Christoph Schwab: Catalogue data in Spring Semester 2016 |
| Name | Prof. Dr. Christoph Schwab |
| Field | Mathematik |
| Address | Seminar für Angewandte Mathematik ETH Zürich, HG G 57.1 Rämistrasse 101 8092 Zürich SWITZERLAND |
| Telephone | +41 44 632 35 95 |
| Fax | +41 44 632 10 85 |
| christoph.schwab@sam.math.ethz.ch | |
| URL | http://www.sam.math.ethz.ch/~schwab |
| Department | Mathematics |
| Relationship | Full Professor |
| Number | Title | ECTS | Hours | Lecturers | |
|---|---|---|---|---|---|
| 401-1652-10L | Numerical Analysis I | 6 credits | 3V + 2U | C. Schwab | |
| Abstract | This course will give an introduction to numerical methods, aimed at mathematics majors. It covers numerical linear algebra, quadrature, interpolation and approximation methods as well as their error analysis and implementation. | ||||
| Learning objective | Knowledge of the fundamental numerical methods as well as `numerical literacy': application of numerical methods for the solution of application problems, mathematical foundations of numerical methods, and basic mathematical methods of the analysis of stability, consistency and convergence of numerical methods, MATLAB implementation. | ||||
| Content | Rounding errors, solution of linear systems of equations, nonlinear equations, interpolation (polynomial as well as trigonometric), least squares problems, extrapolation, numerical quadrature, elementary optimization methods. | ||||
| Lecture notes | Lecture Notes and reading list will be available. | ||||
| Literature | Lecture Notes (german or english) will be made available to students of ETH BSc MATH. Quarteroni, Sacco and Saleri, Numerische Mathematik 1 + 2, Springer Verlag 2002 (in German). There is an English version of this text, containing both German volumes, from the same publisher. If you feel more comfortable with English, you can follow this text as well. Content and Indexing are identical in the German and the English text. | ||||
| Prerequisites / Notice | Admission Requirements: Completed course Linear Algebra I, Analysis I in ETH BSc MATH Parallel enrolment in Linear Algebra II, Analysis II in ETH BSc MATH Weekly homework assignments involving MATLAB programming are an integral part of the course. Turn-in of solutions will be graded. | ||||
| 401-3650-16L | Numerical Analysis of Data Assimilation in High Dimension  Number of participants limited to 6. | 4 credits | 2S | C. Schwab | |
| Abstract | Mathematical foundations of numerical methods for the efficient approximation of high- and infinite-dimensional problems. | ||||
| Learning objective | Mathematical foundations of numerical methods for the efficient approximation of high- and infinite-dimensional problems. | ||||
| Content | The seminar will survey the most widely used numerical methods for computational data assimilation, for evolution problems on high dimensional state and parameter spaces. Focus is on their mathematical analysis and on algorithmic aspects, via prototypical implementations in MATLAB. Examples are: Particle-, Kalman-, extended Kalman, and ensemble Kalman Filters, 3DVAR, Metropolis-Hastings, MCMC algorithms, etc. in discrete and in continuous time. Students will elaborate the mathematical theory from the literature, as well as demonstrate a working implementation of the algorithm to the group, and produce a written mathematical summary. | ||||
| Lecture notes | Original research papers from the past four years on the topic of the course, on N-term polynomial chaos approximations of SPDEs, Multilevel Monte Carlo and Quasi Monte Carlo Methods, Multivariate Decomposition Methods, Dimension adaptive numerical integration methods. | ||||
| Literature | Original research papers from the past four years on the topic of the course, on N-term polynomial chaos approximations of SPDEs, Multilevel Monte Carlo and Quasi Monte Carlo Methods, Multivariate Decomposition Methods, Dimension adaptive numerical integration methods. | ||||
| Prerequisites / Notice | The number of participants of the seminar is limited to 6. The preference will be given to ETH students of the following programs [in this order]: 1. ETH MSc Applied Math, 2. ETH MSc RW/CSE, 3. ETH MSc MATH. 4. ETH BSc MATH, The prerequisites are: (*) for students taking the seminar for ETH BSc MATH: completed BSc examinations in Numerische Mathematik I+II, Numerical Methods for Elliptic and Parabolic PDEs. (*) for students taking the seminar for ETH MSc Math, Applied Math, RW/CSE: completed exam in courses Numerical Methods for Elliptic and Parabolic PDEs, OR NumPDEs for RW/CSE, Numerical Analysis of Stochastic PDEs. Topics will be discussed and selected for each participant during the first meeting on <time/place tba>. "MyCopy Softcover Edition" allows to purchase a hardcopy of the text for 25 CHF. Link only accessible from nethz acct. http://link.springer.com/book/10.1007/978-3-319-20325-6 | ||||
| 401-4658-00L | Computational Methods for Quantitative Finance: PDE Methods | 6 credits | 3V + 1U | C. Schwab | |
| Abstract | Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB programming and knowledge of numerical mathematics at ETH BSc level. | ||||
| Learning objective | Introduce 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. | ||||
| Content | 1. 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. | ||||
| Lecture notes | There will be english, typed lecture notes as well as MATLAB software for registered participants in the course. | ||||
| Literature | R. 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. | ||||
| Prerequisites / Notice | The 2009 title of this course unit was "Computational Methods for Quantitative Finance II: Finite Element and Finite Difference Methods". | ||||
| 401-5650-00L | Zurich Colloquium in Applied and Computational Mathematics  | 0 credits | 2K | R. Abgrall, H. Ammari, P. Grohs, R. Hiptmair, A. Jentzen, S. Mishra, S. Sauter, C. Schwab | |
| Abstract | Research colloquium | ||||
| Learning objective | |||||