Methods For Ill-posed Problems Research Articles

On an iterative fractional Tikhonov–Landweber method for ill-posed problems

Abstract Recently, fractional Landweber and iterative fractional Tikhonov method are well investigated for solving ill-posed problems. In this paper, based on a special discretization scheme of the gradient flow equation, a general iterative scheme is obtained. We call it iterative fractional Tikhonov–Landweber method. Error analysis is conducted for this method. A preliminary numerical test shows that the proposed method has its advantage.

Regularization Methods for Ill-Posed Problems in Quantum Optics

Regularization Methods for Ill-Posed Problems in Quantum Optics

Regularization methods for ill-posed problems of general physics

On the example of a specific physical problem of noise reduction associated with losses, dark counts, and background radiation, a summary of methods for regularizing ill-posed problems is given in the statistics of photocounts of quantum light. The mathematical formulation of the problem is presented by an operator equation of the first kind. The operator is generated by a matrix with countable elements. In the sense of Hadamard, the problem of reconstructing the number of photons of quantum light is due to the compactness of the operator of the mathematical model. A rigorous definition of a regularizing operator (regularizer) is given. The problem of stable approximation to the exact solution of the operator equation with inaccurately given initial data can be overcome by one of the most well-known regularization methods, the theoretical foundations of which were laid in the works of A.N. Tikhonov. The selection of an important class of regularizing algorithms is based on the construction of a parametric family of functions that are Borel measurable on the semiaxis and satisfy some additional conditions. The set of regularizers in this family includes most of the known regularization methods. The main ones are given in the work.

A New Gradient Method for Ill-Posed Problems

ABSTRACTIn this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method.

Spectral method for ill-posed problems based on the balancing principle

In this article, we consider a class of ill-posed problems in two-dimensional setting which can be formulated as the problem of solving ill-posed operator equation. The spectral method is a simple and effective regularization method for such a class of ill-posed problems. For the spectral method, we obtain the error estimates with optimal order under an adaptive a posteriori parameter choice rule called balancing principle. Two ill-posed examples including an inverse diffraction problem and a deblurring problem are provided.

An effective method for solving nonlinear equations and its application

The linearized partial differential equation from the nonlinear partial differential equation which was proposed by Rudin, Osher and Fatemi [L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms] for solving image decomposition was introduced by Chambolle [A. Chambolle, An algorithm for total variation minimization and applications] and R. Acar and C.R. Vogel [R. Acar and C. R. Vogel, Analysis of bounded variation penalty methods for ill-posed problems]. In this paper, we propose a method for solving the linearized partial differential equation and we show numerical results for denoising which demonstrate a significant improvement over other previous works.

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

AbstractIn this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).MSC:65F22, 65J15, 65J22, 65M30, 47A52.

On a Level-Set Method for Ill-Posed Problems with Piecewise Nonconstant Coefficients

We investigate a level-set-type method for solving ill-posed problems, with the assumption that the solutions are piecewise, but not necessarily constant functions with unknown level sets and unknown level values. In order to get stable approximate solutions of the inverse problem, we propose a Tikhonov-type regularization approach coupled with a level-set framework. We prove the existence of generalized minimizers for the Tikhonov functional. Moreover, we prove convergence and stability for regularized solutions with respect to the noise level, characterizing the level-set approach as a regularization method for inverse problems. We also show the applicability of the proposed level-set method in some interesting inverse problems arising in elliptic PDE models.

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.

A continuous regularization method of the first order for nonlinear monotone equations

Ax = f (0.1) has a nonempty solution set N in X, x∗ is the normal solution to (1) (a solution with the minimal norm). Under these assumptions one cannot establish a continuous dependence of a solution to (1) on perturbances ofA and f . Therefore one should consider the problem on finding a solution to equation (1) as an ill-posed one and solve it with the help of a certain regularization method. Recently continuous solution methods for ill-posed problems arouse much interest. These methods are reduced to the solution of the Cauchy problem for a differential equation of a certain order. The order of the differential equation is said to be the order of a continuous method. For the case, when X = H is a Hilbert space, continuous methods are investigated rather well [1]–[6]. In [7], [8], one studies the convergence of a continuous method of the first order for equation (1) with a monotone and accretive operator A in a Banach space, assuming that the operator A is differentiable. Note that in the investigation of the convergence of a continuous method in a Banach space not only the properties of the operator A, but also the geometric properties of the spaces X and X∗ play an essential role. The objective of this paper is to establish the conditions which are sufficient for the convergence of a continuous method of the first order in a Banach space without the assumption on the differentiability of the monotone operator A. Let δX(s) be the module of convexity of the space X. Assume that this function is continuous and grows on [0, 2], δX(0) = 0 ([9], p. 49; [10]). Consequently, the inverse function δ−1 X (e) exists. Since the space X is uniformly smooth, we have that X∗ is also uniformly convex ([9], p. 34). Let δX∗(s) be the module of convexity of X∗. Define the function gX∗(s) = δX∗(s)/s. It is well known [10] that this function is continuous and grows on [0, 2], gX∗(0) = 0. Thus we can construct the function g−1 X∗(e). Assume that the operator of the dual mapping J : X → X∗ is defined by the correlations ([11], p. 311) ‖Jx‖ = ‖x‖, 〈Jx, x〉 = ‖x‖ ∀x ∈ X. (0.2) The properties of this operator are defined by the geometric properties of the spacesX andX∗. Note that under the above assumptions about the spaceX the operator J is monotone, bounded, and continuous ([11], pp. 313, 330, 331). In addition, the following inequality is true (see [8], [12])

