AIME@CZ - Czech workshop on applied mathematics in engineering
15-16 November 2011
Charles Square Campus of the
Czech Technical University in Prague, Czech Republic
Scope
The workshop, organized by Didier Henrion
and Zdenek Hurak
of the Department of Control Engineering
of the Faculty of
Electrical Engineering of the Czech Technical University
in Prague, aims at reporting recent achievements
in applied mathematics in engineering on the Czech scene.
Date and venue
The workshop takes place on Tuesday 15 and Wednesday 16 November 2011
in room K14, ground floor of building E of the Charles Square Campus
of the Czech Technical University in Prague, Czech Republic. To reach
building E, refer to these instructions.
Tentative schedule
Without Czech accents, sorry..
See below for titles and abstracts.
Tuesday 15 November 2011
8:30-9:30 - Celikovsky
10:00-11:00 - Hamhalter
11:30-12:30 - Stebel
14:00-15:00 - Kruzik
15:30-16:30 - Nemcova
Wednesday 16 November 2011
8:30-9:30 - Pajdla/Kukelova
10:00-11:00 - Ratschan
11:30-12:30 - Sebek/Hurak
14:00-15:00 - Sir
15:30-16:30 - Hron
Titles and abstracts
Sergej Celikovsky
(Institute of Information Theory and Automation,
Academy of Sciences of the Czech Republic, Prague)
Singular Perturbation Based Solution to Optimal Microalgal Growth
Problem and its Infinite Time Horizon Analysis
The problem of the optimal microalgal growth of the so-called
photosynthetic factory (PSF) is considered here. The objective is
to maximize the photosynthetic production rate (the specific
growth rate of microalgae) by manipulating the irradiance. Using
the singular perturbation based reduction, an analytical solution
of such an optimal control problem is obtained and its infinite
horizon analysis shows that the optimal solution on large time
intervals tends to the optimal steady state of PSF.
This is a mathematical
confirmation of the hypothesis often mentioned in biotechnological
literature. Mentioned analytical solution is based on Pontryagin maximum principle, which will be also shortly
repeated and discussed.
Jan Hamhalter
(Department of Mathematics, Faculty of Electrical Engineering,
Czech Technical University in Prague)
C*-algebras and noncommutative mathematics
Recent development in mathematics shows a clear trend to built new mathematical theories based on operator
algebras. This process, sometimes called quantisation of mathematics, consists in replacing a commutative
structure of functions by a noncommutative structure of matrices or operators acting on a Hilbert space. In
this way "virtual spaces" are arising as noncommutative counterpart of classical mathematical objects, such
as topological spaces, probability spaces, etc. The aim of this lecture is to provide a nontechnical
intuitive introduction to this area. We state recent results of the author on the abelian subalgebras. We
discuss relevance of operator algebras to quantum theory.
Jan Stebel
(Institute of Mathematics, Academy of Sciences of
the Czech Republic, Prague)
Shape optimization for generalized
Navier-Stokes equations
I will present a mathematical model for the problem of optimal shape
design of a dividing manifold called the header whose purpose is to
distribute a mixture of water and wood fibres in the paper making
process. The aim is to find a shape which a priori ensures the given
velocity profile on the outlet part. The mathematical formulation
leads to an optimal control problem in which the control variable is
the shape of the domain representing the header, the state problem is
represented by a generalized stationary Navier-Stokes system with
nontrivial mixed boundary conditions. We show how the existence of
solutions both to the generalized Navier-Stokes system and to the
shape optimization problem is proved, describe a convergent
approximation scheme, its implementation to a computer code and show
results of example computations. The talk is based on a joint work
with M. Bulicek, J. Haslinger and J. Malek.
Martin Kruzik
(Institute of Information Theory and Automation,
Academy of Sciences of the Czech Republic, Prague)
An efficient approach to the numerical solution of nonconvex variational problems
We propose a new approach to the numerical treatment of nonconvex static/evolutionary problems in continuum mechanics of
solids. The main idea is to replace the original microscopic energy density by its polyconvexification. For this problem,
first-order optimality conditions are derived and used in finding a discrete solution. The effectiveness of the method is
illustrated with some numerical experiments. This is a joint work with Soeren Bartels (Bonn).
