Search result: Catalogue data in Spring Semester 2021
Mathematics Master | ||||||
Electives For the Master's degree in Applied Mathematics the following additional condition (not manifest in myStudies) must be obeyed: At least 15 of the required 28 credits from core courses and electives must be acquired in areas of applied mathematics and further application-oriented fields. | ||||||
Electives: Applied Mathematics and Further Application-Oriented Fields ¬ | ||||||
Selection: Numerical Analysis | ||||||
Number | Title | Type | ECTS | Hours | Lecturers | |
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401-4658-00L | Computational Methods for Quantitative Finance: PDE Methods | W | 6 credits | 3V + 1U | C. Marcati, A. Stein | |
Abstract | Introduction to principal methods of option pricing. Emphasis on PDE-based methods. Prerequisite MATLAB and Python programming and knowledge of numerical mathematics at ETH BSc level. | |||||
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 and Python. 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 lecture notes as well as MATLAB or Python software for registered participants in the course. | |||||
Literature | Main reference (course text): N. Hilber, O. Reichmann, Ch. Schwab and Ch. Winter: Computational Methods for Quantitative Finance, Springer Finance, Springer, 2013. Supplementary texts: 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. | |||||
Prerequisites / Notice | Knowledge of Numerical Analysis/ Scientific Computing Techniques corresponding roughly to BSc MATH or BSc RW/CSE at ETH is expected. Basic programming skills in MATLAB or Python are required for the exercises, and are _not_ taught in this course. | |||||
401-4656-21L | Deep Learning in Scientific Computing Aimed at students in a Master's Programme in Mathematics, Engineering and Physics. | W | 6 credits | 2V + 1U | S. Mishra | |
Abstract | Machine Learning, particularly deep learning is being increasingly applied to perform, enhance and accelerate computer simulations of models in science and engineering. This course aims to present a highly topical selection of themes in the general area of deep learning in scientific computing, with an emphasis on the application of deep learning algorithms for systems, modeled by PDEs. | |||||
Objective | The objective of this course will be to introduce students to advanced applications of deep learning in scientific computing. The focus will be on the design and implementation of algorithms as well as on the underlying theory that guarantees reliability of the algorithms. We will provide several examples of applications in science and engineering where deep learning based algorithms outperform state of the art methods. | |||||
Content | A selection of the following topics will be presented in the lectures. 1. Issues with traditional methods for scientific computing such as Finite Element, Finite Volume etc, particularly for PDE models with high-dimensional state and parameter spaces. 2. Introduction to Deep Learning: Artificial Neural networks, Supervised learning, Stochastic gradient descent algorithms for training, different architectures: Convolutional Neural Networks, Recurrent Neural Networks, ResNets. 3. Theoretical Foundations: Universal approximation properties of the Neural networks, Bias-Variance decomposition, Bounds on approximation and generalization errors. 4. Supervised deep learning for solutions fields and observables of high-dimensional parametric PDEs. Use of low-discrepancy sequences and multi-level training to reduce generalization error. 5. Uncertainty Quantification for PDEs with supervised learning algorithms. 6. Deep Neural Networks as Reduced order models and prediction of solution fields. 7. Active Learning algorithms for PDE constrained optimization. 8. Recurrent Neural Networks and prediction of time series for dynamical systems. 9. Physics Informed Neural networks (PINNs) for the forward problem for PDEs. Applications to high-dimensional PDEs. 10. PINNs for inverse problems for PDEs, parameter identification, optimal control and data assimilation. All the algorithms will be illustrated on a variety of PDEs: diffusion models, Black-Scholes type PDEs from finance, wave equations, Euler and Navier-Stokes equations, hyperbolic systems of conservation laws, Dispersive PDEs among others. | |||||
Lecture notes | Lecture notes will be provided at the end of the course. | |||||
Literature | All the material in the course is based on research articles written in last 1-2 years. The relevant references will be provided. | |||||
