Suchergebnis: Katalogdaten im Frühjahrssemester 2017

Mathematik Master Information
Wahlfächer
Für das Master-Diplom in Angewandter Mathematik ist die folgende Zusatzbedingung (nicht in myStudies ersichtlich) zu beachten: Mindestens 15 KP der erforderlichen 28 KP aus Kern- und Wahlfächern müssen aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten stammen.
Wahlfächer aus Bereichen der reinen Mathematik
Auswahl: Algebra, Topologie, diskrete Mathematik, Logik
NummerTitelTypECTSUmfangDozierende
401-4142-17LAlgebraic CurvesW6 KP3GR. Pandharipande
KurzbeschreibungI will discuss the classical theory of algebraic curves. The topics will include:
divisors, Riemann-Roch, linear systems, differentials, Clifford's theorem,
curves on surfaces, singularities, curves in projective space, elliptic curves,
hyperelliptic curves, families of curves, moduli, and enumerative geometry.
There will be many examples and calculations.
Lernziel
InhaltLecture homepage: Link
LiteraturForster, "Lectures on Riemann Surfaces"

Arbarello, Cornalba, Griffiths, Harris, "Geometry of Algebraic Curves"

Mumford, "Curves and their Jacobians"
Voraussetzungen / BesonderesFor background, a semester course in algebraic geometry should be
sufficient (perhaps even if taken concurrently). You should know the definitions
of algebraic varieties and algebraic morphisms and their basic properties.
401-3106-17LClass Field TheoryW6 KP2V + 1UJ. Fresán
KurzbeschreibungClass Field Theory aims at describing the Galois group of the maximal abelian extension of global and local fields.
Lernziel
Literatur[1] D. Harari, Cohomologie galoisienne et théorie du corps de classes, EDP Sciences, CNRS Éditions, Paris, 2017.
[2] K. Kato, N. Kurokawa, T. Saito, Number theory 2. Introduction to class field theory, Translations of Mathematical Monographs 240, AMS, 2011.
[3] J. S. Milne, Class Field Theory (available at Link)
[4] J-P. Serre, Local fields, Grad. Texts Math. 67. Springer-Verlag, 1979.
401-3033-00LDie Gödel'schen SätzeW8 KP3V + 1UL. Halbeisen
KurzbeschreibungDie Vorlesung besteht aus drei Teilen:
Teil I gibt eine Einführung in die Syntax und Semantik der Prädikatenlogik erster Stufe.
Teil II behandelt den Gödel'schen Vollständigkeitssatz
Teil III behandelt die Gödel'schen Unvollständigkeitssätze
LernzielDas Ziel dieser Vorlesung ist ein fundiertes Verständnis der Grundlagen der Mathematik zu vermitteln.
InhaltSyntax und Semantik der Prädikatenlogik
Gödel'scher Vollständigkeitssatz
Gödel'sche Unvollständigkeitssätze
LiteraturErgänzende Literatur wird in der Vorlesung angegeben.
401-3058-00LKombinatorik IW4 KP2GN. Hungerbühler
KurzbeschreibungDer Kurs Kombinatorik I und II ist eine Einfuehrung in die abzaehlende Kombinatorik.
LernzielDie Studierenden sind in der Lage, kombinatorische Probleme einzuordnen und die adaequaten Techniken zu deren Loesung anzuwenden.
InhaltInhalt der Vorlesungen Kombinatorik I und II: Kongruenztransformationen der Ebene, Symmetriegruppen von geometrischen Figuren, Eulersche Funktion, Cayley-Graphen, formale Potenzreihen, Permutationsgruppen, Zyklen, Lemma von Burnside, Zyklenzeiger, Saetze von Polya, Anwendung auf die Graphentheorie und isomere Molekuele.
Voraussetzungen / BesonderesWer 401-3052-00L Kombinatorik (letztmals im FS 2008 gelesen) für den Bachelor- oder Master-Studiengang Mathematik anrechnen lässt, darf 401-3058-00L Kombinatorik I nur noch fürs Mathematik Lehrdiplom oder fürs Didaktik-Zertifikat Mathematik anrechnen lassen.
401-3112-17LIntroduction to Number TheoryW4 KP2VC. Busch
KurzbeschreibungThis course gives an introduction to number theory. The focus will be on algebraic number theory.
Lernziel
InhaltThe following subjects will be covered:
- Euclidean algorithm, greatest common divisor, ...
- Congruences, Chinese Remainder Theorem
- Quadratic residues, Legendre symbol, law of quadratic reciprocity
- Quadratic number fields, integers and primes
- Units of quadratic number fields, Pell's equation, Dirichlet unit theorem
- Continued fractions and quadratic irrationalities, Theorem of Euler Lagrange, relation to units.
Literatur- A. Fröhlich, M.J. Taylor, Algebraic number theory, Cambridge studies in advanced mathematics 27, Cambridge University Press, 1991
- S. Lang, Algebraic Number Theory, Second Edition, Graduate Texts in Mathematics, 110, Springer, 1994
- J. Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften 322, Springer 1999
- R. Remmert, P. Ullrich, Elementare Zahlentheorie, Grundstudium Mathematik, Basel Birkhäuser, 2008
- P. Samuel, Algebraic Theory of Numbers, Kershaw Publishing Company LTD, 1972 (Original edition in French at Hermann)
- J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer 1973 (Original edition in French at Presses Universitaires de France)
Voraussetzungen / BesonderesBasic knowledge of Algebra as taught in a course Algebra I + II.
Auswahl: Geometrie
NummerTitelTypECTSUmfangDozierende
401-4206-17LGroup Actions on TreesW4 KP2VN. Lazarovich
KurzbeschreibungAs a main theme, we will explain how an action of a group on a tree enables us to break the group into smaller pieces, and thus gain better understanding of its structure. After introducing the general theory, we will cover various topics in this general theme.
LernzielIntroduction to the general theory of group actions on trees, also known as Bass-Serre theory, and various important results on decompositions of groups.
InhaltDepending on time we will cover some of the following topics.
- Free groups and their subgroups.
- The general theory of actions on trees, i.e, Bass-Serre theory.
- Trees as 1-dimensional buildings.
- Stallings' theorem.
- Grushko's and Dunwoody's accessibility results.
- Actions on R-trees and the Rips machine.
LiteraturJ.-P. Serre, Trees. (Translated from the French by John Stillwell). Springer-Verlag, 1980. ISBN 3-540-10103-9

