REGULARIZATION METHODS FOR NONLINEAR ILL-POSED PROBLEMS WITH ACCRETIVE OPERATORS

This article is devoted to the regularization of nonlinear ill-posed problems with accretive operators in Banach spaces. The data involved are assumed to be known approximately. The authors concentrate their discussion on the convergence rates of regular solutions.

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

Newton type methods are one kind of the efficient methods to solve nonlinear ill-posed problems, which have attracted extensive attention. However, computational cost of Newton type methods is high because practical problems are complicated. We propose a mixed Newton-Tikhonov method, i.e., one step Newton-Tikhonov method with several other steps of simplified Newton-Tikhonov method. Convergence and stability of this method are proved under some conditions. Numerical experiments show that the proposed method has obvious advantages over the classical Newton method in terms of computational costs.

Arnoldi Projection Fractional Tikhonov for Large Scale Ill-Posed Problems

It is well known that Tikhonov regularization in standard form may determine approximate solutions that are too smooth for ill-posed problems,so fractional Tikhonov methods have been introduced to remedy this shortcoming.And Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining apartial Arnoldi decomposition of the given matrix.In this paper,we propose a new method to compute an approximate solution of large scale linear discrete ill-posed problems which applies projection fractional Tikhonov regularization in Krylov subspace via Arnoldi process.The projection fractional Tikhonov regularization combines the fractional matrices and orthogonal projection operators.A suitable value of the regularization parameter is determined by the discrepancy principle.Numerical examples with application to image restoration are carried out to examine that the performance of the method.

Nonlinear implicit iterative method for solving nonlinear ill-posed problems

In the paper, we extend the implicit iterative method for linear ill-posed operator equations to solve nonlinear ill-posed problems. We show that under some conditions the error sequence of solutions of the nonlinear implicit iterative method is monotonically decreasing and, with this monotonicity, prove convergence of the new method for both the exact and perturbed equations.

A Newton-type method for nonlinear ill-posed problems with A-smooth regularization

In this paper we present a regularized Newton-type method for ill-posed problems, by using the A-smooth regularization to solve the linearized ill-posed equations. For noisy data a proper a posteriori stopping rule is used that yields convergence of the Newton iteration to a solution, as the noise level goes to zero, under certain smoothness conditions on the nonlinear operator. Some appropriate assumptions on the closedness and smoothness of the starting value and the solution are shown to lead to optimal convergence rates.

Fast Multilevel Methods for Solving Ill-posed Problems

1 Introduetion Many industrial and engineering applieations require numerieally solving ill-posed Problems. Regularization methods are employed to find approximate solutions of these problems.The choice of regularization

On the Regularization Method for Solving Ill-Posed Problems with Unbounded Operators

Let be a linear, closed, and densely defined unbounded operator, where X and Y are Hilbert spaces. Assume that A is not boundedly invertible. Suppose the equation Au=f is solvable, and instead of knowing exactly f only know its approximation satisfies the condition: In this paper, we are interested a regularization method to solve the approximation solution of this equation. This approximation is a unique global minimizer of the functional , for any , defined as follows: . We also study the stability of this method when the regularization parameter is selected a priori and a posteriori. At the same time, we give an application of this method to the weak derivative operator equation in Hilbert space.

A UNIFIED TREATMENT OF REGULARIZATION METHODS FOR LINEAR ILL-POSED PROBLEMS

By presenting a general framework, some regularization methods for solving linear ill-posed problems are considered in a unified manner. Applications to some specific approaches are illustrated.

A New Regularized Solution to Ill-Posed Problem in Coordinate Transformation

Coordinates transformation is generally required in GPS applications. If the transformation parameters are solved with the known coordinates in a small area using the Bursa model, the precision of transformed coordinates is generally very poor. Since the translation parameters and rotation parameters are highly correlated in this case, a very large condition number of the coefficient matrix A exists in the linear system of equations. Regularization is required to reduce the effects caused by the intrinsic ill-conditioning of the problem and noises in the data, and to stabilize the solution. Based on advanced regularized methods, we propose a new regularized solution to the ill-posed coordinate transformation problem. Simulation numerical experiments of coordinate transformation are given to shed light on the relationship among different regularization approaches. The results indicate that the proposed new method can obtain a more reasonable resolution with higher precision and/or robustness.

MULTILEVEL ITERATION METHODS FOR SOLVING LINEAR ILL-POSED PROBLEMS

In this paper we develop multilevel iteration methods for solving linear systems resulting from the Galerkin method and Tikhonov regularization for ill-posed problems. The algorithm and its convergence analysis are presented in an abstract framework.

Optimal Shape Factor and Fictitious Radius in the MQ-RBF:Solving Ill-Posed Laplacian Problems

To solve the Laplacian problems,we adopt a meshless method with the multiquadric radial basis function(MQRBF)as a basis whose center is distributed inside a circle with a fictitious radius.A maximal projection technique is developed to identify the optimal shape factor and fictitious radius by minimizing a merit function.A sample function is interpolated by theMQ-RBF to provide a trial coefficient vector to compute the merit function.We can quickly determine the optimal values of the parameters within a preferred rage using the golden section search algorithm.The novel method provides the optimal values of parameters and,hence,an optimal MQ-RBF;the performance of the method is validated in numerical examples.Moreover,nonharmonic problems are transformed to the Poisson equation endowed with a homogeneous boundary condition;this can overcome the problem of these problems being ill-posed.The optimal MQ-RBF is extremely accurate.We further propose a novel optimal polynomial method to solve the nonharmonic problems,which achieves high precision up to an order of 10^(−11).

