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Relevant bibliographies by topics / Inverses Problem; Regularisierung

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Academic literature on the topic 'Inverses Problem; Regularisierung'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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Dissertations / Theses on the topic "Inverses Problem; Regularisierung"

1

Bonesky, Thomas. "Regularization of inverse problems and inexact operator evaluations." Berlin Logos-Verl, 2009. http://d-nb.info/997726814/04.

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2

Trede, Dennis. "Inverse problems with sparsity constraints convergence rates and exact recovery." Berlin Logos-Verl, 2010. http://d-nb.info/1002361532/04.

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3

Flemming, Jens. "Quadratic Inverse Problems and Sparsity Promoting Regularization." Doctoral thesis, Universitätsbibliothek Chemnitz, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-232402.

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Ill-posed inverse problems with quadratic structure are introduced, studied and solved. As an example an inverse problem appearing in laser optics is solved numerically based on a new regularized inversion algorithm. In addition, the theory of sparsity promoting regularization is extended to situations in which sparsity cannot be expected and also to equations with non-injective operators.

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4

Freitag, Melina. "On the Influence of Multiplication Operators on the Ill-posedness of Inverse Problems." Master's thesis, Universitätsbibliothek Chemnitz, 2004. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200401504.

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In this thesis we deal with the degree of ill-posedness of linear operator equations in Hilbert spaces, where the operator may be decomposed into a compact linear integral operator with a well-known decay rate of singular values and a multiplication operator. This case occurs for example for nonlinear operator equations, where the local degree of ill-posedness is investigated via the Frechet derivative. If the multiplier function has got zeroes, the determination of the local degree of ill-posedness is not trivial. We are going to investigate this situation, provide analytical tools as well as their limitations. By using several numerical approaches for computing the singular values of the operator we find that the degree of ill-posedness does not change through those multiplication operators. We even provide a conjecture, verified by several numerical studies, how these multiplication operators influence the singular values of the operator equation. Finally we analyze the influence of those multiplication operators on the opportunities of Tikhonov regularization and corresponding convergence rates. In this context we also provide a short summary on the relationship between nonlinear problems and their linearizations
Diese Arbeit beschaeftigt sich mit dem Grad der Inkorrektheit linearer Operatorgleichungen in Hilbertraeumen, die sich als Komposition eines vollstetigen linearen Integraloperators mit bekannter Abklingrate der Singulaerwerte und eines Multiplikationsoperators darstellen lassen. Dieser Fall tritt beispielsweise bei nichtlinearen Operatorgleichungen auf, wobei der lokale Inkorrektheitsgrad ueber die Frechetableitung bestimmt wird. Falls die Multiplikatorfunktion Nullstellen hat, so ist die Bestimmung des lokalen Grades der Inkorrektheit nicht einfach. Moeglichkeiten und Grenzen der Analysis fuer diese Situation werden betrachtet. Unterschiedliche numerische Ansaetze fuer die Bestimmung der Singulaerwerte liefern, dass der Grad der Inkorrektheit durch die Multiplikationsoperatoren nicht veraendert wird. Es wird sogar ein Zusammenhang angegeben, wie Multiplikationsoperatoren die Singulaerwerte beeinflussen. Schliesslich werden Moeglichkeiten der Tikhonov-Regularisierung unter Einfluss der Multiplikationsoperatoren untersucht. In diesem Zusammenhang wird auch eine kurze Zusammenfassung zur Beziehung von nichtlinearen Problemen und ihren Linearisierungen gegeben

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5

Krämer, Romy, Matthias Richter, and Bernd Hofmann. "Parameter estimation in a generalized bivariate Ornstein-Uhlenbeck model." Universitätsbibliothek Chemnitz, 2005. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200501307.

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In this paper, we consider the inverse problem of calibrating a generalization of the bivariate Ornstein-Uhlenbeck model introduced by Lo and Wang. Even though the generalized Black-Scholes option pricing formula still holds, option prices change in comparison to the classical Black-Scholes model. The time-dependent volatility function and the other (real-valued) parameters in the model are calibrated simultaneously from option price data and from some empirical moments of the logarithmic returns. This gives an ill-posed inverse problem, which requires a regularization approach. Applying the theory of Engl, Hanke and Neubauer concerning Tikhonov regularization we show convergence of the regularized solution to the true data and study the form of source conditions which ensure convergence rates.

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6

Hein, Torsten. "On solving implicitly defined inverse problems by SQP-approaches." Universitätsbibliothek Chemnitz, 2007. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200702148.

