Invertible smoothing preconditioners for linear discrete ill-posed problems

Cascadic multilevel methods for the solution of linear discrete ill-posed problems with noise-reducing restriction and prolongation operators recently have been developed for the restoration of blur- and noise-contaminated images. This is a particular ill-posed problem. The multilevel methods were found to determine accurate restorations with fairly little computational work. This paper describes noise-reducing multilevel methods for the solution of general linear discrete ill-posed problems.

An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure

Combining approximate solutions for linear discrete ill-posed problems

This chapter discusses iterative methods for the solution of very large severely ill-conditioned linear systems of equations that arise from the discretization of linear ill-posed problems. The right-hand side vector represents the given data and is assumed to be contaminated by errors. Solution methods proposed in the literature employ some form of filtering to reduce the influence of the error in the right-hand side on the computed approximate solution. The amount of filtering is determined by a parameter often referred to as the regularization parameter. We discuss how the filtering affects the computed approximate solution and consider the selection of regularization parameter. Methods in which a suitable value of the regularization parameter is determined during the computation without user intervention are emphasized. New iterative solution methods based on expanding explicitly chosen filter functions in terms of Chebyshev polynomials are presented. The properties of these methods are illustrated with applications to image restoration.

A stopping criterion for iterative regularization methods

The Landweber iteration method is one of the most popular methods for the solution of linear discrete ill-posed problems. The diversity of physical problems and the diversity of operators that result from them leads us to think about updating the main methods and algorithms to achieve the best results. We considered in this work the linear operator equation and the use of a new version of the Landweber iterative method as an iterative solver. The main goal of updating the Landweber iteration method is to make the iteration process fast and more accurate. We used a polar decomposition to achieve a symmetric positive definite operator instead of an identity operator in the classical Landweber method. We carried out the convergence and other necessary analyses to prove the usability of the new iteration method. The residual method was used as an analysis method to rate the convergence of the iteration. The modified iterative method was compared to the classical Landweber method. A numerical experiment illustrates the effectiveness of this method by applying it to solve the inverse boundary value problem of the heat equation (IBVP).

The purpose of this paper is to bring a series of rather surprising results which have appeared in the technical literature to the attention of a wider audience. We describe the results informally here; precise definitions and results will be given below. Our subject is the solution of an ill-posed problem on a digital computer. Such a problem can at best only be solved approximately. We say a problem is solvable if we can compute an "-approximation for any positive " and is unsolvable if we cannot compute an "-approximation, even for arbitrarily large ". It was shown in [17] that a linear problem is unsolvable iff it is ill-posed. Are there circumstances under which one can avoid this negative conclusion? It is common to associate an operator S with an ill-posed problem. If S is linear, then a problem is ill-posed iff S is unbounded. Note that unboundedness is a worst case concept; the supremum of S operating on elements in the unit ball of its domain is infinite. This suggests introducing the concept of a problem’s being ill-posed on the average if the expectation of S with respect to a measure is infinite and being well-posed on the average if this expectation is finite. These concepts were introduced in [17], where it was shown that if the measure is Gaussian and the linear operator S is measurable, then a linear problem is unsolvable on the average iff it is ill-posed on the average. A natural next step is to find a linear ill-posed problem which is also ill-posed on the average. Several attempts to do this for Gaussian measures were unsuccessful. This suggested the question: Is every linear problem well-posed on the average for any Gaussian measure? This question was answered in the affirmative in [5] and [15]. The conclusion in the title follows immediately from this and the preceding result. What if the measure is non-Gaussian? Now the situation is more complicated. In particular, there are problems that are even ill-posed on the average. In this paper, we round out the picture by presenting examples of such problems for non-Gaussian measures.

An iterative Lagrange method for the regularization of discrete ill-posed inverse problems

On stability and convergence of a three-layer semi-discrete scheme for an abstract analogue of the Ball integro-differential equation

Tikhonov regularization is a popular method to approximate solutions of linear discrete ill-posed problems when the observed or measured data is contaminated by noise. Multiparameter Tikhonov regularization may improve the quality of the computed approximate solutions. We propose a new iterative method for large-scale multiparameter Tikhonov regularization with general regularization operators based on a multidirectional subspace expansion. The multidirectional subspace expansion may be combined with subspace truncation to avoid excessive growth of the search space. Furthermore, we introduce a simple and effective parameter selection strategy based on the discrepancy principle and related to perturbation results.

Geophysical Inverse Theory and Regularization Problems

An ill-posed problem which involves heterogonous data can yield good results if the weight of observations is properly introduced into the adjustment model. Variance component estimation can be used in this respect to update and improve the weights based on the results of the adjustment. The variance component estimation will not be as simple as that is in an ordinary adjustment problem, because the result of the solution of an ill-posed problem contains a bias due to stabilizing the adjustment model. This paper investigates the variance component estimation in those ill-posed problems solved by the truncation singular value decomposition. The biases of the variance components are analyzed and the biased-corrected and the biased-corrected non-negative estimators of the variance components are developed. The derivations show that in order to estimate unbiased variance components, it suffices to estimate and remove the bias from the estimated residuals.

Iterative Tikhonov regularization of tensor equations based on the Arnoldi process and some of its generalizations

A Highly Accurate Technique for Interpolations Using Very High-Order Polynomials, and Its Applications to Some Ill-Posed Linear Problems

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method.