Vortrag UNIX-Stammtisch 04/98

Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.

Zugl.: Bielefeld, Univ., Diss., 2009

Zugl.: Chemnitz, Techn. Univ., Diss., 2005. / Ersch. im Verl. Kovač, (ISBN 3-8300-1884-3, 978-3-8300-1884-1), in der Schriftenr.: Chemnitzer Beiträge zur Sozialpädagogik, Bd. 2.

Zugl.: Bielefeld, Universiẗat, Diss., 2009.

Universiẗat, Diss., 2004--Bielefeld.

Please read the abstract in the section 00front of this document / Dissertation (MEd)--University of Pretoria, 2009. / Orthopaedic Surgery / unrestricted

The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.

The authors study parameter choice strategies for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depend on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader the authors review in an appendix a few instances where the validity of a variational inequality can be established.

Thesis (MDram (Drama))--Stellenbosch University, 2009. / ENGLISH ABSTRACT: Kenyan culture is deeply rooted in tradition. Many problems arise from the traditional role and conceptions of the Kenyan woman. These include female genital mutilation, polygamy and lack of scholastic education. A definite correlation can be found between the problem of HIV/AIDS and other sexually transmitted infections and certain social traditions as well as the unscripted prohibition of open conversations about sex and death. Kenya does not have an indigenous tradition of puppetry, and Community Health and Awareness Puppeteers (CHAPS) have committed themselves to developing interactive puppetry to address these tradition- orientated social problems. Puppetry makes use of three sign systems, namely sound, design and movement. This sign systems, together with the imagination of the audience, brings life to a lifeless object and serves as a meaningful vehicle of communication. CHAPS use the inherent qualities of the puppet. The puppet’s symbolic and double nature demeanor gives this art form exceptional freedom of speech, which makes it possible discuss sensitive issues. The visual nature of puppetry and general principals such as exaggeration and simplification make it possible to convey a message to people irrespective of their literacy level. The absence of television in rural and impoverished areas in Kenya ensures that people from any age group relate to puppetry as a visual art form. / AFRIKAANSE OPSOMMING: Kenia is ‘n baie tradisievaste land. Uit veral die tradisionele posisie van die vrou spruit ‘n aantal sosiale probleme, onder andere vroulike geslagsskending, poligamie en gebrekkige skolastiese opvoeding. Die problematiek rondom MIV/VIGS en oordraaglke geslagssiektes hou verband met sekere tradisionele sosiale gebruike en die ongeskrewe verbod om oor seks en dood te praat. Kenia het nie ‘n inheemse poppeteatertradisie nie en Community Health and Awareness Puppeteers (CHAPS) beywer hulle daarvoor om poppeteater in Kenia te bevorder en bogenoemde tradisie-verwante sosiale probleme deur middel van interaktiewe poppeteater aan te spreek. Poppeteater maak van drie tekensisteme gebruik, naamlik klank, ontwerp en beweging. Hierdie tekens, tesame met die gehoor se verbeelding, gee ‘n verbeelde lewe aan die toneelpop as ‘n lewelose voorwerp en dien as betekenisdraende kommunikasiemiddele. CHAPS benut die inherente eienskappe van die toneelpop. Die toneelpop se simboliese en dubbele aard gee aan die kunsvorm ‘n uitsonderlike vryheid van spraak wat dit moontlik maak om oor sensitiewe kwessies te praat. Die visuele aard van die poppeteater en spelbeginsels soos oordrywing en vereenvoudiging maak die kunsvorm toeganklik vir mense van enige geletterdheidsvlak. In die afwesigheid van televisie in die landelike en armoedige stedelike gebiede in Kenia, vind poppeteater as nuwigheid en visueelgerigte kunsvorm aanklank by enige ouderdomsgroep.