Lectures
Winter Term 2008/09
Potentialtheorie
Prof. Freeden
Di, 8:15 - 9:45, 48-538 (Beginn: 21.10.2008, 13 Termine)
Do, 17:15 - 18:45, 48-538 (Beginn: 23.10.2008, 11 Termine)
Regelmäßige Termine für Übungen:
Mo, 11:45 - 13:15, 48-538 (Beginn: 3.11.2008, 13 Termine)
Anzahl der SWS: 4 Std. + 2 Std., (4 ECTS)
Unterrichtssprache:
Content:
The lecture will give an introduction into potential theory under geoscientifically relevant aspects. First, the following topics are treated in classical way: Harmonicity, (multi-layer) potentials, jump and limit relations, boundary value problems, integral equations methods. In the second part the lecture deals with new developments such as ill-posed problems of "downward harmonic continuation", Runge approximation by inner/outer harmonics, splines, and wavelets. Finally, numerical aspects of multiscale modelling are discussed in problems of meass (re-)distribution and geoidal determination.
Literatur
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W. Freeden (1999): Multiscale Modelling of Spaceborne Geodata. B.G. Teubner, Stuttgart, Leipzig.
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W. Freeden, V. Michel (2003): Multiscale Potential Methods (With Applications to Earth's Sciences), Birkhäuser Verlag, Boston, Basel, Berlin .
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W. Freeden, M. Schreiner (2008): Spherical Functions in Mathematical Geosciences, Springer (in preparation).
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O.D. Kellogg (1929): Foundations of Potential Theory. Frederick Ungar Publishing Company, New York.
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C.W. Misner, K.S. Thorne, J.A. Wheeler (1973): Gravitation. Freeman, San Francisco.
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W. Walter (1971). Einführung in die Potentialtheorie. BI Hochschulskripten 765/765a.
Voraussetzungen
Analysis
Leistungsnachweis und Prüfungen
'Uebungsschein' for successful participation in the tutorial; 4 credits can be obtained in an oral examination after the lecture.
Sonstiges
There is a growing public concern about the future of our planet, its climate, its environment and about expected shortage of natural resources. Any consistent and efficient strategy of protection against these threats depends on a profound understanding of the Earth system. In particular, the knowledge of the Earth mass (re-)distribution is of crucial importance for the exploration of processes driving deformation of the Earth surface and influencing ocean surface topography. Closely interrelated with mass transport and mass anomalies is the Earth's gravity field and its variances. In consequence, potential theory has become a renewed importance; gravity variations play a prominant role in modern research (for example in the DFG Priority Research Programme (2006-2012) in which the Geomathematics Group is involved. In fact, the geoid is viewed as an almost static reference for many rapidly changing processes (e.g., sea level heights, hydrological phenomena, etc) and at the same time as a "frozen picture" of tectonic processes that evolved over geological time spans.
Numerische Integration
Prof. Freeden
Di, 17:15 - 18:45, 48-582 (Beginn: 21.10.2008, 15 Termine)
Do, 8:15 - 9:45, 48-538 (Beginn: 23.10.2008, 15 Termine)
Regelmäßige Termine für Übungen:
Mo, 15:30 - 17:00, 13-370 (Beginn: 27.10.2008, 14 Termine)
Anzahl der SWS: 4 Std. + 2 Std., (4 ECTS)
Unterrichtssprache:
Content:
The problem of finding the numerical value of an integral, because of its geometrical meaning, is often for simplicity called quadrature/cubature. In this lecture we study methods of quadrature/cubature which are used to approximately evaluate integrals by means of a finite number of functional values. These methods can be applied where others integration techniques fail. In many cases the procedures also require less work than other calculations.
Interpolatory quadrature, Gauß quadrature, and Romberg (extrapolation) methods will be discussed in more detail. Particular aim is their generalization to the multivariate case, e.g., sphere, torus, general (regular) surfaces (relevant for geomathematical purposes).
Literatur
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Stoer, J. (1994), Numerische Mathematik I, Springer.
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Hämmerlin, G. Hoffmann, K.-H. (1992), Numerische Mathematik, Springer.
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Freeden, W. (1980). On Integral Formulas of the (Unit) Sphere and Their Application to Numerical Integration, Computing (25), 131-146.
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Freeden, W. (1982). Multidimensional Euler Summation Formulas and Numerical Cubature, ISNM, 77-99.
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Freeden, W. (1980). Über die Gaußsche Methode zur angenäherten Berechnung von Integralen. Math. Meth. Appl. Sci. (2), 397-409.
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Freeden, W., Fleck, J. (1987). Numerical Integration by Means of Adapted Euler Summation Fomrulas. Numer. Math., (51), 37-64.
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Freeden, W., T. Gerven, M. Schreiner (1998). Constructive Approximation on the Sphere (With Application to Geomathematics), Clarendon Press, Oxford.
Voraussetzungen
Analysis
Leistungsnachweis und Prüfungen
'Uebungsschein' for successful participation in the tutorial; 4 credits can be obtained in an oral examination after the lecture.
Proseminar zur Geomathematik (Seminar)
Prof. Freeden
Inhalt:
In den Geowissenschaften werden eine Vielzahl grundlegender mathematische Methoden zur Beschreibung und Lösung der betrachteten Probleme benötigt. Solche Methoden sollen in diesem Proseminar behandelt werden, dazu gehören unter anderem Integraltransformationen, Funktionensysteme wie Kugelflächenfunktionen und Legendrefunktionen, sowie Differentialoperatoren. Desweiteren sollen deren Anwendungen z.B. in der der Gravitation und Magnetik vorgestellt werden.
Vorbesprechung:
wird noch bekanntgegeben
Termin:
Blockveranstaltung am Ende des Semesters
Voraussetzungen
Interesse an modernen mathematischen Verfahren und Methoden sowie deren Anwendung; Analysis; Lineare Algebra
Leistungsnachweis und Prüfungen
Vortrag und aktive Teilnahme bei den Vorträgen.
Prof. Dr. Willi Freeden,
Dr. Carsten Mayer,
HDoz. Dr. V. Michel,
Dr. habil. M. Schreiner
Anzahl der SWS: 2 Std., (0 ECTS), n. V..
Sonstiges: Diplomanden- und Doktorandenseminar (ergänzt durch Vorträge renommierter Gäste).
Inhalt:
Diskussion forschungsrelevanter Themen der Geomathematik.
Content
Discussion of research topics in geomathematics.
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