Prerequisites / Notice | The students should be familiar with numerical methods for PDEs, for instance in courses such as Numerical Methods for PDEs for CSE, Numerical analysis of Elliptic and Parabolic PDEs, Numerical methods for hyperbolic PDEs, Computational methods for Engineering Applications. Some familiarity with basic concepts in machine learning will be beneficial. The exercises in the course rely on standard machine learning frameworks such as KERAS, TENSORFLOW or PYTORCH. So, competence in Python is helpful. | |||||
401-4652-21L | Nonlocal Inverse Problems | W | 4 credits | 2V | J. Railo | |
Abstract | This course is an introduction to the Calderón problem and nonlocal inverse problems for the fractional Schrödinger equation. These are examples of nonlinear inverse problems. The classical Calderón problem models electrical impedance tomography (EIT) and fractional operators appear, for example, in some mathematical models in finance. | |||||
Objective | Students become familiar with the Calderón problem and some nonlocal phenomena related to the fractional Laplacian. Advanced students should be able to read research articles on the fractional Calderón problems after the course. | |||||
Content | In the beginning of the course, we will introduce some basic theory for the classical Calderón problem. The focus of the course will be in the study of nonlocal inverse problems for the fractional Schrödinger equation with lower order perturbations. We discuss necessary preliminaries on Sobolev spaces, Fourier analysis, functional analysis and theory of PDEs. Our scope will be in the uniqueness properties. Classical Calderón problem (about 1/3): Conductivity and Schrödinger equations, Dirichlet-to-Neumann maps, Cauchy data, and related boundary value inverse problems. The methods include, for example, complex geometric optics (CGO) solutions. Fractional Calderón problem (about 2/3): Nonlocal unique continuation principles (UCP), Runge approximation properties, and uniqueness for the fractional Calderón problem. The methods include, for example, Caffarelli-Silvestre extensions, the fractional Poincaré inequality and Riesz transforms. | |||||
Lecture notes | Lecture notes and exercises | |||||
Literature | 1. M. Salo: Calderón problem. Lecture notes, University of Helsinki (2008). (Available at Link.) 2. T. Ghosh, M. Salo, G. Uhlmann: The Calderón problem for the fractional Schrödinger equation. Analysis & PDE 13 (2020), no. 2, 455-475. 3. A. Rüland, M. Salo: The fractional Calderón problem: low regularity and stability. Nonlinear Analysis 193 (2020), special issue "Nonlocal and Fractional Phenomena", 111529. 4. Other literature will be specified in the course. | |||||
Prerequisites / Notice | Functional Analysis I & II or similar knowledge. Any additional knowledge of Fourier analysis, Sobolev spaces, distributions and PDEs will be an asset. | |||||
401-3426-21L | Time-Frequency Analysis | W | 4 credits | 2G | R. Alaifari | |
Abstract | This course gives a basic introduction to time-frequency analysis from the viewpoint of applied harmonic analysis. | |||||
Objective | By the end of the course students should be familiar with the concept of the short-time Fourier transform, the Bargmann transform, quadratic time-frequency representations (ambiguity function and Wigner distribution), Gabor frames and modulation spaces. The connection and comparison to time-scale representations will also be subject of this course. | |||||
Content | Time-frequency analysis lies at the heart of many applications in signal processing and aims at capturing time and frequency information simultaneously (as opposed to the classical Fourier transform). This course gives a basic introduction that starts with studying the short-time Fourier transform and the special role of the Gauss window. We will visit quadratic representations and then focus on discrete time-frequency representations, where Gabor frames will be introduced. Later, we aim at a more quantitative analysis of time-frequency information through modulation spaces. At the end, we touch on wavelets (time-scale representation) as a counterpart to the short-time Fourier transform. | |||||
Literature | Gröchenig, K. (2001). Foundations of time-frequency analysis. Springer Science & Business Media. | |||||
Prerequisites / Notice | Functional analysis, Fourier analysis, complex analysis, operator theory |
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