Martin Balažovjech, Peter Frolkovič and Karol Mikula

Abstract: We present a second order accurate finite volume scheme for level set equation. The key idea is combination of a Crank-Nicolson type of the time discretization for curvature term and an Inflow Implicit and Outflow Explicit scheme for advection term. Moreover we present derivation of new exact solution, which is used to verify the accuracy of our scheme.

Michal Beneš

Abstract: The contribution discusses several details of ice surface melting under contact with edges of skates or skis. The phenomena given by phase diagram of ice, dynamic heating and liquid thin layer are presented and compared. The understanding of subtleties of this problem helps in development of mathematical models of phase and structural transitions in porous media.

Radek Fučík, Jakub Klinkovský

Abstract:

We deal with mathematical modelling of two-phase compositional flow in porous media and an efficient solution of the resulting model using advanced numerical methods for modern parallel computational architectures. Firstly, we describe the physical context and the mathematical model of two-phase flow in porous media. Then, we present a multidimensional numerical scheme for a general system of n partial differential equations in a general coefficient form, which can be used to describe many specific problems depending on the choice of the coefficients. The scheme is based on the combination of the mixed-hybrid finite element and discontinuous Galerkin methods for the spatial discretization, the temporal discretization is performed using the Euler method and the semi-implicit approach of the frozen coefficients method is used to linearize the scheme in time. The scheme can work with both structured and unstructured meshes. In this work, we use structured meshes consisting of rectangles in 2D and cuboids in 3D and unstructured meshes consisting of triangles in 2D and tetrahedrons in 3D. The scheme is stabilized by the upwind technique and in the case of rectangular and cuboidal elements, we also use the mass- lumping technique. The next part of the work deals with the implementation of the numerical scheme for modern parallel architectures of GPUs and multicore CPUs. The solver is implemented in the C++ language with the help of the TNL library, the CUDA framework for parallelization on GPUs and OpenMP for parallelization on CPUs. A detailed description of notable steps of the computational algorithm, such as the assembly of the sparse matrix and the solution of the linear system, is presented. We focus on the key aspects of the efficient implementation for GPU, notably the importance of proper data layout in memory and the influence of unstructured mesh ordering on the computational efficiency. To improve the efficiency of the linear system solver on GPU, we present a modification of the GMRES method using the orthogonalization by Householder transformations and the compact WY representation. In the last part of the work, we describe benchmark problems for two-phase and two-phase compositional flow in porous media and use their known exact solutions to verify the convergence of the numerical scheme by means of the analysis of the experimental order of convergence. Both benchmark problems are computed on structured and unstructured meshes in 2D and 3D and an efficiency analysis of the parallel computation on GPU and multicore CPU for all mesh types is performed.

Viera Kleinová, Peter Frolkovič

Abstract: The optical flow methods are based on searching a deformation of one image toward the second one. The most popular ones are so-called differential methods. We are interested in two approaches. The first approach is based on mathematical model created by Lucas and Kanade. This method assumes that the optical flow is constant within some neighborhood of pixels. The second one is new method based on level-set motion that is motivated by Vemuri et. al.. The main goal is to show difference between these approaches and characterize for which type of problems which method is preferable. Some representative results of the methods on 2D synthetic and real data will be presented.

Jaromír Kukal

Abstract: There are many various definitions of fractional derivatives which produce various solutions of mass balance equations. But only Riesz and Feller derivatives are suitable for the realization of fractional Laplacian and fractional gradient. Begining with Riesz potential we can easily formulate Fick's law for anomalous diffusion flow and therefore continue in mass ballance formulation. This will be demonstrated on unstady closed onedimensional systems including novel analytical solutions which could help in both Monte Carlo and FVM simulations of more complex systems.

Michal Sněhota

Abstract: TBA

Pavel Strachota, Marek Pavlíček, Světlana Smrčková, Dávid Bortňák, Zuzana Ruttkayová

Abstract:

Let's make it less math and more fun this time. Apart from mathematical modeling, I'm working with my students on topics from computer graphics. We focus on algorithms for procedural generation of models ranging from individual objects to complete virtual environments. In this contribution, I would like to present the summary of the achieved results together with a very brief insight into the used methods. With the objective of creating a whole virtual city landscape, it is necessary to deal with the modeling of terrain, road network, lands, buildings and vegetation, to name the most important parts. I will also comment on some advanced ideas that may introduce more realism into the resulting models.

Robert Straka, Keerti V. Sharma

Abstract: In this talk, the cascaded version of a collision operator for D$_2$Q$_5$ lattice and one conservation law will be derived and applied to thermal flows. Comparisons of our and benchmark solutions for thermal problems will be presented together with other results of forced and natural convection in complex arrays of cylindrical obstacles. In the end, the stability of CLBM and other collision operators (SRT-BGKW, MRT) will be presented for the case of a double shear flow and the Rayleigh-Taylor instability.

Quang Van Tran, Jaromír Kukal

Abstract: Today searching algorithms are indispensable tools used to find the global solution to an optimization problem. For these algorithms, their searching effectiveness heavily depends on the quality of randomization of new mutations in the search domain. So far, this objective is realized through the called Levy flights which are random non-Gaussian step sizes following the alpha stable distribution. However, as this random variable is difficult to be generated in the multivariate case, the step size generation procedure has never been investigated accordingly. To examine the impact of the general usage of this possibility, this paper is to introduce the multivariate alpha stable random variable step size generation technique into several novel searching algorithms as Random Descent with Levy flights (RDLF), Cuckoo Search (CS) and Modified Cuckoo Search algorithms (MCS). These algorithms are then used to find the global optimum for several well- known benchmark functions. The results we obtained show that the inclusion of alpha stably distributed mutations has substantially improved the performance of these algorithms.

Róbert Špir, Karol Mikula

Abstract: We present numerical algorithm and postprocessing for an automated cell tracking and cell lineage tree reconstruction from large-scale 3D+time two-photon laser scanning microscopy images of early stages of zebrafish (Danio rerio) embryo development. The cell trajectories are extracted as centered paths inside segmented spatio-temporal tree structures representing cell movements and divisions. Such paths are found by using a suitably designed and computed constrained distance functions and by a backtracking in steepest descent direction of a potential field based on these distance functions combination. Then we can compare the results with ground truth tracking obtained by manual checking of cell links by biologists and measure the accuracy of our algorithm. Using visualization tool displaying our results, ground truth and original 3D images simultaneously we can easily verify the correctness of the tracking.