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Fri, 10 Nov 2017 10:53:46 +0100Fri, 10 Nov 2017 10:53:46 +0100Information Complexity of Multivariate Fredholm Equations in Sobolev Classes
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5062
In this paper, the complexity of full solution of Fredholm integral equations of the second kind with data from the Sobolev class \(W^r_2\) is studied. The exact order of information complexity is derived. The lower bound is proved using a Gelfand number technique. The upper bound is shown by providing a concrete algorithm of optimal order, based on a specific hyperbolic cross approximation of the kernel function. Numerical experiments are included, comparing the optimal algorithm with the standard Galerkin method.Karin Frank; Stefan Heinrich; Sergei Pereverzevreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/5062Fri, 10 Nov 2017 10:53:46 +0100Optimization of Projection Methods for Solving Ill-Posed Problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4995
Optimization of Projection Methods for Solving ill-posed Problems. In this paper we propose a modification of the projection scheme for solving ill-posed problems. We show that this modification allows to obtain the best possible order of accuracy of Tikhonov Regularization using an amount of information which is far less than for the standard projection technique.Sergei V. Pereverzevreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/4995Mon, 30 Oct 2017 11:16:00 +0100Regularization without Preliminary Knowledge of Smoothness and Error Behavior
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1585
The mathematical formulation of many physical problems results in the task of inverting a compact operator. The only known sensible solution technique is regularization which poses a severe problem in itself. Classically one dealt with deterministic noise models and required both the knowledge of smoothness of the solution function and the overall error behavior. We will show that we can guarantee an asymptotically optimal regularization for a physically motivated noise model under no assumptions for the smoothness and rather weak assumptions on the noise behavior which can mostly obtained out of two input data sets. An application to the determination of the gravitational field out of satellite data will be shown.Frank Bauer; Sergei Pereverzevpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1585Tue, 09 Nov 2004 15:32:13 +0100On the adaptive selection of the parameter in regularization of ill-posed problems
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1445
We study a possiblity to use 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 (1964), and to an adaptation scheme proposed in a statistical context by Lepskii (1990), 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 non-linear ill-posed problems and an inverse problem of the two-dimensional profile reconstruction.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1445Mon, 10 Nov 2003 10:38:04 +0100The finite-section approximation for ill-posed integral equations on the half-line
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260
Integral equations on the half of line are commonly approximated by the finite-section approximation, in which the infinite upper limit is replaced by apositie number called finite-section parameter. In this paper we consider the finite-section approximation for first kind intgral equations which are typically ill-posed and call for regularization. For some classes of such equations corresponding to inverse problems from optics and astronomy we indicate the finite-section parameters that allows to apply standard regularization techniques. Two discretization schemes for the finite-section equations ar also proposed and their efficiency is studied.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1260Fri, 07 Sep 2001 00:00:00 +0200On the efficient discretization of integral equations of the third kind
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/825
We propose a new discretization scheme for solving ill-posed integral equations of the third kind. Combining this scheme with Morozov's discrepancy principle for Landweber iteration we show that for some classes of equations in such method a number of arithmetic operations of smaller order than in collocation method is required to appoximately solve an equation with the same accuracy.Sergei V. Pereverzev; Eberhard Schock; Sergei G. Solodkypreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/825Mon, 03 Apr 2000 00:00:00 +0200Brakhage's implicit iteration method and Information Complexity of equations with operators having closed range
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/828
An a posteriori stopping rule connected with monitoringthe norm of second residual is introduced forBrakhage's implicit nonstationary iteration method, applied to ill-posed problems involving linear operatorswith closed range. It is also shown that for someclasses of equations with such operators the algorithmconsisting in combination of Brakhage's method withsome new discretization scheme is order optimal in the sense of Information Complexity.Sergei V. Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/828Mon, 03 Apr 2000 00:00:00 +0200Morozov's discrepancy principle for Tikhonov regularization of severely ill-posed problems in finite-dimensional subspaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/854
Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/854Tue, 08 Feb 2000 00:00:00 +0100Error estimates for band-limited spherical regularization wavelets in some inverse problems of satellite geodesy
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/800
In this paper we discuss a special class of regularization methods for solving the satellite gravity gradiometry problem in a spherical framework based on band-limited spherical regularization wavelets. Considering such wavelets as a reesult of a combination of some regularization methods with Galerkin discretization based on the spherical harmonic system we obtain the error estimates of regularized solutions as well as the estimates for regularization parameters and parameters of band-limitation.Sergei Pereverzev; Eberhard Schockpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/800Thu, 09 Dec 1999 00:00:00 +0100