential equations are demonstrated to show the usefulness of our theoretical results.

Convergence analysis of inexact Newton–Landweber iteration under Hölder stability

In this paper, we focus on a class of inverse problems with Lipschitz continuous Fréchet derivatives both in Hilbert spaces and Banach spaces. The convergence and convergence rate of the inexact Newton–Landweber method (INLM) for such problems are presented under some assumptions. For the inverse problems in Hilbert spaces, we revisit the convergence result and the convergence rate of the INLM under Lipschitz condition and Hölder stability. Furthermore, the INLM for nonlinear inverse problems in Banach spaces is also considered. By using a Hölder stability corresponding to the Bregman distance, we derive the convergence property and convergence rate of the method.

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gâteaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (2016 Numer. Math. 133 655–684) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the non-existing Gâteaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

Analysis of a generalized regularized Gauss–Newton method under heuristic rule in Banach spaces

In this work, we propose a generalized regularized Gauss–Newton method to solve nonlinear inverse problems in Banach spaces and consider its heuristic stopping rule. The proposed method only requires a generalized operator which could be independent of the Fréchet derivative of the forward operator, thus it can solve smooth as well as non-smooth inverse problems. In addition, the proposed heuristic rule is fully data driven and does not require any information on the noise level. Therefore, it can cope with inverse problems whose noise levels are unknown. An a posteriori error for this method under heuristic rule is obtained and general convergence result is given under some standard assumptions. Numerical experiments are presented to illustrate the efficiency of the proposed method.

An accelerated homotopy perturbation iteration for nonlinear ill-posed problems in Banach spaces with uniformly convex penalty

We propose and analyze an accelerated homotopy perturbation iteration based on Nesterov strategy for nonlinear inverse problems in Banach spaces. The method allows to use L 1-like and total variation-like penalty terms. Therefore, it can cope with the situation where the sought solution is sparse and piece-wise constant. Under some standard assumptions, we show the convergence and regularization properties of the method. Numerical experiments are presented to show the effectiveness as well as the acceleration effect of the proposed method.

A novel two-point gradient method for regularization of inverse problems in Banach spaces

In this paper, we introduce a novel two-point gradient method for solving the ill-posed problems in Banach spaces and study its convergence analysis. The method is based on the well known iteratively regularized Landweber iteration method together with an extrapolation strategy. The general formulation of iteratively regularized Landweber iteration method in Banach spaces excludes the use of certain functions such as total variation like penalty functionals, functions, etc. The novel scheme presented in this paper allows to use such non-smooth penalty terms that can be helpful in practical applications involving the reconstruction of several important features of solutions such as piecewise constancy and sparsity. We carefully discuss the choices for important parameters, such as combination parameters and step sizes involved in the design of the method. Additionally, we discuss an example to validate our assumptions.

REGINN-IT method with general convex penalty terms for nonlinear inverse problems

In the spirit of the method in Margotti and Rieder (An inexact Newton regularization in Banach spaces based on the non-stationary iterated Tikhonov method. J Inverse Ill-Posed Probl. 2015;23:373–392), we propose an inexact Newton regularization which is based on the non-stationary iterated Tikhonov method for non-smooth solutions of nonlinear inverse problems in Banach spaces. The method consists of an inner iteration and an outer iteration. The outer iteration is terminated by the discrepancy principle and consists of an inexact Newton regularization method. The inner iteration is performed by the non-stationary iterated Tikhonov method. The remarkable point is that the penalty terms can be general convex functions. Under certain assumptions on nonlinear operators, the convergence analysis of our method is given by making use of tools from convex analysis. Furthermore, numerical simulations are provided to support the theoretical results and test the performance of the method.

Iteratively regularized Landweber iteration method: Convergence analysis via Hölder stability

In this paper, the local convergence of Iteratively regularized Landweber iteration method is investigated for solving non-linear inverse problems in Banach spaces. Our analysis mainly relies on the assumption that the inverse mapping satisfies the Hölder stability estimate locally. We consider both noisy as well as non-noisy data in our analysis. Under the a-priori choice of stopping index for noisy data, we show that the iterates remain in a certain ball around exact solution and obtain the convergence rates. The convergence of the Iteratively regularized Landweber iterates to the exact solution is shown under certain assumptions in the case of non-noisy data and as a by-product, under different conditions, two different convergence rates are obtained.

