General Convex Penalty Research Articles

Two-point Landweber-type method with convex penalty terms for nonsmooth nonlinear inverse problems

Abstract In this work, we propose a two-point Landweber-type method with general convex penalty terms for solving nonsmooth nonlinear inverse problems. The design of our method can cope with nonsmooth nonlinear inverse problems or the nonlinear inverse problems whose data are contaminated by various types of noise. The method consists of the two-point acceleration strategy and inner solvers. Inner solvers are used to solve the minimization problems with respect to the penalty terms in each steps. If the minimization problems can be solved explicitly, the inner solvers will be chosen to be the exact solvers. Otherwise, we will use inexact solvers as inner solvers. Convergence results are given without utilizing the Gâteaux differentiability of the forward operator or the reflexivity of the image space. Numerical simulations are given to test the performance of the proposed method.

Levenberg–Marquardt method with general convex penalty for nonlinear inverse problems

We consider a Levenberg–Marquardt method for solving nonlinear inverse problems in Hilbert spaces. The proposed method uses general convex penalty terms to reconstruct nonsmooth solutions of inverse problems. Instead of an a priori choice, the regularization parameter in each iteration is chosen by solving an equation which depends on the residual. We utilize the discrepancy principle to terminate the iteration and give the convergence results. In addition, numerical simulations are presented to test the performance of the method.

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.

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.

Penalty-based smoothness conditions in convex variational regularization

Abstract The authors study Tikhonov regularization of linear ill-posed problems with a general convex penalty defined on a Banach space. It is well known that the error analysis requires smoothness assumptions. Here such assumptions are given in form of inequalities involving only the family of noise-free minimizers along the regularization parameter and the (unknown) penalty-minimizing solution. These inequalities control, respectively, the defect of the penalty, or likewise, the defect of the whole Tikhonov functional. The main results provide error bounds for a Bregman distance, which split into two summands: the first smoothness-dependent term does not depend on the noise level, whereas the second term includes the noise level. This resembles the situation of standard quadratic Tikhonov regularization in Hilbert spaces. It is shown that variational inequalities, as these were studied recently, imply the validity of the assumptions made here. Several examples highlight the results in specific applications.

A converse result for Banach space convergence rates in Tikhonov-type convex regularization of ill-posed linear equations

Abstract We consider Tikhonov-type variational regularization of ill-posed linear operator equations in Banach spaces with general convex penalty functionals. Upper bounds for certain error measures expressing the distance between exact and regularized solutions, especially for Bregman distances, can be obtained from variational source conditions. We prove that such bounds are optimal in case of twisted Bregman distances, a specific a priori parameter choice, and low regularity of the exact solution, that is, the rate function is also an asymptotic lower bound for the error measure. This result extends existing converse results from Hilbert space settings to Banach spaces without adhering to spectral theory.

Landweber-Kaczmarz method in Banach spaces with inexact inner solvers

In recent years the Landweber-Kaczmarz method has been proposed for solving nonlinear ill-posed inverse problems in Banach spaces using general convex penalty functions. The implementation of this method involves solving a (nonsmooth) convex minimization problem at each iteration step and the existing theory requires its exact resolution which in general is impossible in practical applications. In this paper we propose a version of the Landweber-Kaczmarz method in Banach spaces in which the minimization problem involved in each iteration step is solved inexactly. Based on the -subdifferential calculus we give a convergence analysis of our method. Furthermore, using Nesterov's strategy, we propose a possible accelerated version of the Landweber-Kaczmarz method. Numerical results on computed tomography and parameter identification in partial differential equations are provided to support our theoretical results and to demonstrate our accelerated method.

Alternating Direction Method of Multipliers for Linear Inverse Problems

In this paper we propose an iterative method using alternating direction method of multipliers (ADMM) strategy to solve linear inverse problems in Hilbert spaces with general convex penalty term. When the data is given exactly, we give a convergence analysis of our ADMM algorithm without assuming the existence of Lagrange multiplier. In case the data contains noise, we show that our method is a regularization method as long as it is terminated by a suitable stopping rule. Various numerical simulations are performed to test the efficiency of the method.

Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data

In this paper, we study a Tikhonov-type method for ill-posed nonlinear operator equations g† = F(u†), where g† is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density tg†, where t > 0 may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations and phase retrieval problems in optics. Our approach uses a Kullback–Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as t → ∞ both for an a priori and for a Lepskii-type parameter choice rule.

Morozov's discrepancy principle for Tikhonov-type functionals with nonlinear operators

In this paper we deal with Morozov's discrepancy principle as an a posteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for nonlinear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for suitable penalty terms the regularized solutions converge to the true solution in the topology induced by Ψ. It is illustrated that for this parameter choice rule it holds α → 0, δq/α → 0 as the noise level δ goes to 0. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.

Regularization of ill-posed problems in Banach spaces: convergence rates

This paper deals with quantitative aspects of regularization for ill-posed linear equations in Banach spaces, when the regularization is done using a general convex penalty functional. The error estimates shown here by means of Bregman distances yield better convergences rates than those already known for maximum entropy regularization, as well as for total variation regularization.