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In this paper two basic SQP-approaches for solving implicitly defined inverse problems are presented. Such problems often arises in parameter identification for differential equations. We also include regularization strategies which differ from similar problems in Optimal control. The main focus is on formulating saddle point problems for calculating the next iterate. Conditions for the unique and stable solvability of these problems are presented. The analytical considerations are illustrated by two examples including their discretizations and a numerical case study.

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7

Hein, Torsten. "Analytische und numerische Studien zu inversen Problemen der Optionspreisbildung." Doctoral thesis, [S.l. : s.n.], 2003. http://deposit.ddb.de/cgi-bin/dokserv?idn=969860986.

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8

Anzengruber, Stephan W., Bernd Hofmann, and Peter Mathé. "Regularization properties of the discrepancy principle for Tikhonov regularization in Banach spaces." Universitätsbibliothek Chemnitz, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-99353.

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The stable solution of ill-posed non-linear operator equations in Banach space requires regularization. One important approach is based on Tikhonov regularization, in which case a one-parameter family of regularized solutions is obtained. It is crucial to choose the parameter appropriately. Here, a variant of the discrepancy principle is analyzed. In many cases such parameter choice exhibits the feature, called regularization property below, that the chosen parameter tends to zero as the noise tends to zero, but slower than the noise level. Here we shall show such regularization property under two natural assumptions. First, exact penalization must be excluded, and secondly, the discrepancy principle must stop after a finite number of iterations. We conclude this study with a discussion of some consequences for convergence rates obtained by the discrepancy principle under the validity of some kind of variational inequality, a recent tool for the analysis of inverse problems.

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9

Shao, Yuanyuan. "Beiträge zur Regularisierung inverser Probleme und zur bedingten Stabilität bei partiellen Differentialgleichungen." Doctoral thesis, Universitätsbibliothek Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-102801.

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Wir betrachten die lineare inverse Probleme mit gestörter rechter Seite und gestörtem Operator in Hilberträumen, die inkorrekt sind. Um die Auswirkung der Inkorrektheit zu verringen, müssen spezielle Lösungsmethode angewendet werden, hier nutzen wir die sogenannte Tikhonov Regularisierungsmethode. Die Regularisierungsparameter wählen wir aus das verallgemeinerte Defektprinzip. Eine typische numerische Methode zur Lösen der nichtlinearen äquivalenten Defektgleichung ist Newtonverfahren. Wir schreiben einen Algorithmus, die global und monoton konvergent für beliebige Startwerte garantiert. Um die Stabilität zu garantieren, benutzen wir die Glattheit der Lösung, dann erhalten wir eine sogenannte bedingte Stabilität. Wir demonstrieren die sogenannte Interpolationsmethode zur Herleitung von bedingten Stabilitätsabschätzungen bei inversen Problemen für partielle Differentialgleichungen.

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10

Bürger, Steven, and Bernd Hofmann. "About a deficit in low order convergence rates on the example of autoconvolution." Universitätsbibliothek Chemnitz, 2013. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-130630.

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We revisit in L2-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 L2-functions with a fixed and uniformly bounded modulus function.

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More sources

Book chapters on the topic "Inverses Problem; Regularisierung"

1

Rieder, Andreas. "Tikhonov—Phillips-Regularisierung." In Keine Probleme mit Inversen Problemen, 93–105. Wiesbaden: Vieweg+Teubner Verlag, 2003. http://dx.doi.org/10.1007/978-3-322-80234-7_4.

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Rieder, Andreas. "Diskretisierung und Regularisierung." In Keine Probleme mit Inversen Problemen, 153–222. Wiesbaden: Vieweg+Teubner Verlag, 2003. http://dx.doi.org/10.1007/978-3-322-80234-7_6.

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3

Rieder, Andreas. "Iterative Regularisierungen." In Keine Probleme mit Inversen Problemen, 106–52. Wiesbaden: Vieweg+Teubner Verlag, 2003. http://dx.doi.org/10.1007/978-3-322-80234-7_5.

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4

Richter, Mathias. "Regularisierung linearer inverser Probleme." In Inverse Probleme, 67–99. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-45811-2_4.

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5

Richter, Mathias. "Regularisierung nichtlinearer inverser Probleme." In Inverse Probleme, 101–18. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-662-45811-2_5.

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6

Rieder, Andreas. "Regularisierung linearer Probleme und Optimalität." In Keine Probleme mit Inversen Problemen, 52–92. Wiesbaden: Vieweg+Teubner Verlag, 2003. http://dx.doi.org/10.1007/978-3-322-80234-7_3.

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