Secant-type iteration for nonlinear ill-posed equations in Banach space

In this paper, we study secant-type iteration for nonlinear ill-posed equations involving 𝑚-accretive mappings in Banach spaces. We prove that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on the first Fréchet derivative of the operator. Further, using a general Hölder-type source condition, we obtain an optimal error estimate. We also use the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter.

On the Parameter Choice in the Multilevel Augmentation Method

In this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

A linear regularization method for a parameter identification problem in heat equation

Recently, Nair and Roy (2017) considered a linear regularization method for a parameter identification problem in an elliptic PDE. In this paper, we consider similar procedure for identifying the diffusion coefficient in the heat equation, modifying the Sobolev spaces involved appropriately. We derive error estimates under appropriate conditions and also consider the finite-dimensional realization of the method, which is essential for practical application. In the analysis of finite-dimensional realization, we give a procedure to obtain finite-dimensional subspaces of an infinite-dimensional Hilbert space {L^{2}(0,T;H^{1}(\Omega))} by doing double discretization, that is, discretization corresponding to both the space and time domain. Also, we analyze the parameter choice strategy and obtain an a posteriori parameter which is order optimal.

A Multilevel Iteration Method for Solving a Coupled Integral Equation Model in Image Restoration

The problem of out-of-focus image restoration can be modeled as an ill-posed integral equation, which can be regularized as a second kind of equation using the Tikhonov method. The multiscale collocation method with the compression strategy has already been developed to discretize this well-posed equation. However, the integral computation and solution of the large multiscale collocation integral equation are two time-consuming processes. To overcome these difficulties, we propose a fully discrete multiscale collocation method using an integral approximation strategy to compute the integral, which efficiently converts the integral operation to the matrix operation and reduces costs. In addition, we also propose a multilevel iteration method (MIM) to solve the fully discrete integral equation obtained from the integral approximation strategy. Herein, the stopping criterion and the computation complexity that correspond to the MIM are shown. Furthermore, a posteriori parameter choice strategy is developed for this method, and the final convergence order is evaluated. We present three numerical experiments to display the performance and computation efficiency of our proposed methods.

Adaptive multi-parameter regularization approach to construct the distribution function of relaxation times

Determination of the distribution function of relaxation times (DFRT) is an approach that gives us more detailed insight into system processes, which are not observable by simple electrochemical impedance spectroscopy (EIS) measurements. DFRT maps EIS data into a function containing the timescale characteristics of the system under consideration. The extraction of such characteristics from noisy EIS measurements can be described by Fredholm integral equation of the first kind that is known to be ill-posed and can be treated only with regularization techniques. Moreover, since only a finite number of EIS data may actually be obtained, the above-mentioned equation appears as after application of a collocation method that needs to be combined with the regularization. In the present study, we discuss how a regularized collocation of DFRT problem can be implemented such that all appearing quantities allow symbolic computations as sums of table integrals. The proposed implementation of the regularized collocation is treated as a multi-parameter regularization. Another contribution of the present work is the adjustment of the previously proposed multiple parameter choice strategy to the context of DFRT problem. The resulting strategy is based on the aggregation of all computed regularized approximants, and can be in principle used in synergy with other methods for solving DFRT problem. We also report the results from the experiments that apply the synthetic data showing that the proposed technique successfully reproduced known exact DFRT. The data obtained by our techniques is also compared to data obtained by well-known DFRT software (DRTtools).

Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

A parameter choice strategy for a multilevel augmentation method in iterated Lavrentiev regularization

In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

Multiscale Galerkin methods for the nonstationary iterated Tikhonov method with a modified posteriori parameter selection

In this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.