Charles Square Campus of the Czech Technical University in Prague, Czech Republic

Tuesday 15 November 2011

Wednesday 16 November 2011

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.

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.

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.

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).

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.

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.

- 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.

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.

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).

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.