ProgramLectures 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.
|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.
|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
|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
|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