Optimal control as a regularization method for ill-posed problems

We describe two regularization techniques based on optimal control for solving two types of ill-posed problems. We include convergence proofs of the regularization method and error estimates. We illustrate our method through problems in signal processing and parameter identification using an efficient Riccati solver. Our numerical results are compared to the same examples solved using Tikhonov regularization.

A Restarted Conjugate Gradient Method for Ill-posed Problems

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.

On theregularizing properties of a full multigrid method forill-posed problems

For the stable solution of a linear ill-posed operator equation\U0001d4afx = ywe project the equation onto finite-dimensionalspaces and apply a full multigrid method (MGM) with the smootherproposed by King. After giving level-independentcontraction factor estimates for the corresponding symmetricmultigrid operators, we study the regularizing properties ofthe corresponding full MGM with an a priori chosenfinest level, proving convergence, with optimal rates underrespective sourcewise representation conditions. We showapplicability of the method to a parameter identificationproblem from inverse groundwater filtration, and give numericalresults.

Fuzzy regularization method for ill-posed problems on ECT with Laplace transform BEM

This paper presents an identification method of crack shapes on eddy current testing. In order to use the impulse excitation, numerical analysis is performed by the boundary element method with Laplace transform. The shapes of cracks are identified directly from the integral equation of BEM. The singular value decomposition and regularization method by means of fuzzy inference are applied in the coefficient matrix for the unknown variables to obtain proper crack shapes.

NEW BAYESIAN METHODS FOR ILL POSED PROBLEMS

NEW BAYESIAN METHODS FOR ILL POSED PROBLEMS

Conjugate Gradient Type Methods for Ill-Posed Problems (Martin Hanke)

Conjugate Gradient Type Methods for Ill-Posed Problems (Martin Hanke)

A class of regularization methods for ill-posed problems with nonexact data

The paper deals with approximate methods for solving ill-posed problems with nonselfadjoint bounded linear operators acting in a Hilbert space. The case of nonexact right hand side is considered. A posteriori choice of a method parameter n(δ) depending on the error of data is based on the value of the residuum. The convergence of approximate solution when δ → 0 is investigated and the parameter n(δ) is estimated. The simple iteration method is considered in more detail.

Torsion of elastic shaft of revolution embedded in an elastic half space

The problem of torsion of elastic shaft of revolution embedded in an elastic half space is studied by the Line-Loaded Integral Equation Method (LLIEM). The problem is reduced to a pair of one-dimensional Fredholm integral equations of the first kind due to the distributions of the fictitious loads “Point Ring Couple (PRC)” and “Point Ring Couple in Half Space (PRCHS)” on the axis of symmetry in the interior and external ranges of the shaft occupied respectively. The direct discrete solution of this integral equations may be unstable, i.e. an ill-posed case occurs. In this paper, such an ill-posed Fredholm integral equation of first kind is replaced by a Fredholm integral equation of the second kind with small parameter, which provides a stable solution. This method is simpler and easier to carry out on a computer than the Tikhonov's regularization method for ill-posed problems. Numerical examples for conical, cylindrical, conical-cylindrical, and parabolic shafts are given.

A minimal error conjugate gradient method for ill-posed problems

For the ill-posed operator equationTx=y in Hilbert space, we introduce a modification of the usual conjugate gradient method which minimizes the error, not the residual, at each step. Moreover, the error is minimized over the same finite-dimensional subspace that is associated with the usual method.

An iterative method for ill-posed problems with inequalities

Ill-posed problems for integral and operator equations with nonnegativity and band inequality constraints arise in a wide range of applications. The effect and propagation of data perturbations in mathematical programming problems are highly dramatized in the area of ill-posed problems. In this note an iterative method for solving an ill-posed integral inequality and its moment discretization is described.