C. T. C. Wall. The geometry of abstract groups and their splittings. Revista Matemática Complutense vol. 16(2003), no. 1, pp. 5-101
Voraussetzungen / BesonderesFamiliarity with the basics of fundamental group (and covering theory).
401-4148-17LReading Course: Introduction to the Moduli of Maps and Gromow-Witten InvariantsW2 KP4AG. Bérczi
KurzbeschreibungEnumerative questions motivated the development of algebraic geometry for centuries. This course is a short tour to some ideas which have revolutionised enumerative geometry in the last 30 years: stable maps, Gromov-Witten invariants and quantum cohomology.
LernzielThe aim of the course is to understand the concept of stable maps, their moduli and quantum cohomology. We prove Kontsevich's celebrated formula on the number of plane rational curves of degree d passing through 3d-1 given points in general position.
InhaltTopics covered:
1) Brief survey on moduli spaces: fine and coarse moduli.
2) Stable n-pointed curves
3) Stable maps
4) Enumerative geometry via stable maps
5) Gromov-Witten invariants
6) Quantum cohomology and quantum product
7) Kontsevich's formula
LiteraturThe main reference for the course is:
J. Kock and I.Vainsencher: Kontsevich's Formula for Rational Plane Curves
Link

Background material:
-Algebraic varieties: I. R. Shafarevich, Basic Algebraic geometry 1 & 2, Springer-Verlag.
-Moduli of curves: Joe Harris and Ian Morrison, Moduli of Curves, Graduate Texts in Mathematics 187, Springer-Verlag
-Moduli spaces (fine and coarse): Peter. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes, Springer-Verlag
Voraussetzungen / BesonderesSome minimal background in algebraic geometry (varieties, line bundles, Grassmannians, curves).
Basic concepts of moduli spaces (fine and coarse) and group actions will be explained mainly through examples.
401-3056-00LEndliche Geometrien I
Findet dieses Semester nicht statt.
W4 KP2GN. Hungerbühler
KurzbeschreibungEndliche Geometrien I, II: Endliche Geometrien verbinden Aspekte der Geometrie mit solchen der diskreten Mathematik und der Algebra endlicher Körper. Inbesondere werden Modelle der Inzidenzaxiome konstruiert und Schliessungssätze der Geometrie untersucht. Anwendungen liegen im Bereich der Statistik, der Theorie der Blockpläne und der Konstruktion orthogonaler lateinischer Quadrate.
LernzielEndliche Geometrien I, II: Die Studierenden sind in der Lage, Modelle endlicher Geometrien zu konstruieren und zu analysieren. Sie kennen die Schliessungssätze der Inzidenzgeometrie und können mit Hilfe der Theorie statistische Tests entwerfen sowie orthogonale lateinische Quadrate konstruieren. Sie sind vertraut mit Elementen der Theorie der Blockpläne.
InhaltEndliche Geometrien I, II: Endliche Körper, Polynomringe, endliche affine Ebenen, Axiome der Inzidenzgeometrie, Eulersches Offiziersproblem, statistische Versuchsplanung, orthogonale lateinische Quadrate, Transformationen endlicher Ebenen, Schliessungsfiguren von Desargues und Pappus-Pascal, Hierarchie der Schliessungsfiguren, endliche Koordinatenebenen, Schiefkörper, endliche projektive Ebenen, Dualitätsprinzip, endliche Möbiusebenen, selbstkorrigierende Codes, Blockpläne
Literatur- Max Jeger, Endliche Geometrien, ETH Skript 1988

