A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets

Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples

Das Konzept "Virtuelle Fachbibliothek" : Resümee und Ausblick

Die gegenwärtigen medientechnischen Entwicklungen, der schrankenlose weltweite Daten- und Kommunikationsfluss, für den das Internet steht, fordern traditionelle Informationsversorgungseinrichtungen heraus und verlangen nach neuen Denkweisen und Strategien. Als viel versprechendes Konzept gelten seit Ende der 1990er Jahre so genannte Virtuelle Fachbibliotheken (ViFa), die in der Regel von einem Konsortium aus Sondersammelgebiets- Bibliotheken, Fachgesellschaften und anderen einschlägig spezialisierten Institutionen deutschlandweit realisiert werden. Über einen zentralen WWW-Einstiegspunkt versammeln sie ausgewählte konventionelle und elektronische Medien zu einem Fachgebiet, bereiten diese systematisch auf und bieten diese einer ausgewiesenen Scientific Community über differenzierte Zugriffsmöglichkeiten an. Der Beitrag zeichnet die Entwicklung des ViFa-Konzepts kursorisch nach und stellt - in Auswahl - problematische Aspekte vor, die sich im Laufe der Realisierung ergaben und die noch immer aktuell sind.The current developments in media technique, the borderless and worldwide flow of data and communication is challenging the traditional institutions of information supply und requests new ways of thinking and strategies. Since the end of the nineties the so called ,"Virtuelle Fachbibliotheken" (virtual discipline based libraries), which are to be realized by special collection libraries and learning societies have to be seen as a promising concept in Germany. Via a centralized WWW-starting point selected printed and digital media are collected, prepared systematically and offered to a special scientific community via different access possibilities. This contribution regards briefly the development of the ViFA concept and presents some problematic aspects which arose during their development and are still On the fable

Theoretical aspects of a multiscale analysis of the eigenoscillations of the earth

The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy-Navier equation. Using a standard approach in seismology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy-Navier equation into two non-coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations using the Mie representation. Those solutions are denoted by the Hansen vectors Ln;j , Mn;j , and Nn;j in geophysics. Next we apply the inverse Fourier transform to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy-Navier equation.The elastic behaviour of the Earth, including its eigenoscillations, is usually described by the Cauchy­Navier equation. Using a standard approach in seis mology we apply the Helmholtz decomposition theorem to transform the Fourier transformed Cauchy­Navier equation into two non­coupled Helmholtz equations and then derive sequences of fundamental solutions for this pair of equations us ing the Mie representation. Those solutions are denoted by the Hansen vectors Ln,j , Mn,j , and Nn,j in geophysics. Next we apply the inverse Fourier trans form to obtain a function system depending on time and space. Using this basis for the space of eigenoscillations we construct scaling functions and wavelets to obtain a multiresolution for the solution space of the Cauchy­Navier equation. 2000 Mathematics Sub ject Classification: 35J05, 42C40

Das Sondersammelgebiet "Allgemeine und Vergleichende Literaturwissenschaft" an der Universitätsbibliothek Frankfurt am Main

Studierende und Lehrende des Faches Allgemeine und Vergleichende Literaturwissenschaft, die in den Beständen ihrer Bibliothek nicht fündig geworden sind, werden sich vielleicht fragen, warum viele ihrer Fernleihen von der Universitätsbibliothek Frankfurt am Main stammen. Dass sich ausgerechnet dort das Sondersammelgebiet für Allgemeine und Vergleichende Literaturwissenschaft befindet, mag überraschen, denn schließlich wurde an der Goethe-Universität ein eigenes Institut für Komparatistik erst im Sommersemester 2001 gegründet. [...] Die bedeutenden Bestände der Rothschildschen Bibliothek, die durch rechtzeitige Auslagerung ohne größere Verluste den Zweiten Weltkrieg überstanden haben, waren u. a. ausschlaggebend dafür, dass 1949 der damaligen Stadt- und Universitätsbibliothek Frankfurt von der Deutschen Forschungsgemeinschaft die Sondersammelgebiete »Allgemeine und Vergleichende Sprachwissenschaft« (SSG 7.11), »Germanistik« (SSG 7.20) sowie »Allgemeine und Vergleichende Literaturwissenschaft« (SSG 7.12) zugewiesen wurden. [...] So sehr es auch Utopie bleiben wird, gerade im Blick auf eine per se fachübergreifende Disziplin, so besteht doch der Anspruch, die wissenschaftliche Literatur eines Faches (auch und gerade die im Ausland erscheinende) möglichst umfassend und vollständig zu sammeln […]. Publikationen in digitaler Form sind solchen in konventioneller Gestalt gleichgestellt. Jede wissenschaftliche Publikation ist damit im Idealfall zumindest einmal in Deutschland vorhanden

Das Sondersammelgebiet Germanistik an der Universitätsbibliothek Frankfurt am Main

Innerhalb des DFG-Verteilungsplans zur überregionalen Literaturversorgung betreut die Frankfurter Universitätsbibliothek seit über sechzig Jahren das Sondersammelgebiet "Germanistik, Deutsche Sprache und Literatur". Der Beitrag stellt dieses kurz vor, ebenso die vom SSG verantworteten Informationsdienstleistungsangebote "Bibliographie der deutschen Sprach- und Literaturwissenschaft (BDSL)", "OLC Germanistik" sowie die "Virtuelle Fachbibliothek Germanistik – Germanistik im Netz (GiN)"

Time-Dependent Cauchy-Navier Splines and their Application to Seismic Wave Front Propagation

In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves

Harmonic Spline-Wavelets on the 3-dimensional Ball and their Application to the Reconstruction of the Earth´s Density Distribution from Gravitational Data at Arbitrarily Shaped Satellite Orbits

We introduce splines for the approximation of harmonic functions on a 3-dimensional ball. Those splines are combined with a multiresolution concept. More precisely, at each step of improving the approximation we add more data and, at the same time, reduce the hat-width of the used spline basis functions. Finally, a convergence theorem is proved. One possible application, that is discussed in detail, is the reconstruction of the Earth´s density distribution from gravitational data obtained at a satellite orbit. This is an exponentially ill-posed problem where only the harmonic part of the density can be recovered since its orthogonal complement has the potential 0. Whereas classical approaches use a truncated singular value decomposition (TSVD) with the well-known disadvantages like the non-localizing character of the used spherical harmonics and the bandlimitedness of the solution, modern regularization techniques use wavelets allowing a localized reconstruction via convolutions with kernels that are only essentially large in the region of interest. The essential remaining drawback of a TSVD and the wavelet approaches is that the integrals (i.e. the inner product in case of a TSVD and the convolution in case of wavelets) are calculated on a spherical orbit, which is not given in reality. Thus, simplifying modelling assumptions, that certainly include a modelling error, have to be made. The splines introduced here have the important advantage, that the given data need not be located on a sphere but may be (almost) arbitrarily distributed in the outer space of the Earth. This includes, in particular, the possibility to mix data from different satellite missions (different orbits, different derivatives of the gravitational potential) in the calculation of the Earth´s density distribution. Moreover, the approximating splines can be calculated at varying resolution scales, where the differences for increasing the resolution can be computed with the introduced spline-wavelet technique