Regularization Of Ill-posed Problems Research Articles

Regularization of ill-posed problems in Sobolev space <I>W</I><SUB>1</SUB><SUP>1</SUP>

Regularization of ill-posed problems in Sobolev space <I>W</I><SUB>1</SUB><SUP>1</SUP>

Regularization in Hilbert scales under general smoothing conditions

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high- and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience.

A descent method for regularization of ill-posed problems

In this paper, we describe an iterative algorithm, called descent-TCG, based on truncated conjugate gradients iterations to compute Tikhonov regularized solutions of linear ill-posed problems. The sequence of approximate solutions and regularization parameters, computed by the algorithm, is shown to decrease the value of the Tikhonov functional. Suitable termination criteria are built-up to define an inner–outer iteration scheme that computes a regularized solution. Numerical experiments are performed to compare this algorithm with other well established regularization methods. We observe that the best descent-TCG results occur for highly noised data and we always get fairly reliable solutions, thus it prevents the dangerous error growth often appearing in other well established regularization methods. Finally, the descent-TCG method is computationally advantageous especially for large size problems.

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.

Convergence rates in the Prokhorov metric for assessing uncertainty in ill-posed problems

In the convergence theory of regularization methods for ill-posed problems, so far deterministic error concepts have dominated, which leads to worst-case error estimates. Since this is sometimes not desirable, we aim at providing a framework for proving convergence rates in the Prokhorov metric for the regularization of ill-posed problems with stochastic noise. This allows us to assess uncertainty in the sense of a confidence region for the probability that the deviation between the exact and regularized solutions stays below a given bound with given probability. We exemplify this method for the special case of Tikhonov regularization for linear ill-posed problems and apply the result to the problem of deblurring an image contaminated by random blurring.

On the Adaptive Selection of the Parameter in Regularization of Ill-Posed Problems

We study the possibility of using the structure of the regularization error for a posteriori choice of the regularization parameter. As a result, a rather general form of a selection criterion is proposed, and its relation to the heuristical quasi-optimality principle of Tikhonov and Glasko [Z. Vychisl. Mat. Mat. Fiz., 4 (1964), pp. 564-571] and to an adaptation scheme proposed in a statistical context by Lepskii [Theory Probab. Appl., 36 (1990), pp. 454-466] is discussed. The advantages of the proposed criterion are illustrated by using such examples as self-regularization of the trapezoidal rule for noisy Abel-type integral equations, Lavrentiev regularization for nonlinear ill-posed problems, and an inverse problem of the two-dimensional profile reconstruction.

Spatial decay of transient end effects for nonstandard linear diffusion problems

In this paper we investigate the spatial decay of transient end effects for some nonstandard linear diffusion problems. We consider classes of linear parabolic equations on three-dimensional semi-infinite cylinders with non-zero boundary conditions only on the near end. If zero initial conditions were imposed, it is well-known that solutions decay exponentially with distance from the end and many results on the rate of spatial decay have been established. Here we consider a different type of initial condition that arises in regularization of ill-posed problems: namely, that the field quantity at time t = T is assumed to be proportional to that at t = 0. A spatial decay estimate for solutions of such problems is obtained with optimal decay rate explicit in the proportionality parameter. Continuous dependence on this parameter is established. The results are extended to a pseudo-parabolic equation. Binary mixtures of rigid solids are then considered. The governing PDEs are a coupled system of two parabolic equations for the temperature in each phase. Differential inequality arguments are used to obtain decay estimates for cross-sectional norms of the solution pair.

Comparing different types of approximators for choosing the parameters in the regularization of ill-posed problems

Regularized approximations to the solutions of ill-posed problems typically vary from over-smoothed, inaccurate reconstructions to under-smoothed and unstable solutions as the regularization parameter varies about its optimal value. It thus makes sense to compare two (or more) regularized approximations, and seek the parametervalues where the distance between the approximations is a minimum. This paper advances the theory and practice of this methodology. The method appears to workvery well and be very stable, particularly in the presence of extreme error levels.

