Steffen Seeger

Random walks on finitely ramified Sierpinski carpets

A new algorithm is presented that allows an efficient computer simulation of random walks on finitely ramified Sierpinski carpets. Instead of using a bitmap of the nth iteration of the carpet to determine allowed neighbor sites, neighbourhood relations are stored in small lookup tables and a hierarchical coordinate notation is used to give the random walker position. The resulting algorithm has low memory requirements, shows no surface effects even for extremely long walks and is well suited for modern computer architectures.

AN EFFICIENT IMPLEMENTATION OF THE EXACT ENUMERATION METHOD FOR RANDOM WALKS ON SIERPINSKI CARPETS

In the following, we present a highly efficient algorithm to iterate the master equation for random walks on effectively infinite Sierpinski carpets, i.e. without surface effects. The resulting probability distribution can, for instance, be used to get an estimate for the random walk dimension, which is determined by the scaling exponent of the mean square displacement versus time. The advantage of this algorithm is a dynamic data structure for storing the fractal. It covers only a little bit more than the points of the fractal with positive probability and is enlarged when needed. Thus the size of the considered part of the Sierpinski carpet need not be fixed at the beginning of the algorithm. It is restricted only by the amount of available computer RAM. Furthermore, all the information which is needed in every step to update the probability distribution is stored in tables. The lookup of this information is much faster compared to a repeated calculation. Hence, every time step is speeded up and the total computation time for a given number of time steps is decreased.

Modeling anomalous superdiffusion

Continuous models for anomalous diffusion have previously been tested in the subdiffusive case by making comparisons to diffusion on a Sierpinski gasket. This paper extends this discussion to the superdiffusive case by comparing performance to diffusion on a tree model. Although there is reasonable agreement within limited regimes for all four models, one model, due to Compte and Jou, stands out as being consistently sound over all regimes studied.

Random walks on random Koch curves

Diffusion processes in porous materials are often modeled as random walks on fractals. In order to capture the randomness of the materials random fractals are employed, which no longer show the deterministic self-similarity of regular fractals. Finding a continuum differential equation describing the diffusion on such fractals has been a long-standing goal, and we address the question of whether the concepts developed for regular fractals are still applicable. We use the random Koch curve as a convenient example as it provides certain technical advantages by its separation of time and space features. While some of the concepts developed for regular fractals can be used unaltered, others have to be modified. Based on the concept of fibers, we introduce ensemble-averaged density functions which produce a differentiable estimate of probability explicitly and compare it to random walk data.

Task pool teams implementation of the master equation approach for random sierpinski carpets

We consider the use of task pool teams in implementation of the master equation on random Sierpinski carpets. Though the basic idea of dynamic storage of the probability density reported earlier applies straightforward to random carpets, the randomized construction breaks up most of the simplifications possible for regular carpets. In addition, parallel implementations show highly irregular communication patterns. We compare four implementations on three different Beowulf-Cluster architectures, mainly differing in throughput and latency of their interconnection networks. It appears that task pool teams provide a powerful programming paradigm for handling the irregular communication patterns that arise in our application and show a promising approach to efficiently handle the problems that appear with such randomized structures. This will allow for highly improved modelling of anomalous diffusion in porous media, taking the random structure of real materials into account.

Beschleunigung physikalischer Simulationen durch Grafikprozessoren (Accelerating Physical Simulations Using Graphics Processing Units).

Abstract Graphics processors are used in many fields of applications that require high computational power. Especially in scientific computing, the programming of graphics processing units is an active field of research. Because of their hardware characteristics, graphics processors are well-suited for regular parallelism, however the implementation of irregular problems requires more advanced strategies. In this article, the hardware architecture of graphics processors and different frameworks for graphics processor programming, such as CAL, Brook+, CUDA and OpenCL with their specific properties, are presented. Additionally, an overview of different physical applications that have been implemented successfully on graphics processors is given. The parallel implementation of a specific irregular physical application on graphics processors is presented in more detail. This application simulates anomalous diffusion in porous media using random walk on Random Sierpinski Carpets. Zusammenfassung Grafikprozessoren werden in vielen Anwendungsbereichen, in denen es auf hohe Rechenleistung ankommt, genutzt. Auch im Wissenschaftlichen Rechnen sind parallele Implementierungen auf Grafikprozessoren Gegenstand aktueller Forschung. Obwohl Grafikprozessoren besonders für reguläre Parallelität geeignet sind, werden zunehmend auch irreguläre Anwendungsprobleme betrachtet. Dieser Artikel stellt die Eigenschaften von Grafikprozessoren und verschiedene Frameworks zur Grafikprozessorprogrammierung wie CAL, Brook+, CUDA und OpenCL mit ihren spezifischen Eigenheiten vor. Weiter gibt der Artikel einen Überblick über verschiedene physikalische Anwendungen, die bereits erfolgreich auf Grafikprozessoren implementiert wurden. Die parallele Implementierung einer speziellen irregulären physikalischen Anwendung auf Grafikprozessoren wird detaillierter vorgestellt. Diese simuliert anomale Diffusion in porösen Materialien durch Random Walk auf zufälligen Sierpinski-Teppichen.

Diffusion on Fractals — Efficient algorithms to compute the random walk dimension

Abstract Self-similar fractals are used as a simple model for porous media in order to describe diffusive processes. The diffusion or Brownian motion of particles on a fractal is approximated by random walks on pre-fractals. Since there are a lot of holes in the fractal, where a random walker is not allowed to move in, the mean square displacement scales with time t asymptotically as t2/dw, where the random walk dimension dw is usually greater than 2. This dimension is an important quantity to characterize diffusion properties. In this paper three efficient methods to calculate the random walk dimension of finitely ramified Sierpinski carpets are presented: First a simulation of random walks on pre-carpets, where an efficient storing scheme decreases the needed amount of memory and speeds up the computation. Secondly we iterate the master equation describing the time evolution of the probability distribution. Thirdly a resistance scaling algorithm is presented which yields a resistance scaling exponent. This exponent is related to the random walk dimension via the Einstein relation, using analogies between random walks on graphs and resistor networks.

Simulating anomalous diffusion on graphics processing units

The computational power of modern graphics processing units (GPUs) has become an interesting alternative in high performance computing. The specialized hardware of GPUs delivers a high degree of parallelism and performance. Various applications in scientific computing have been implemented such that computationally intensive parts are executed on GPUs. In this article, we present a GPU implementation of an application for the simulation of diffusion processes using random fractal structures. It is shown how the irregular computational structure that is inherent to the application can be implemented efficiently in the regular computing environment of a GPU. Performance results are shown to demonstrate the benefits of the chosen implementation approaches.

On symbolic derivation of the cumulant equations

Abstract We discuss the application of Mathematica for automated, symbolic calculation of the cumulant equations of arbitrary order. Like moment equations, these partial differential equations—describing fluid motion on a mesoscopic scale—may be considered an approximation to the Boltzmann equation, a highly nonlinear integro-differential equation that describes the motion of gases at a microscopic scale. Though the cumulant method provides a simple and compact presentation of the theory, actual calculation of very high order equations turns out to be a challenging task.