# Search result: Catalogue data in Autumn 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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401-4657-00L | Numerical Analysis of Stochastic Ordinary Differential Equations Alternative course title: "Computational Methods for Quantitative Finance: Monte Carlo and Sampling Methods" | W | 6 credits | 3V + 1U | A. Stein | |

Abstract | Course on numerical approximations of stochastic ordinary differential equations driven by Wiener processes. These equations have several applications, for example in financial option valuation. This course also contains an introduction to random number generation and Monte Carlo methods for random variables. | |||||

Objective | The aim of this course is to enable the students to carry out simulations and their mathematical convergence analysis for stochastic models originating from applications such as mathematical finance. For this the course teaches a decent knowledge of the different numerical methods, their underlying ideas, convergence properties and implementation issues. | |||||

Content | Generation of random numbers Monte Carlo methods for the numerical integration of random variables Stochastic processes and Brownian motion Stochastic ordinary differential equations (SODEs) Numerical approximations of SODEs Applications to computational finance: Option valuation | |||||

Lecture notes | There will be English, typed lecture notes for registered participants in the course. | |||||

Literature | P. Glassermann: Monte Carlo Methods in Financial Engineering. Springer-Verlag, New York, 2004. P. E. Kloeden and E. Platen: Numerical Solution of Stochastic Differential Equations. Springer-Verlag, Berlin, 1992. | |||||

Prerequisites / Notice | Prerequisites: Mandatory: Probability and measure theory, basic numerical analysis and basics of MATLAB/Python programming. a) mandatory courses: Elementary Probability, Probability Theory I. b) recommended courses: Stochastic Processes. Start of lectures: Wednesday September 22, 2021. | |||||

401-4785-00L | Mathematical and Computational Methods in Photonics | W | 8 credits | 4G | H. Ammari | |

Abstract | The aim of this course is to review new and fundamental mathematical tools, computational approaches, and inversion and optimal design methods used to address challenging problems in nanophotonics. The emphasis will be on analyzing plasmon resonant nanoparticles, super-focusing & super-resolution of electromagnetic waves, photonic crystals, electromagnetic cloaking, metamaterials, and metasurfaces | |||||

Objective | The field of photonics encompasses the fundamental science of light propagation and interactions in complex structures, and its technological applications. The recent advances in nanoscience present great challenges for the applied and computational mathematics community. In nanophotonics, the aim is to control, manipulate, reshape, guide, and focus electromagnetic waves at nanometer length scales, beyond the resolution limit. In particular, one wants to break the resolution limit by reducing the focal spot and confine light to length scales that are significantly smaller than half the wavelength. Interactions between the field of photonics and mathematics has led to the emergence of a multitude of new and unique solutions in which today's conventional technologies are approaching their limits in terms of speed, capacity and accuracy. Light can be used for detection and measurement in a fast, sensitive and accurate manner, and thus photonics possesses a unique potential to revolutionize healthcare. Light-based technologies can be used effectively for the very early detection of diseases, with non-invasive imaging techniques or point-of-care applications. They are also instrumental in the analysis of processes at the molecular level, giving a greater understanding of the origin of diseases, and hence allowing prevention along with new treatments. Photonic technologies also play a major role in addressing the needs of our ageing society: from pace-makers to synthetic bones, and from endoscopes to the micro-cameras used in in-vivo processes. Furthermore, photonics are also used in advanced lighting technology, and in improving energy efficiency and quality. By using photonic media to control waves across a wide band of wavelengths, we have an unprecedented ability to fabricate new materials with specific microstructures. The main objective in this course is to report on the use of sophisticated mathematics in diffractive optics, plasmonics, super-resolution, photonic crystals, and metamaterials for electromagnetic invisibility and cloaking. The book merges highly nontrivial multi-mathematics in order to make a breakthrough in the field of mathematical modelling, imaging, and optimal design of optical nanodevices and nanostructures capable of light enhancement, and of the focusing and guiding of light at a subwavelength scale. We demonstrate the power of layer potential techniques in solving challenging problems in photonics, when they are combined with asymptotic analysis and the elegant theory of Gohberg and Sigal on meromorphic operator-valued functions. In this course we shall consider both analytical and computational matters in photonics. The issues we consider lead to the investigation of fundamental problems in various branches of mathematics. These include asymptotic analysis, spectral analysis, mathematical imaging, optimal design, stochastic modelling, and analysis of wave propagation phenomena. On the other hand, deriving mathematical foundations, and new and efficient computational frameworks and tools in photonics, requires a deep understanding of the different scales in the wave propagation problem, an accurate mathematical modelling of the nanodevices, and fine analysis of complex wave propagation phenomena. An emphasis is put on mathematically analyzing plasmon resonant nanoparticles, diffractive optics, photonic crystals, super-resolution, and metamaterials. | |||||

401-5003-71L | At the Interface Between Semiclassical Analysis and Numerical Analysis of Wave-Scattering Problems | W | 4 credits | 2V | E. Spence | |

Abstract | Postgraduate degree lecture | |||||

Objective | ||||||

Content | Semiclassical analysis (SCA) is a branch of microlocal analysis concerned with rigorously analysing PDEs with large (or small) parameters. On the other hand, numerical analysis (NA) seeks to design numerical methods that are accurate, efficient, and robust, with theorems guaranteeing these properties. In the context of high-frequency wave scattering, both SCA and NA share the same goal – that of understanding the behaviour of the scattered wave – but these two fields have operated largely in isolation, mainly because the tools and techniques of the two fields are somewhat disjoint. This by-and-large self-contained course focuses on the Helmholtz equation, which is arguably the simplest possible model of wave propagation. Our first goal will be to show how even relatively-simple tools from semiclassical analysis can be used to prove fundamental results about the numerical analysis of finite-element method applied to the high-frequency Helmholtz equation. | |||||

Prerequisites / Notice | The course will aim at being accessible both to students coming from a numerical-analysis/applied-maths background and to students coming from an analysis background. |

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