On Lavrentiev Regularization for Ill-Posed Problems in Hilbert Scales

In this article we study the problem of identifying the solution x † of linear ill-posed problems Ax = y in a Hilbert space X where instead of exact data y noisy data y δ ∈ X are given satisfying with known noise level δ. Regularized approximations are obtained by the method of Lavrentiev regularization in Hilbert scales, that is, is the solution of the singularly perturbed operator equation where B is an unbounded self-adjoint strictly positive definite operator satisfying . Assuming the smoothness condition we prove that the regularized approximation provides order optimal error bounds (i) in case of a priori parameter choice for and (ii) in case of Morozov's discrepancy principle for s ≥ p. In addition, we provide generalizations, extend our study to the case of infinitely smoothing operators A as well as to nonlinear ill-posed problems and discuss some applications.

Steepest Descent, CG, and Iterative Regularization of Ill-Posed Problems

The state of the art iterative method for solving large linear systems is the conjugate gradient (CG) algorithm. Theoretical convergence analysis suggests that CG converges more rapidly than steepest descent. This paper argues that steepest descent may be an attractive alternative to CG when solving linear systems arising from the discretization of ill-posed problems. Specifically, it is shown that, for ill-posed problems, steepest descent has a more stable convergence behavior than CG, which may be explained by the fact that the filter factors for steepest descent behave much less erratically than those for CG. Moreover, it is shown that, with proper preconditioning, the convergence rate of steepest descent is competitive with that of CG.

Regularization of ill-posed problems with unbounded operators

Variational regularization and the method of quasisolutions are justified for unbounded closed operators.

Optimal order results for a class of regularization methods using unbounded operators

A class of regularization methods using unbounded regularizing operators is considered for obtaining stable approximate solutions for ill-posed operator equations. With an a posteriori as well as an a priori parameter choice strategy, it is shown that the method yields the optimal order. Error estimates have also been obtained under stronger assumptions on the generalized solution. The results of the paper unify and simplify many of the results available in the literature. For example, the optimal results of the paper include, as particular cases for Tikhonov regularization, the main result of Mair (1994) with an a priori parameter choice, and a result of Nair (1999) with an a posteriori parameter choice. Thus the observations of Mair (1994) on Tikhonov regularization of ill-posed problems involving finitely and infinitely smoothing operators is applicable to various other regularization procedures as well. Subsequent results on error estimates include, as special cases, an optimal result of Vainikko (1987) and also some recent results of Tautenhahn (1996) in the setting of Hilbert scales.

Lavrent'ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation

In this paper the method of Lavrent'ev for regularization of ill-posed problems with near-to-monotone operators is considered. An a priori rule of choice of the parameter of regularization is presented and the corresponding error estimate is derived. The theory is applied to the autoconvolution equation. Numerical examples are provided.

Regularization of Ill-Posed Problems Via the Level Set Approach

We introduce a new formulation for the motion of curves in R2 (easily extendable to the motion of surfaces in R3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy--Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill-posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature ofthe regularization, including the development of bubbles for curves evolving under Cauchy--Riemann flow.

Application of functions of several variables with limited variations for piecewise uniform regularization of ill-posed problems

Application of functions of several variables with limited variations for piecewise uniform regularization of ill-posed problems

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography.

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

Regularization of Ill-Posed Problems by Envelope Guided Conjugate Gradients

On a certain class of operator equations with small parameter and regularization of ill-posed problems

On a certain class of operator equations with small parameter and regularization of ill-posed problems

A class of discrepancy principles for the simplified regularization of ill-posed problems

AbstractA class of discrepancy principles for the choice of parameters for the simplified regularization of ill-posed problems is proposed. This procedure does not require knowledge of the unknown solution, and if the smoothness of the unknown solution is known then the convergence rate obtained is optimal. The results of this paper include the Arcangeli's method considered by Groetsch and Guacaneme (1987) for which the convergence rate was not known and also of a result of Guacaneme (1988) for which there is a gap in the proof.

An a posteriori parameter choice for simplified regularization of ill-posed problems

An a posteriori parameter choice strategy is proposed for the simplified regularization of ill-posed problems where no information about the smoothness of the unknown solution is required. If the smoothness of the solution is known then, as a particular case, the optimal rate is achieved. Our result also includes a recent result of Guacanme (1990).