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

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 L1-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.

An accelerated Kaczmarz type method for nonlinear inverse problems in Banach spaces with uniformly convex penalty

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 L1 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.

A revisit on Landweber iteration

In this paper we revisit the discrepancy principle for Landweber iteration for solving linear as well as nonlinear inverse problems in Banach spaces and prove a new convergence result which requires neither the Gâteaux differentiability of the forward operator nor the reflexivity of the image space. Therefore, we expand the applied range of the discrepancy principle for Landweber iteration to cover non-smooth ill-posed inverse problems and to handle the situation that the data is contaminated by various types of noise.

Fast subspace optimization method for nonlinear inverse problems in Banach spaces with uniformly convex penalty terms

In this paper, we consider the subspace optimization method for solving nonlinear inverse problems in Banach spaces, which is based on the sequential Bregman projections with uniformly convex penalty term. The penalty term is allowed to be non-smooth, including L1 and total variation like penalty functionals, to reconstruct the special features of solutions such as sparsity and discontinuities. Instead of just utilizing the current gradient like the Landweber iteration, the method uses multiple search directions in each iteration to accelerate the convergence. Moreover, their step lengths are calculated by the projection onto the subspace that contains the solution set of the unperturbed problem. Under certain assumptions, we present the detailed convergence analysis when the data is given exactly. For the data containing noise, we use the discrepancy principle as stopping rule and then obtain the regularization result of the method. Finally, some numerical simulations for parameter identification problems are provided to illustrate the effectiveness of capturing the property of exact solutions and the acceleration effect of the method.

Regularization of inverse problems by two-point gradient methods in Banach spaces

In this paper, we propose and analyze a two-point gradient method for solving inverse problems in Banach spaces which is based on the Landweber iteration and an extrapolation strategy. The method allows to use non-smooth penalty terms, including the $$L^1$$ -like and the total variation-like penalty functionals, which are significant in reconstructing special features of solutions such as sparsity and piecewise constancy in practical applications. The design of the method involves the choices of the step sizes and the combination parameters which are carefully discussed. Numerical simulations are presented to illustrate the effectiveness of the proposed method.

An iteration regularization method with general convex penalty for nonlinear inverse problems in Banach spaces

This paper investigates the construction, analysis and implementation of a novel iterative regularization scheme with general convex penalty term for nonlinear inverse problems in Banach spaces. The method is formulated by combining the homotopy perturbation technique. The penalty term is allowed to be nonsmooth to include L1-norm or total variation-like penalty functionals, especially suitable for detecting the special features of the sought solutions such as sparsity or piecewise constant. By using tools from convex analysis in Banach spaces, we establish the detailed convergence and stability results. Numerical simulations for one-dimensional and two-dimensional parameter identification problems are performed to validate that our approach is competitive in terms of reducing the overall computational time in comparison with the existing Landweber iteration with general convex penalty. Moreover, our approach is used to the non-Gaussian noisy data to illustrate the iterative regularization in Banach spaces.

Heuristic rule for non-stationary iterated Tikhonov regularization in Banach spaces

In this paper we propose a heuristic rule of Hanke–Raus type for non-stationary iterated Tikhonov regularization for solving ill-posed inverse problems in Banach spaces. This heuristic rule does not need any information on the noise level and is fully data driven. Under certain conditions on the noisy data, we obtain a convergence result. Various numerical simulations are provided to illustrate the efficiency of the proposed heuristic rule.

A fast subspace optimization method for nonlinear inverse problems in Banach spaces with an application in parameter identification

We introduce and analyze a fast iterative method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. The key idea, in contrast to the standard Landweber method, is to use multiple search directions per iteration in combination with a regulation of the step width in order to reduce the total number of iterations. This method is suitable for both exact and noisy data. In the latter case, we obtain a regularization method. An algorithm with two search directions is used for the numerical identification of a parameter in an elliptic boundary value problem.

Existence of variational source conditions for nonlinear inverse problems in Banach spaces

Abstract Variational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.