- Albrecht Beutelspacher: Einführung in die endliche Geometrie I,II. Bibliographisches Institut 1983

- Margaret Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press

- Dembowski: Finite Geometries.
401-3574-61LIntroduction to Knot Theory Information
Findet dieses Semester nicht statt.
W6 KP3G
KurzbeschreibungIntroduction to the mathematical theory of knots. We will discuss some elementary topics in knot theory and we will repeatedly centre on how this knowledge can be used in secondary school.
LernzielThe aim of this lecture course is to give an introduction to knot theory. In the course we will discuss the definition of a knot and what is meant by equivalence. The focus of the course will be on knot invariants. We will consider various knot invariants amongst which we will also find the so called knot polynomials. In doing so we will again and again show how this knowledge can be transferred down to secondary school.
InhaltDefinition of a knot and of equivalent knots.
Definition of a knot invariant and some elementary examples.
Various operations on knots.
Knot polynomials (Jones, ev. Alexander.....)
LiteraturAn extensive bibliography will be handed out in the course.
Voraussetzungen / BesonderesPrerequisites are some elementary knowledge of algebra and topology.
Auswahl: Analysis
NummerTitelTypECTSUmfangDozierende
401-4832-17LMathematical Themes in General Relativity IIW4 KP2VA. Carlotto
KurzbeschreibungSecond part of a one-year course offering a rigorous introduction to general relativity, with special emphasis on aspects of current interest in mathematical research. Topics covered include: initial value formulation of the Einstein equations, causality theory and singularities, constructions of data sets by gluing or conformal methods, asymptotically flat spaces and positive mass theorems.
LernzielAcquisition of a solid and broad background in general relativity and mastery of the basic mathematical methods and ideas developed in such context and successfully exploited in the field of geometric analysis.
InhaltAnalysis of Jang's equation and application to the proof of the spacetime positive energy theorem; the conformal method for the Einstein constraint equations and links with the Yamabe problem; gluing methods for the Einstein constraint equations: canonical asymptotics, N-body solutions, gravitational shielding.
SkriptLecture notes written by the instructor will be provided to all enrolled students.
Voraussetzungen / BesonderesThe content of the basic courses of the first three years at ETH will be assumed. In particular, enrolled students are expected to be fluent both in Differential Geometry (at least at the level of Differentialgeometrie I, II) and Functional Analysis (at least at the level of Funktionalanalysis I, II). Some background on partial differential equations, mainly of elliptic and hyperbolic type, (say at the level of the monograph by L. C. Evans) would also be desirable.
**This course is the sequel of the one offered during the first semester.**
401-3352-09LAn Introduction to Partial Differential EquationsW6 KP3VF. Da Lio
KurzbeschreibungThis course aims at being an introduction to first and second order partial differential equations (in short PDEs).
We will present the so called method of characteristics to solve quasilinear PDEs and some basic properties of classical solutions to second order linear PDEs.
Lernziel
InhaltA preliminary plan is the following
- Laplace equation, fundamental solution, harmonic functions and main properties, maximum principle. Poisson equation. Green functions. Perron method for the solution of the Dirichlet problem.
- Weak and strong maximum principle for elliptic operators.
- Heat equation, fundamental solution, existence of solutions to the Cauchy problem and representation formulas, main properties, uniqueness by maximum principle, regularity.
- Wave equation, existence of the solution, D'Alembert formula, solutions by spherical means, main properties, uniqueness by energy methods.
- The Method of characteristics for first order equations, linear and nonlinear, transport equation, Hamilton-Jacobi equation, scalar conservation laws.
- A brief introduction to viscosity solutions.
SkriptThe teacher provides the students with personal notes.
LiteraturBibliography
- L.Evans Partial Differential Equations, AMS 2010 (2nd edition)
- D. Gilbarg, N.S. Trudinger Elliptic Partial Differential Equations of Second Order, Springer, 1998.
- E. Di Benedetto Partial Differential Equations, Birkauser, 2010 (2nd edition).
- W. A. Strauss Partial Differential Equations. An Introduction, Wiley, 1992.
Voraussetzungen / BesonderesDifferential and integral calculus for functions of several variables; elementary theory of ordinary differential equations, basic facts of measure theory.
401-3496-17LTopics in the Calculus of VariationsW4 KP2VA. Figalli
Kurzbeschreibung
Lernziel
Auswahl: Weitere Gebiete
NummerTitelTypECTSUmfangDozierende
401-3502-17LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W2 KP4AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3503-17LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W3 KP6AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
401-3504-17LReading Course Belegung eingeschränkt - Details anzeigen
DIE BELEGUNG ERFOLGT DURCH DAS STUDIENSEKRETARIAT.

