On the condition number of the antireflective transform

Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for deconvolution problems is a regularization method. Moreover, we establish upper bounds for the regularization error of the reblurring strategy that hold uniformly with respect to the size n of the algebraic system, even though the condition number of the antireflective transform grows with n. We briefly sketch how our results extend to higher space dimensions.     

Multilevel algorithms for ill-posed problems

Summary In this paper new multilevel algorithms are proposed for the numerical solution of first kind operator equations. Convergence estimates are established for multilevel algorithms applied to Tikhonov type regularization methods. Our theory relates the convergence rate of these algorithms to the minimal eigenvalue of the discrete version of the operator and the regularization parameter. The algorithms and analysis are presented in an abstract setting that can be applied to first kind integral equations.Dedicated to Jim Bramble on the occasion of his sixtieth birthday     

A modified Newton–Lavrentiev regularization for nonlinear ill-posed Hammerstein-type operator equations

Recently, a new iterative method, called Newton–Lavrentiev regularization (NLR) method, was considered by George (2006) for regularizing a nonlinear ill-posed Hammerstein-type operator equation in Hilbert spaces. In this paper we introduce a modified form of the NLR method and derive order optimal error bounds by choosing the regularization parameter according to the adaptive scheme considered by Pereverzev and Schock (2005).     

Sylvester Tikhonov-regularization methods in image restoration

In this paper, we consider large-scale linear discrete ill-posed problems where the right-hand side contains noise. Regularization techniques such as Tikhonov regularization are needed to control the effect of the noise on the solution. In many applications such as in image restoration the coefficient matrix is given as a Kronecker product of two matrices and then Tikhonov regularization problem leads to the generalized Sylvester matrix equation. For large-scale problems, we use the global-GMRES method which is an orthogonal projection method onto a matrix Krylov subspace. We present some theoretical results and give numerical tests in image restoration.     

A multi-grid method with a priori and a posteriori level choice for the regularization of nonlinear ill-posed problems

Summary In this paper we study a multi-grid method for the numerical solution of nonlinear systems of equations arising from the discretization of ill-posed problems, where the special eigensystem structure of the underlying operator equation makes it necessary to use special smoothers. We provide uniform contraction factor estimates and show that a nested multigrid iteration together with an a priori or a posteriori chosen stopping index defines a regularization method for the ill-posed problem, i.e., a stable solution method, that converges to an exact solution of the underlying infinite-dimensional problem as the data noise level goes to zero, with optimal rates under additional regularity conditions. Supported by the Fonds zur F?rderung der wissenschaftlichen Forschung under grant T 7-TEC and project F1308 within Spezialforschungsbereich 13     

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.     

A wavelet multilevel method for ill-posed problems stabilized by Tikhonov regularization

Summary. An additive Schwarz iteration is described for the fast resolution of linear ill-posed problems which are stabilized by Tikhonov regularization. The algorithm and its analysis are presented in a general framework which applies to integral equations of the first kind discretized either by spline functions or Daubechies wavelets. Numerical experiments are reported on to illustrate the theoretical results and to compare both discretization schemes. Received March 6, 1995 / Revised version received December 27, 1995     

On the regularizing properties of the GMRES method

Summary. The GMRES method is a popular iterative method for the solution of large linear systems of equations with a nonsymmetric nonsingular matrix. However, little is known about the behavior of this method when it is applied to the solution of nonsymmetric linear ill-posed problems with a right-hand side that is contaminated by errors. We show that when the associated error-free right-hand side lies in a finite-dimensional Krylov subspace, the GMRES method is a regularization method. The iterations are terminated by a stopping rule based on the discrepancy principle. Received November 10, 2000 / Revised version received April 11, 2001 / Published online October 17, 2001     

Regularisierung schlecht gestellter Probleme durch Projektionsverfahren

Summary The numerical solution of ill-posed problems by projection methods is considered. Regularization is carried out simply by choosing an optimal discretization parameter. It is shown by asymptotic estimates and by numerical examples that this kind of regularization is as efficient as the method of Tikhonov and Phillips.

     

Computation of the inverse Laplace transform based on a collocation method which uses only real values

We develop a numerical algorithm for inverting a Laplace transform (LT), based on Laguerre polynomial series expansion of the inverse function under the assumption that the LT is known on the real axis only. The method belongs to the class of Collocation methods (C-methods), and is applicable when the LT function is regular at infinity. Difficulties associated with these problems are due to their intrinsic ill-posedness. The main contribution of this paper is to provide computable estimates of truncation, discretization, conditioning and roundoff errors introduced by numerical computations. Moreover, we introduce the pseudoaccuracy which will be used by the numerical algorithm in order to provide uniform scaled accuracy of the computed approximation for any x   with respect to e^σx${\mathrm{e}}^{\sigma x}$. These estimates are then employed to dynamically truncate the series expansion. In other words, the number of the terms of the series acts like the regularization parameter which provides the trade-off between errors.     