Jana Nemcova
(Department of Mathematics, Institute of Chemical
Technology, Prague)
Realization theory - from linear to Nash systems
The notion of realization was introduced by R.E. Kalman to deal with
relations between linear dynamical systems and impulse response
functions. It refers to an "internal" state-space representation of an
observed input-output relation. In years this concept has developed
further to relate input-output maps and various types of systems.
In this talk I focus on the class of analytic semi-algebraic systems,
so-called Nash systems, which arise in systems biology as models of
metabolic and gene-regulatory networks. For this class of systems I
formulate the realization problem and provide a partial solution to it.
Possible contributions of the presented results to system identification
and model reduction of Nash systems will be discussed.
Tomas Pajdla
and Zuzana Kukelova
(Center for Machine Perception, Faculty of Electrical Engineering,
Czech Technical University in Prague)
Solving Minimal Problems
in Computer Vision
Many problems in computer vision can be formulated using systems of algebraic
equations. Often, these systems are not trivial and therefore special
algorithms have to be designed to achieve numerical robustness and
computational efficiency when solving them. In this talk we will briefly
discuss two methods for creating such efficient solvers for computer vision
problems. One is based on Groebner basis methods for solving systems of
polynomial equations and one on polynomial eigenvalue problems and resultants.
We will also introduce the automatic generator of Groebner basis solvers which
could be used even by non-experts to solve problems resulting in systems of
polynomial equations. Finally we will show several new solutions to absolute
and relative pose problems which we have created using the two presented
methods.
Stefan Ratschan
(Institute of Computer Science, Academy of Sciences
of the Czech Republic, Prague)
Computational Methods for Global Optimization and Constraint Solving
- Finding a solution of a system of equations vs.
finding all solutions or proving that none exists.
- Finding a local optimizer of an elementary function vs.
finding a global optimizer.
- Given a parametric ordinary differential equation,
finding a solution that fulfills certain additional conditions vs.
finding all such solutions or proving that none exists.
- Given a parametric ordinary differential equation,
finding a locally optimal solution vs.
finding a globally optimal solution.
Most of classical computational mathematics follows along the lines of
the respective first parts of this dichotomy. In the talk, we will
discuss current research on the second parts, its motivation from
industrial applications, its available computational techniques, and its
underlying theory.
Michael Sebek and
Zdenek Hurak
(Department of Control Engineering, Faculty of Electrical Engineering,
Czech Technical University in Prague)
Distributed control
of platoons of vehicles using n-D systems theory
Practically motivated
by critical transportation problems, this research aims
to answer some basic questions related to distributed control of very long
platoons (or strings) of vehicles. For instance, is it possible to design and
implement distributed controllers onboard each vehicle that only measure the
distance to the vehicle ahead, and yet attenuate distrurbances acting locally
on some vehicles in the platoon ? The problem will be formulated using the
compact formalism of joint Laplace and z-transforms, which give rise to
description of such dynamical systems using two-variable transfer functions.
Alternatively, this can be viewed as a partial differential/difference equation.
Stability of long platoons is studied using concepts from n-D systems theory.
It appears, however, that the core problem is far from contrained to vehicular
systems; for example, some analogy with modeling and controling flow can be
found. Finally, a few laboratory experiments based on racing slotcars and Lego
Mindstorms NXT set will be reported. More info on the research, including
papers and videos can be found here.
Zbynek Sir
(Faculty of Mathematics and Physics, Charles University in Prague)
Rational Representations in Geometric Modeling
Geometric modeling and all geometrical applications are based
on piecewise polynomial and rational representations, e.g. line or
circular splines, Bezier curves and surfaces, NURBs curves and surfaces
etc. But many natural geometrical operations such as offsetting,
sweeping, convolution do not preserve the rationality of the input data.
We will present several approaches solving this problem. In particular
we will discuss Pythagorean Hodograph curves and dually represented
curves and surfaces (as envelopes of families of hyperplanes or
hyperspheres).
Jaroslav Hron
(Mathematical Institute, Charles University in Prague)
Monolithic solver for fluid-structure interaction problems
A monolithic approach for solving the fluid-structure interaction problem with motivation by biological flows will be presented. It is based on the ALE formulation of the balance equations for the fluid and solid in the time dependent domain.
The discretization is done by the finite element method. Our treatment of the problem as a one system suggests to use the same finite elements on both, the solid part and the fluid region. The discretized system of non-linear algebraic equations is solved using approximate Newton method with line-search strategy as the basic iteration combined with geometric multigrid as linear solver.
Last updated on 28 November 2011.