A computational method for the evaluation of highly resolved dos functions from APS measurements

Schleicher, Klaus-Thomas; Schulz, Sebastian; Gmeiner, R.; Chun, H.-U.

doi:10.1016/0368-2048(83)85012-9

Cited by 15 publications

(12 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where Q X D H s with s > task (see, e.g., [11,70,108]). A series of studies on deautoconvolution and regularization have been published for the setting (58); see, for example, [34,100].…”

Section: Example V (Identification Of Potential In An Elliptic Equatimentioning

confidence: 99%

Regularization Methods for Ill-Posed Problems

Cheng

Hofmann

2014

Handbook of Mathematical Methods in Imaging

View full text Add to dashboard Cite

In this chapter are outlined some aspects of the mathematical theory for direct regularization methods aimed at the stable approximate solution of nonlinear ill-posed inverse problems. The focus is on Tikhonov type variational regularization applied to nonlinear ill-posed operator equations formulated in Hilbert and Banach spaces. The chapter begins with the consideration of the classical approach in the Hilbert space setting with quadratic misfit and penalty terms, followed by extensions of the theory to Banach spaces and present assertions on convergence and rates concerning the variational regularization with general convex penalty terms. Recent results refer to the interplay between solution smoothness and nonlinearity conditions expressed by variational inequalities. Six examples of parameter identification problems in integral and differential equations are given in order to show how to apply the theory of this chapter to specific inverse and ill-posed problems.

show abstract

“…where Q X D H s with s > task (see, e.g., [11,70,108]). A series of studies on deautoconvolution and regularization have been published for the setting (58); see, for example, [34,100].…”

Section: Example V (Identification Of Potential In An Elliptic Equatimentioning

confidence: 99%

Regularization Methods for Ill-Posed Problems

Cheng

Hofmann

2014

Handbook of Mathematical Methods in Imaging

View full text Add to dashboard Cite

show abstract

“…In the subsequent analysis we will relate the nonlinear autoconvolution equation (1) to the following linear Volterra equation. We note that for F from (1) its Frèchet-derivative F (x) : L 2 (0, 1) → L 2 (0, 1), at element x, is given by (10) […”

Section: 2mentioning

confidence: 99%

“…The solution to such equations attracted attention both from applications, cf. [10,1], and from the mathematical side, see e.g. [6,5,3], and [2], where different approaches are taken.…”

Section: Introduction Problem Formulationmentioning

confidence: 99%

Discretized Lavrent’ev regularization for the autoconvolution equation

Burger

Mathé

2016

Applicable Analysis

View full text Add to dashboard Cite

Abstract. Lavrent ev regularization for the autoconvolution equation was considered by J. Janno in Lavrent ev regularization of illposed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Problems, 16(2):333-348, 2000. Here this study is extended by considering discretization of the Lavrent ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

show abstract

“…This operator equation of quadratic type occurs in physics of spectra, in optics and in stochastics, often as part of a more complex task (see, e.g., [2,6,35]). A series of studies on deautoconvolution and regularization have been published for the setting (2.1), see for example [7,24,25,32].…”

Section: Autoconvolution For Real Functions On the Unit Intervalmentioning

confidence: 99%

About a deficit in low-order convergence rates on the example of autoconvolution

Burger

Hofmann

2014

Applicable Analysis

View full text Add to dashboard Cite

Abstract. We revisit in L 2 -spaces the autoconvolution equation x * x = y with solutions which are real-valued or complex-valued functions x(t) defined on a finite real interval, say t ∈ [0, 1]. Such operator equations of quadratic type occur in physics of spectra, in optics and in stochastics, often as part of a more complex task. Because of their weak nonlinearity deautoconvolution problems are not seen as difficult and hence little attention is paid to them wrongly. In this paper, we will indicate on the example of autoconvolution a deficit in low order convergence rates for regularized solutions of nonlinear ill-posed operator equations F (x) = y with solutions x † in a Hilbert space setting. So for the real-valued version of the deautoconvolution problem, which is locally ill-posed everywhere, the classical convergence rate theory developed for the Tikhonov regularization of nonlinear ill-posed problems reaches its limits if standard source conditions using the range of F (x † ) * fail. On the other hand, convergence rate results based on Hölder source conditions with small Hölder exponent and logarithmic source conditions or on the method of approximate source conditions are not applicable since qualified nonlinearity conditions are required which cannot be shown for the autoconvolution case according to current knowledge. We also discuss the complex-valued version of autoconvolution with full data on [0, 2] and see that ill-posedness must be expected if unbounded amplitude functions are admissible. As a new detail, we present situations of local well-posedness if the domain of the autoconvolution operator is restricted to complex L 2 -functions with a fixed and uniformly bounded modulus function.

show abstract

A computational method for the evaluation of highly resolved dos functions from APS measurements

Cited by 15 publications

References 31 publications

Regularization Methods for Ill-Posed Problems

Regularization Methods for Ill-Posed Problems

Discretized Lavrent’ev regularization for the autoconvolution equation

About a deficit in low-order convergence rates on the example of autoconvolution

Contact Info

Product

Resources

About