Workshop on Scientific Computing

June 15-18, 2012
Department of Mathematics, FNSPE CTU in Prague
Děčín, Czech Republic

Employment, Production and Consumption Model: Patterns of Phase Transitions

Hynek Lavička


Abstract:We have simulated the model of Employment, Production and Consumption (EPC) using Monte Carlo. The EPC model is an agent-based model that mimics very basic rules of industrial economy. From perspective of physics, the nature of the interactions in the EPC model represents multi-agent interactions where the relations among agents follow the key laws for circulation of capital and money. Monte Carlo simulations of the stochastic model reveal phase transition in the model economy, i.e., the structure exhibits strong non-linear behavior as a response to the change of the exogenous parameters.

Optimal MPI parallel computing on systems with NUMA architecture

Róbert Špir

SvF STU, Bratislava

Abstract:All current multiprocessor servers based on x86 architecture are NUMA (non-uniform memory access) systems. This memory design has many benefits compared to older, single common bus and better scales with higher number of processors. But to fully utilize the computational power of these systems, one must be familiar with architecture of these servers and exploit optimal memory locality usage manually, programmatically using special functions. We tested this approach on multiple real-life parallel programs in areas of image processing and geodetic boundary value problems using finite volume methods and boundary element methods and we achieved notable computational speedup in all cases.

Finite volume method approach for solving oblique derivative boundary-value problem

Marek Macák, Karol Mikula

SvF STU, Bratislava

Abstract:Talk deals with the fi rst approach for solving non-simpli ed oblique derivative geodetic boundary value problems (GBVP). From the mathematical point of view, GBVP is formulated in the form of the Laplace partial di erential equation for the unknown potential in the external domain. Various boundary conditions (BC) defi ned on the Earth surface are considered, e.g. a Newton BC are prescribed, if the so-called gravity anomalies are used, or a Neumann-type BC are prescribed, if the so-called gravity disturbances are used. Old approach considers that the normal derivative of the unknown potential field is given on the Earth surface. We presented new approaches for solving geodetic boundary value problems involving the boundary conditions of the unknown potential fi eld with prescribed derivative in an oblique direction. We tested this approach numerically and we showed its second order accuracy in several experiments.

Numerical Solution for Surface Diffusion

Michal Beneš


Abstract:The article studies the motion of graphs by the surface Laplacian of the mean curvature by means of the numerical algorithm based on the method of lines with the finite differences in space. The semi-discrete scheme is analysed from the viewpoint of several integral properties and is then used for the computations. We also present the numerical convergence results and investigate the nonlinear dynamics of the problem.

Incompressible flow simulation based on the vorticity-stream function formulation in complex geometries.

Sebastian Reuther

TU Dresden

Abstract:For the simulation of viscous flows the vorticity-stream function formulation is often used. This formulation is much easier to handle than the full Navier-Stokes-System. Otherwise there exists some issues, which come up with the numerical treatment of this formulation. One of these is the exact definition of the boundary conditions for both variables – the vorticity and the stream function. We present a new method that approximates the boundary conditions for the two variables using a modified phase-field method. In addition we present a validation of our model with a popular benchmark problem and an application on the two-dimensional sphere.

Radiative association of molecular ions of astrochemical interest

Lucie Augustovičová

MFF UK, Praha

Abstract:Spontaneous radiative association of the molecular ion LiHe+ is investigated including three electronic states X1+, A1+, and a3+. Cross sections for four processes of radiative association X → X, A → A, A → X, and a → a are calculated as functions of collision energy. The corresponding rates of formation are derived as functions of temperature. The A → X radiative association process exhibits the largest cross sections and rate coefficients because of the huge amount of the de-excitation energy from the A to X state.

Stability and Dynamics of Liquid Crystalline Phases in the Phase-Field-Crystal Model

Simon Praetorius

TU Dresden

Abstract:The Phase-Field-Crystal model for liquid crystals is solved numerically in two spatial dimensions, using a finite element discretization in space and a semi-implicit discretization in time. The model is formulated in two variables, the reduced translational density and the symmetric and traceless nematic order tensor, that encapsulates the nematic order parameter and the nematic director. This formulation has the advantage, to allow for minimization of the free energy without constraints. For different system parameters an initial configuration is relaxed up to equilibrium and compared qualitatively with the equilibrium solutions of [1]. All phases could be reproduced with our dynamic model. Also some dynamic evolutions of the order parameter fields are considered, i.e. phase transitions between plastic triangular crystals with non-vanishing nematic order parameter and columnar/smectic A phase, as well as transition between plastic triangular crystals and square crystals. The work is done in cooperation with Hartmut Löwen [**], Axel Voigt [*] and Raphael Wittkowski [**]. [1] Cristian Vasile Achim, Raphael Wittkowski and Hartmut Löwen, Stability of liquid crystalline phases in the phase-field-crystal model, Phys. Rev. E 83, 061712, 2011 [*] Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, D-01062 Dresden, Germany [**] Insitut für Theoretische Physik II, Weiche Materie, Heinrich-Heine-Universität Düsseldorf, D-40225 Düsseldorf, Germany

Lattice Boltzmann Method for heat transfer and fluid flow

Robert Straka

AGH, Krakow

Abstract:Lattice Boltzmann Method (LBM) is recently gaining more and more popularity, mainly due to it's mesoscopic treatment of fluid and heat transfer, which allows for straighforward implementation of complex geometry of the system solved. Another excellent property is it's locallity, thus easy transformation from sequential to parallel codes. Fundamentals of the LBM and several cases of fluid flow and heat transfer problems will be presented.

