Lectures will run in the mornings and will be accompanied by hands-on training sessions in the afternoons. (If possible, bring your own laptop with MATLAB® or Octave!) An excursion to Split will be organized on Wednesday afternoon.


Zlatko Drmač
Numerical Algorithms in Control
Control theory provides interesting and challenging problems to numerical linear algebra. Modern theoretical developments and exciting engineering applications demand efficient and numerically sound algorithms implemented as robust and accurate numerical software. The concepts, problem formulations and solutions are often stated in terms of matrix decompositions (Hessenberg and Schur forms with generalizations, singular and eigenvalue decompositions with generalizations, staircase forms, contragredient diagonalization of grammians, rank revealing QR factorization, etc.). We will study how some recent developments in accurate linear algebra (accurate algorithms for eigenvalues and singular values of matrices, matrix products and quotients, and corresponding theory) improve numerical computations of control--theoretic decompositions, and contribute to control software improvements. It will be shown that in some cases even the tiniest quantities can be computed with low relative error. Few separate topics will be used as case studies. Starting with system--theoretic notions, we will discuss corresponding matrix formulations, analyze numerical algorithms using state of the art perturbation theory, and finally discuss the fine details of software implementation (using Matlab, LAPACK, SLICOT). It will be shown e.g. how and why changing compiler options can completely change the output for inputs sufficiently close to singularity, and how to detect and resolve the problem using numerical analysis.

Mark Embree
Pseudospectra and the Behavior of Dynamical Systems
The analysis of dynamical systems has long revolved around eigenvalues: from stability assessment to modal truncation for reducing dimension, we look to eigenvalues to aid understanding and computation. However, for many problems analysis based on eigenvalues alone can be misleading. In these lectures, we shall point out where such situations arise, and describe one helpful alternative to eigenvalues: pseudospectra. We will introduce the fundamental properties of pseudospectra, then show how these sets relate to the transient behavior of dynamical systems and affect performance of algorithms for computing the matrix exponential and reduced order models. Theory will be illustrated with examples from mechanics and fluids dynamics. Our focus will be on linear systems, with some emphasis on generalized and quadratic eigenvalue problems. Algorithms for the computation of pseudospectra will be discussed as time permits.

Matlab files of exercises
Peter Benner
Model Reduction for Linear Dynamical Systems
Model reduction is an 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: 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. 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.

Matlab solutions of exercises
Daniel Kressner
Low-Rank Tensor Techniques for High-Dimensional Problems
This lecture will give an introduction to low-rank tensor techniques for coping with high-dimensional problems on a linear algebra level. In particular, we will focus on the hierarchical Tucker format, a storage-efficient scheme to approximate and represent tensors of possibly high order. Illustrative examples and hands-on experience will be provided with the recently released Matlab toolbox htucker (joint work with Christine Tobler, ETH Zurich). We will discuss the methodology and algorithms behind htucker, which not only allows for the efficient storage and manipulation of tensors but also offers a set of tools for the development of higher-level algorithms. Several examples for the use of low-rank tensors and the toolbox are given. This includes simple algorithms, such as an iterative method for orthogonalizing tensors, as well as more complex applications, such as the solution of parameter-dependent and stochastic elliptic PDEs.

Exercises and Solutions