Habib Ammari: Catalogue data in Spring Semester 2020
|Name||Prof. Dr. Habib Ammari|
Seminar für Angewandte Mathematik
ETH Zürich, HG G 57.3
|Telephone||+41 44 633 80 31|
|401-2654-00L||Numerical Analysis II||6 credits||3V + 2U||H. Ammari|
|Abstract||The central topic of this course is the numerical treatment of ordinary differential equations. It focuses on the derivation, analysis, efficient implementation, and practical application of single step methods and pay particular attention to structure preservation.|
|Objective||The course aims to impart knowledge about important numerical methods for the solution of ordinary differential equations. This includes familiarity with their main ideas, awareness of their advantages and limitations, and techniques for investigating stability and convergence. Further, students should know about structural properties of ordinary diferential equations and how to use them as guideline for the selection of numerical integration schemes. They should also acquire the skills to implement numerical integrators in Python and test them in numerical experiments.|
|Content||Chapter 1. Some basics|
1.1. What is a differential equation?
1.2. Some methods of resolution
1.3. Important examples of ODEs
Chapter 2. Existence, uniqueness, and regularity in the Lipschitz case
2.1. Banach fixed point theorem
2.2. Gronwall’s lemma
2.3. Cauchy-Lipschitz theorem
Chapter 3. Linear systems
3.1. Exponential of a matrix
3.2. Linear systems with constant coefficients
3.3. Linear system with non-constant real coefficients
3.4. Second order linear equations
3.5. Linearization and stability for autonomous systems
3.6 Periodic Linear Systems
Chapter 4. Numerical solution of ordinary differential equations
4.2. The general explicit one-step method
4.3. Example of linear systems
4.4. Runge-Kutta methods
4.5. Multi-step methods
4.6. Stiff equations and systems
4.7. Perturbation theories for differential equations
Chapter 5. Geometrical numerical integration methods for differential equation
5.2. Structure preserving methods for Hamiltonian systems
5.3. Runge-Kutta methods
5.4. Long-time behaviour of numerical solutions
Chapter 6. Finite difference methods
6.2. Numerical algorithms for the heat equation
6.3. Numerical algorithms for the wave equation
6.4. Numerical algorithms for the Hamilton-Jacobi equation in one dimension
Chapter 7. Stochastic differential equations
7.2. Langevin equation
7.3. Ornstein-Uhlenbeck equation
7.4. Existence and uniqueness of solutions in dimension one
7.5. Numerical solution of stochastic differential equations
|Lecture notes||Lecture notes including supplements will be provided electronically.|
Please find the lecture homepage here:
All assignments and some previous exam problems will be available for download on lecture homepage.
|Literature||Note: Extra reading is not considered important for understanding the|
Deuflhard and Bornemann: Numerische Mathematik II - Integration gewöhnlicher Differentialgleichungen, Walter de Gruyter & Co., 1994.
Hairer and Wanner: Solving ordinary differential equations II - Stiff and differential-algebraic problems, Springer-Verlag, 1996.
Hairer, Lubich and Wanner: Geometric numerical integration - Structure-preserving algorithms for ordinary differential equations}, Springer-Verlag, Berlin, 2002.
L. Gruene, O. Junge "Gewoehnliche Differentialgleichungen", Vieweg+Teubner, 2009.
Hairer, Norsett and Wanner: Solving ordinary differential equations I - Nonstiff problems, Springer-Verlag, Berlin, 1993.
Walter: Gewöhnliche Differentialgleichungen - Eine Einführung, Springer-Verlag, Berlin, 1972.
Walter: Ordinary differential equations, Springer-Verlag, New York, 1998.
|Prerequisites / Notice||Homework problems involve Python implementation of numerical algorithms.|
|401-4788-16L||Mathematics of (Super-Resolution) Biomedical Imaging|
NOTICE: The exercise class scheduled for 5 March has been cancelled
|8 credits||4G||H. Ammari|
|Abstract||The 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.|
|Objective||Super-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.
|401-5650-00L||Zurich Colloquium in Applied and Computational Mathematics||0 credits||1K||R. Abgrall, R. Alaifari, H. Ammari, R. Hiptmair, S. Mishra, S. Sauter, C. Schwab|