Course Description & Materials
1. Functions of Matrices and Exponential IntegratorsFunctions of matrices are widely used in science, engineering and the social sciences, due to the succinct and insightful way they allow problems to be formulated and solutions to be expressed. New applications involving matrix functions are regularly being found, ranging from small but difficult problems in medicine to huge, sparse systems arising in exponential integrators for the solution of partial differential equations. This course will treat the underlying theory of matrix functions, describe a variety of algorithms, give an overview of some applications, and briefly discuss software issues. It will also treat exponential integrators, giving an introduction to the topic from theory to software and emphasizing the use of matrix functions in the methods.
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2. Matrix Equations and Model Reduction
Model reduction is a ubiquitous tool in analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the order of a given model. Often these methods have been derived in parallel in different disciplines with particular applications in mind. In this course, we will derive some of the most prominent methods used for linear systems: modal truncation based on eigenvalue algorithms, interpolatory methods which construct an approximate model by rational interpolation of the system's transfer function, and balanced truncation - a method based on a best approximation of a certain energy transfer operator related to the system. We will also compare the properties of these approaches and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these new techniques, the range of applicability of some of the methods has considerably widened. Particular emphasis will be given to the numerical solution of matrix equations which is the main computational bottleneck in methods based on balanced truncation. We will also present some ideas in the direction towards nonlinear model reduction at the end of the course. Numerical experiments to be performed in the exercise session will show the efficiency of several approaches when applied to real-world examples from several disciplines.
3. High Performance Linear Solvers and Eigenvalue Computations
Efficient algorithms for matrix functions and equations depend critically on the solutions of underlying linear systems and eigenvalue problems. This course aims at bringing to the students the state-of-the-art high performance methods for sparse linear systems and eigenvalue computations. Many practical aspects will be addressed to deliver high speed and robustness to the users of today's sophisticated high performance computers.