Tracking of cells in drosophila morphogenesis

Michal Smíšek

SvF, STU Bratislava

Abstract:We present a novel algorithm for tracking cells in time lapse confocal microscopy movie of a Drosophila epithelial tissue during pupal morphogenesis.We consider a 2D + time video as a 3D static image, where frames are stacked atop each other, and using a spatio-temporal segmentation algorithm we obtain information about spatio-temporal 3D tubes representing evolutions of cells. The main idea for tracking is the usage of two distance functions - fi rst one from the root cells and second one from segmented boundaries. We track the cells backwards in time. The fi rst distance function attracts the subsequently constructed cell tra- jectories to the root cells and the second one forces them to be close to centerlines of the segmented tubular structures. This makes our tracking algorithm robust against noise and missing spatio-temporal boundaries. This approach can be generalized to a 3D + time video analysis, where spatio-temporal tubes are 4D objects.

Signaling networks and cell motility - a computational approach using a phase fi eld description

Wieland Marth, Axel Voigt

TU Dresden

Abstract:The movement of cells is characterised by two processes, fi rst protruding a cell front and subsequently retracting the cell rear. The processes of protrusion and retraction are both driven by the turnover and reorganisation of the actin cytoskeleton. Within a reaction-diff usion model which combines processes along the cell membrane with processes within the cytoplasm a Turing type instability is used to form the necessary polarity to distinguish between cell front and rear and to form diff erent organisational arrays within the cytoplasm. We combine this model with a classical Helfrich type model to simulate the dynamics of cells. The coupled model is formulated within a phase field approach and solved using adaptive finite elements.

Computing optimal designs of experiments for linear parameter subsystems

Mária Trnovská

Comenius University, Bratislava

Abstract:In the presentation we introduce a relatively new method for computing approximate optimal designs of experiments w.r.t. the criteria of partial D, A and E optimality measuring the quality of designs for estimating linear parameter subsystems. The novelty is mainly in transformation into a different problem for which efficient algorithms are known. The efficiency of these algorithms is essential for branch and bound method we use for constructing the exact optimal designs.

Numerical scheme for solving Hamilton-Jacobi-Bellman equation arising from a constrained optimal allocation problem

Soňa Kiliánová

Comenius University, Bratislava

Abstract:We propose a numerical scheme for solving Hamilton-Jacobi-Bellman (HJB) equations arising in optimal investment problems. We use a Ricatti-type of transformation to pass from the HJB equation to a quasilinear PDE for the optimal response function. In the multiasset setting, the coefficients of the governing PDE are computed by means of a solution to a constrained quadratic program. Accuracy of the numerical scheme is tested for the semiexplicit travelling wave solution. We present results of numerical computations adopted for the case of Slovak pension fund system and optimal allocation problem for the German DAX stock index.

A mixed anisoperimetric inequality and the flow of planar curves minimizing anisoperimetric ratio

Daniel Ševčovič

Comenius University, Bratislava

Abstract:We investigate a gradient flow of closed planar curves minimizing the anisoperimetric ratio. It turns out that such a flow has the normal velocity locally depending on the anisotropic curvature and nonlocally depending on the total interfacial energy and the enclosed area of the evolved curve. In contrast to the gradient flow for the isoperimetric ratio, we show there exist initial curves for which the enclosed area is decreasing with respect to time. To this end, we make use of a mixed anisoperimetric inequality for the product of total interfacial energies corresponding to different anisotropy functions. The proof of the inequality is based on Lagrange multiplier method and interesting duality identity between total interfacial energies corresponding to different anisotropies. We also present a numerical scheme for solving curvature driven flow with nonlocally dependent normal velocities. The scheme is based on a flowing finite volume method combined with a precise scheme for approximation of non-local terms. References: [1] D. Sevcovic and S.Yazaki: On a gradient flow of plane curves minimizing the anisoperimetric ratio, submitted, arXiv:1203.2238, 2012 [2] D. Sevcovic and S.Yazaki: Computational and qualitative aspects of motion of plane curves with a curvature adjusted tangential velocity, Mathematical Methods in the Applied Sciences 2012, arXiv:0711.2568v2, DOI:10.1002/mma.2554

Mathematical modelling of solidification of a binary alloy

Juraj Kyselica

Comenius University, Bratislava

Abstract:We study the two-dimensional solidification of a binary alloy by numerical means. We employ a front-fixing transformation and develop a finite-difference scheme to simulate the evolution of an initially-planar solidification front. The self-similar solution is taken as a seed field in cases when the melt is constitutionally supercooled. In the regime considered, we find that the solid--liquid interface takes the form of a travelling wave. The parametric dependence of the numerical solution is also investigated.