Bitte schicken Sie ein E-Mail an das Studiensekretariat D-MATH <Link> mit folgenden Angaben:
1) welchen Reading Course (60, 90, 120 Arbeitsstunden entsprechend 2, 3, 4 ECTS-Kreditpunkten) Sie belegen möchten;
2) in welchem Semester;
3) für welchen Studiengang;
4) Ihr Name und Vorname;
5) Ihre Studierenden-Nummer;
6) der Name und Vorname des Betreuers/der Betreuerin des Reading Courses.
W4 KP9AProfessor/innen
KurzbeschreibungIn diesem Reading Course wird auf Eigeninitiative und auf individuelle Vereinbarung mit einem Dozenten/einer Dozentin hin ein Stoff durch eigenständiges Literaturstudium erarbeitet.
Lernziel
Wahlfächer aus Bereichen der angewandten Mathematik ...
vollständiger Titel:
Wahlfächer aus Bereichen der angewandten Mathematik und weiteren anwendungsorientierten Gebieten
Auswahl: Numerische Mathematik
NummerTitelTypECTSUmfangDozierende
401-4606-00LNumerical Analysis of Stochastic Partial Differential Equations
Findet dieses Semester nicht statt.
W8 KP4Gkeine Angaben
KurzbeschreibungIn this course solutions of semilinear stochastic partial differential equations (SPDEs) of the evolutionary type and some of their numerical approximation methods are investigated. Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences.
LernzielThe aim of this course is to teach the students a decent knowledge on solutions of semilinear stochastic partial differential equations (SPDEs), on some numerical approximation methods for such equations and on the functional analytic and probabilistic concepts used to formulate and study such equations.
InhaltThe course includes content (i) on the (functional) analytic concepts used to study semilinear stochastic partial differential equations (SPDEs) (e.g., nuclear operators, Hilbert-Schmidt operators, diagonal linear operators on Hilbert spaces, interpolation spaces associated to a diagonal linear operator, semigroups of bounded linear operators, Gronwall-type inequalities), (ii) on the probabilistic concepts used to study SPDEs (e.g., Hilbert space valued random variables, Hilbert space valued stochastic processes, infinite dimensional Wiener processes, stochastic integration with respect to infinite dimensional Wiener processes, infinite dimensional jump processes), (iii) on solutions of SPDEs (e.g., existence, uniqueness and regularity properties of mild solutions of SPDEs, applications involving SPDEs), and (iv) on numerical approximations of SPDEs (e.g., spatial and temporal discretizations, strong convergence, weak convergence). Semilinear SPDEs are a key ingredient in a number of models from economics and the natural sciences. They appear, for example, in models from neurobiology for the approximative description of the propagation of electrical impulses along nerve cells, in models from financial engineering for the approximative pricing of financial derivatives, in models from fluid mechanics for the approximative description of velocity fields in fully developed turbulent flows, in models from quantum field theory for describing the temporal dynamics associated to Euclidean quantum field theories, and in models from chemistry for the approximative description of the temporal evolution of the concentration of an undesired chemical contaminant in the groundwater system.
SkriptLecture notes will be available as a PDF file.
Literatur1. Stochastic Equations in Infinite Dimensions
G. Da Prato and J. Zabczyk
Cambridge Univ. Press (1992)