Error bounds and parameter choice strategies for simplified regularization in Hilbert scales

Simplified regularization in the setting of Hilbert scales has been considered for obtaining stable approximate solutions for ill-posed operator equations. The derived error estimates using an a posteriori as well as an a priori parameter choice strategy are shown to be of optimal order with respect to certain natural assumptions on the ill-posedness of the equation.The work of M. Thamban Nair is partially supported by IC&SR, I.I.T., Madras     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

Convergence rates of iterated Tikhonov regularized solutions of nonlinear III — posed problems

Summary In this paper we investigate iterated Tikhonov regularization for the solution of nonlinear ill-posed problems. In the case of linear ill-posed problems it is well-known that (under appropriate assumptions) then-th iterated regularized solutions can converge likeO(²² /(2n+1)), where denotes the noise level of the data perturbation. We give conditions that guarantee this convergence rate also for nonlinear ill-posed problems, and motivate these conditions by the mapping degree. The results are derived by a comparison of the iterated regularized solutions of the nonlinear problem with the iterated regularized solutions of its linearization. Numerical examples are presented.Supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung,project P-7869 PHY, and by the Christian Doppler Society     

Regularization,GSVD and truncatedGSVD

The purpose of this paper is to analyze Tikhonov regularization in general form by means of generalized SVD (GSVD) in the same spirit as SVD is used to analyze standard-form regularization. We also define a truncated GSVD solution which is of interest in its own right and which sheds light on regularization as well. In addition, our analysis gives insight into a particular numerical method for solving the general-form problem via a transformation to standard form.Part of this work was carried out while visiting the Mathematical Sciences Section, Oak Ridge National Laboratory, Tennessee, during the Numerical Linear Algebra Year 1987–88, and was supported by the Danish Natural Science Foundation.     

Preconditioning Landweber iteration in Hilbert scales

In this paper we investigate convergence of Landweber iteration in Hilbert scales for linear and nonlinear inverse problems. As opposed to the usual application of Hilbert scales in the framework of regularization methods, we focus here on the case s≤0, which (for Tikhonov regularization) corresponds to regularization in a weaker norm. In this case, the Hilbert scale operator L^−2s appearing in the iteration acts as a preconditioner, which significantly reduces the number of iterations needed to match an appropriate stopping criterion. Additionally, we carry out our analysis under significantly relaxed conditions, i.e., we only require instead of which is the usual condition for regularization in Hilbert scales. The assumptions needed for our analysis are verified for several examples and numerical results are presented illustrating the theoretical ones. supported by the Austrian Science Foundation (FWF) under grant SFB/F013     

Wavelet moment method for the Cauchy problem for the Helmholtz equation

The paper is concerned with the problem of reconstruction of acoustic or electromagnetic field from inexact data given on an open part of the boundary of a given domain. A regularization concept is presented for the moment problem that is equivalent to a Cauchy problem for the Helmholtz equation. A method of regularization by projection with application of the Meyer wavelet subspaces is introduced and analyzed. The derived formula, describing the projection level in terms of the error bound of the inexact Cauchy data, allows us to prove the convergence and stability of the method.     

Approximation Methods by Regular Functions

This paper is a survey of approximation results and methods by smooth functions in Banach spaces. The topics considered in the paper are the following: approximation by polynomials by C^k-functions using the method of smooth partitions of unity, approximation by the fine topology, analytic approximation and regularization in Banach spaces using the infimal convolution method.     

The Riemann problem for the Leray-Burgers equation

For Riemann data consisting of a single decreasing jump, we find that the Leray regularization captures the correct shock solution of the inviscid Burgers equation. However, for Riemann data consisting of a single increasing jump, the Leray regularization captures an unphysical shock. This behavior can be remedied by considering the behavior of the Leray regularization with initial data consisting of an arbitrary mollification of the Riemann data. As we show, for this case, the Leray regularization captures the correct rarefaction solution of the inviscid Burgers equation. Additionally, we prove the existence and uniqueness of solutions of the Leray-regularized equation for a large class of discontinuous initial data. All of our results make extensive use of a reformulation of the Leray-regularized equation in the Lagrangian reference frame. The results indicate that the regularization works by bending the characteristics of the inviscid Burgers equation and thereby preventing their finite-time crossing.     

The truncatedSVD as a method for regularization

The truncated singular value decomposition (SVD) is considered as a method for regularization of ill-posed linear least squares problems. In particular, the truncated SVD solution is compared with the usual regularized solution. Necessary conditions are defined in which the two methods will yield similar results. This investigation suggests the truncated SVD as a favorable alternative to standard-form regularization in cases of ill-conditioned matrices with well-determined numerical rank.This work was carried out while the author visited the Dept. of Computer Science, Stanford University, California, U.S.A., and was supported in part by National Science Foundation Grant Number DCR 8412314, by a Fulbright Supplementary Grant, and by the Danish Space Board.     

A saddle point variational formulation for projection–regularized parameter identification

Summary. This paper is concerned with the ill-posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection. As the theory has shown, the ansatz functions for the parameter have to be sufficiently smooth. In this paper we show that these – for a practical implementation unrealistic – smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation. We prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level goes to zero in the case of noisy data, as well as convergence rates under additional smoothness conditions. Received August 4, 2000 / Revised version received March 21, 2001 / Published online October 17, 2001