Frozen Landweber Iteration for Nonlinear Ill-Posed Problems

In this paper we propose a modification of the Landweber iteration termed frozen Landweber iteration for nonlinear ill-posed problems. A convergence analysis for this iteration is presented. The numerical performance of this frozen Landweber iteration for a nonlinear Hammerstein integral equation is compared with that of the Landweber iteration. We obtain a shorter running time of the frozen Landweber iteration based on the same convergence accuracy.

A Restarted Conjugate Gradient Method for Ill-posed Problems

Abstract This paper presents a restarted conjugate gradient iterative algorithm for solving ill-posed problems. The damped Morozov's discrepancy principle is used as a stopping rule. Numerical experiments are given to illustrate the efficiency of the method.

An iterative virtual projection method to improve the reconstruction performance for ill-posed emission tomographic problems

In order to improve the reconstruction performance for ill-posed emission tomographic problems with limited projections, a generalized interpolation method is proposed in this paper, in which the virtual lines of projection are fabricated from, but not linearly dependent on, the measured projections. The method is called the virtual projection（VP） method.Also, an iterative correction method for the integral lengths is proposed to reduce the error brought about by the virtual lines of projection. The combination of the two methods is called the iterative virtual projection（IVP） method. Based on a scheme of equilateral triangle plane meshes and a six asymmetrically arranged detection system, numerical simulations and experimental verification are conducted. Simulation results obtained by using a non-negative linear least squares method,without any other constraints or regularization, demonstrate that the VP method can gradually reduce the reconstruction error and converges to the desired one by fabricating additional effective projections. When the mean square deviation of normal error superimposed on the simulated measured projections is smaller than 0.03, i.e., the signal-to-noise ratio（SNR）for the measured projections is higher than 30.4, the IVP method can further reduce the reconstruction error reached by the VP method apparently. In addition, as the regularization matrix in the Tikhonov regularization method is updated by an iterative correction process similar to the IVP method presented in this paper, or the Tikhonov regularization method is used in the IVP method, good improvement is achieved.

STRUCTURED CONDITION NUMBERS FOR THE TIKHONOV REGULARIZATION OF DISCRETE ILL-POSED PROBLEMS

The possibly most popular regularization method for solving the least squares problem rain ‖Ax - b‖2 with a highly ill-conditioned or rank deficient coefficient matrix A is the x Tikhonov regularization method. In this paper we present the explicit expressions of the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when A has linear structures. The structured condition numbers in the special cases of nonlinear structure i.e. Vandermonde and Cauchy matrices are also considered. Some comparisons between structured condition numbers and unstructured condition numbers are made by numerical experiments. In addition, we also derive the normwise, mixed and componentwise condition numbers for the Tikhonov regularization when the coefficient matrix, regularization matrix and right-hand side vector are all perturbed, which generalize the results obtained by Chu et al. [Numer. Linear Algebra Appl., 18 （2011）, 87-103].

REGULARIZATION OF AN ILL-POSED HYPERBOLIC PROBLEM AND THE UNIQUENESS OF THE SOLUTION BY A WAVELET GALERKIN METHOD

We consider the problem K（x）Uxx = utt , 0 〈 x 〈 1, t 〉 0, with the boundary condition u（O,t） = g（t） E LZ（R） and ux（O,t） = 0, where K（x） is continuous and 0 〈α≤ K （x） 〈 ＋∞. This is an ill-posed problem in the sense that, if the solution exists, it does not depend continuously on g. Considering the existence of a solution u（x, .） E H2（R） and using a wavelet Galerkin method with Meyer multiresolution analysis, we regularize the ill-posedness of the problem. Furthermore we prove the uniqueness of the solution for this problem.

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successful applied to signal/image deblurring problems.When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full.A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations.The smoother has to be an iterative regularization method.The grid transfer operator should preserve the regularization property of the smoother.This paper improves the iterative multigrid method proposed in[11]introducing a wavelet soft-thresholding denoising post-smoother.Such postsmoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects.The resulting iterative multigrid regularization method stabilizes the iterations so that and imprecise(over)estimate of the stopping iteration does not have a deleterious effect on the computed solution.Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.

A FINITE DIMENSIONAL METHOD FOR SOLVINGNONLINEAR ILL-POSED PROBLEMS

We propose a finite dimensional method to compute the solution of nonlinear ill-posed problems approximately and show that under certain conditions, the convergence can be guaranteed. Moreover, we obtain the rate of convergence of our method provided that the true solution satisfies suitable smoothness condition. Finally, we present two examples from the parameter estimation problems of differential equations and illustrate the applicability of our method.

A Note on the GMRES Method for Linear Discrete Ill-Posed Problems

In this paper,we are presenting a proposal for new modified algorithms for RRGMRES and AGMRES.It is known that RRGMRES and AGMRES are viable methods for solving linear discrete ill-posed problems.In this paper we have focused on the residual norm and have come-up with two improvements where successive updates and the stabilization of decreases for the residual norm improve performance respectively.Our numerical experiments confirm that our improved algorithms are effective for linear discrete ill-posed problems.

Numerical estimates for the regularization of nonautonomous ill-posed problems

The regularization of ill-posed problems has become a useful tool in studying initial value problems that do not adhere to certain desired properties such as continuous dependence of solutions on initial data.Because direct computation of the solution becomes difficult in this situation,many authors have alternatively approximated the solution by the solution of a closely defined well posed problem.In this paper,we demonstrate this process of regularization for nonautonomous ill-posed problems including the backward heat equation with a time-dependent diffusion coefficient.In the process,we provide two different approximate well posed models and numerically compare convergence rates of their solutions to a known solution of the original ill-posed problem.