2. Taylor Approximations for Stochastic Partial Differential Equations
A. Jentzen and P.E. Kloeden
Siam (2011)

3. Numerical Solution of Stochastic Differential Equations
P.E. Kloeden and E. Platen
Springer Verlag (1992)

4. A Concise Course on Stochastic Partial Differential Equations
C. Prévôt and M. Röckner
Springer Verlag (2007)

5. Galerkin Finite Element Methods for Parabolic Problems
V. Thomée
Springer Verlag (2006)
Voraussetzungen / BesonderesMandatory prerequisites: Functional analysis, probability theory;
Recommended prerequisites: stochastic processes;
401-4658-00LComputational Methods for Quantitative Finance: PDE Methods Information Belegung eingeschränkt - Details anzeigen W6 KP3V + 1UC. Schwab
KurzbeschreibungIntroduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming
and knowledge of numerical mathematics at ETH BSc level.
LernzielIntroduce the main methods for efficient numerical valuation of derivative contracts in a
Black Scholes as well as in incomplete markets due Levy processes or due to stochastic volatility
models. Develop implementation of pricing methods in MATLAB.
Finite-Difference/ Finite Element based methods for the solution of the pricing integrodifferential equation.
Inhalt1. Review of option pricing. Wiener and Levy price process models. Deterministic, local and stochastic
volatility models.
2. Finite Difference Methods for option pricing. Relation to bi- and multinomial trees.
European contracts.
3. Finite Difference methods for Asian, American and Barrier type contracts.
4. Finite element methods for European and American style contracts.
5. Pricing under local and stochastic volatility in Black-Scholes Markets.
6. Finite Element Methods for option pricing under Levy processes. Treatment of
integrodifferential operators.
7. Stochastic volatility models for Levy processes.
8. Techniques for multidimensional problems. Baskets in a Black-Scholes setting and
stochastic volatility models in Black Scholes and Levy markets.
9. Introduction to sparse grid option pricing techniques.
SkriptThere will be english, typed lecture notes as well as MATLAB software for registered participants in the course.
LiteraturR. Cont and P. Tankov : Financial Modelling with Jump Processes, Chapman and Hall Publ. 2004.

Y. Achdou and O. Pironneau : Computational Methods for Option Pricing, SIAM Frontiers in Applied Mathematics, SIAM Publishers, Philadelphia 2005.

D. Lamberton and B. Lapeyre : Introduction to stochastic calculus Applied to Finance (second edition), Chapman & Hall/CRC Financial Mathematics Series, Taylor & Francis Publ. Boca Raton, London, New York 2008.

J.-P. Fouque, G. Papanicolaou and K.-R. Sircar : Derivatives in financial markets with stochastic volatility, Cambridge Univeristy Press, Cambridge, 2000.

N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013.
Voraussetzungen / BesonderesStart of the lecture: Wednesday, March 1, 2017 (second week of the semester).
401-4788-16LMathematics of (Super-Resolution) Biomedical ImagingW8 KP4GH. Ammari
KurzbeschreibungThe aim of this course is to review different methods used to address challenging problems in biomedical imaging. The emphasis will be on scale separation techniques, hybrid imaging, spectroscopic techniques, and nanoparticle imaging. These approaches allow one to overcome the ill-posedness character of imaging reconstruction in biomedical applications and to achieve super-resolution imaging.
LernzielSuper-resolution imaging is a collective name for a number of emerging techniques that achieve resolution below the conventional resolution limit, defined as the minimum distance that two point-source objects have to be in order to distinguish the two sources from each other.

In this course we describe recent advances in scale separation techniques, spectroscopic approaches, multi-wave imaging, and nanoparticle imaging. The objective is fivefold:
(i) To provide asymptotic expansions for both internal and boundary perturbations that are due to the presence
of small anomalies;
(ii) To apply those asymptotic formulas for the purpose of identifying the material parameters and certain geometric features of the anomalies;
(iii) To design efficient inversion algorithms in multi-wave modalities;
(iv) to develop inversion techniques using multi-frequency measurements;
(v) to develop a mathematical and numerical framework for nanoparticle imaging.

In this course we shall consider both analytical and computational
matters in biomedical imaging. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, inverse problems, mathematical imaging, optimal control, stochastic modelling, and analysis of physical phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in biomedical imaging, requires a deep understanding of the different scales in the physical models, an accurate mathematical modelling of the imaging techniques, and fine analysis of complex physical phenomena.

An emphasis is put on mathematically analyzing acoustic-electric imaging, thermo-elastic imaging, Lorentz force based imaging, elastography, multifrequency electrical impedance tomography, and plasmonic resonant nanoparticles.
252-0504-00LNumerical Methods for Solving Large Scale Eigenvalue Problems Information
Findet dieses Semester nicht statt.
W4 KP3GP. Arbenz
KurzbeschreibungDie Vorlesung behandelt Algorithmen zur Lösung von Eigenwertproblemen
mit grossen, schwach besetzten Matrizen. Die z.T. erst in den letzten Jahren
entwickelten Verfahren werden theoretisch und praktisch mit MATLAB
untersucht.
LernzielKenntnisse der modernen Eigenlöser, ihres numerischen Verhaltens, ihrer Einsatzmöglichkeiten und Grenzen.
InhaltDie Vorlesung beginnt mit verschiedenartigen Beispielen für Anwendungen
in denen Eigenwertprobleme eine wichtige Rolle spielen. Nach einer
Einführung in die Lineare Algebra der Eigenwertprobleme wird ein
Überblick über Verfahren (QR-Algorithmus u.ä.) zur Behandlung kleiner
und mittelgrosser Eigenwertprobleme gegeben.

Danach werden die heute wichtigsten Löser für grosse, typischerweise
schwach-besetzte Matrixeigenwertprobleme vorgestellt und analysiert.
Dabei wird eine Auswahl der folgenden Themen behandelt:

* Vektor- und Teilraumiteration
* Spurminimierungsalgorithmus
* Arnoldi- und Lanczos-Algorithmus (inkl. Varianten mit Neustart)
* Davidson- und Jacobi-Davidson-Algorithmus
* vorkonditionierte inverse Iteration und LOBPCG
* Verfahren für nichtlineaere Eigenwertprobleme

In den Übungen werden diese Algorithmen (in vereinfachter Form) in
MATLAB implementiert und numerisch untersucht.
SkriptLecture notes (Englisch),
Kopien der Folien
LiteraturZ. Bai, J. Demmel, J. Dongarra, A. Ruhe, and H. van der Vorst: Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide. SIAM, Philadelphia, 2000.

Y. Saad: Numerical Methods for Large Eigenvalue Problems. Manchester University Press, Manchester, 1994.

G. H. Golub and Ch. van Loan: Matrix Computations, 3rd ed. Johns Hopkins University Press, Baltimore 1996.
Voraussetzungen / BesonderesVoraussetzung: Lineare Algebra
Auswahl: Wahrscheinlichkeitstheorie, Statistik
NummerTitelTypECTSUmfangDozierende
401-3919-60LAn Introduction to the Modelling of ExtremesW4 KP2VP. Embrechts
KurzbeschreibungThis course yields an introduction into the MATHEMATICAL THEORY of one-dimensional extremes, and this mainly from a more probabilistic point of view.
LernzielIn this course, students learn to distinguish between so-called normal models, i.e. models based on the normal or Gaussian distribution, and so-called heavy-tailed or power-tail models.
They learn to do probabilistic modelling of extremes in one-dimensional data. The probabilistic key theorems are the Fisher-Tippett Theorem and the Balkema-de Haan-Pickands Theorem. These lead to the statistical techniques for the analysis of extremes or rare events known as the Block Method, and Peaks Over Threshold method, respectively.
Inhalt- Introduction to rare or extreme events
- Regular Variation
- The Convergence to Types Theorem
- The Fisher-Tippett Theorem
- The Method of Block Maxima
- The Maximal Domain of Attraction
- The Fre'chet, Gumbel and Weibull distributions
- The POT method
- The Point Process Method: a first introduction
- The Pickands-Balkema-de Haan Theorem and its applications
- Some extensions and outlook
SkriptThere will be no script available, students are required to take notes from the blackboard lectures. The course follows closely Extreme Value Theory as developed in:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
LiteraturThe main text on which the course is based is:
P. Embrechts, C. Klueppelberg and T. Mikosch (1997)
Modelling Extremal Events for Insurance and Finance.
Springer.
Further relevant literature is:
S. I. Resnick (2007) Heavy-Tail Phenomena. Probabilistic and
Statistical Modeling. Springer.
S. I. Resnick (1987) Extreme Values, Regular Variation,
and Point Processes. Springer.
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