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## Über dieses Buch

This book constitutes the refereed post-proceedings of the International Conference on Mathematical Modeling and Computational Physics, MMCP 2011, held in Stará Lesná, Slovakia, in July 2011. The 41 revised papers presented were carefully reviewed and selected from numerous submissions. They are organized in topical sections on mathematical modeling and methods, numerical modeling and methods, computational support of the experiments, computing tools, and optimization and simulation.

## Inhaltsverzeichnis

### Bayesian Automatic Adaptive Quadrature: An Overview

The progress obtained within the Bayesian approach to the automatic adaptive quadrature is reviewed. It is shown that the derivation of reliable Bayesian inferences, both as it concerns the construction of the subrange binary tree with its associated priority queue and the

a priori

validation of the input to the local quadrature rules, can be done provided the well-conditioning criteria for the integrand profile check are implemented taking into account the hardware and software environments at hand.

Gheorghe Adam, Sanda Adam

### Computational Challenges for the CBM Experiment

CBM (“Compressed Baryonic Matter”) is an experiment being prepared to operate at the future Facility for Anti-Proton and Ion Research (FAIR) in Darmstadt, Germany, from 2018 on. CBM will explore the high-density region of the QCD phase diagram by investigating nuclear collisions from 2 to 45 GeV beam energy per nucleon. Its main focus is the measurement of very rare probes (e.g. charmed hadrons), which requires interaction rates of up to 10 MHz, unprecedented in heavy-ion experiments so far. Together with the high multiplicity of charged tracks produced in heavy-ion collisions, this leads to huge data rates (up to 1 TB/s), which must be reduced on-line to a recordable rate of about 1 GB/s.

Moreover, most trigger signatures are complex (e.g. displaced vertices of open charm decays) and require information from several detector sub-systems. The data acquisition is thus being designed in a free-running fashion, without a hardware trigger. Event reconstruction and selection will be performed on-line in a dedicated processor farm. This necessitates the development of fast and precise reconstruction algorithms suitable for on-line data processing. In order to exploit the benefits of modern computer architectures (many-core CPU/GPU), such algorithms have to be intrinsically local and parallel and thus require a fundamental redesign of traditional approaches to event data processing. Massive hardware parallelisation has to be reflected in mathematical and computational optimisation of the algorithms. This is a challenge not only for CBM, but also for current and future experiments, in particular for heavy-ion eperiments like e.g. ALICE at the LHC.

For the development of the proper algorithms, a careful simulation of the input data is required. Such a simulation must reflect the free-running DAQ concept, where data are delivered asynchronously by the detector front-ends on activation, and no association to a physical interaction is given a priori by a hardware trigger. It hence goes beyond traditional event-based software frameworks. In this article, we present the challenges of and the current approaches to simulation, data processing and reconstruction in the CBM experiment.

Volker Friese

### Consistency Analysis of Finite Difference Approximations to PDE Systems

We consider finite difference approximations to systems of polynomially-nonlinear partial differential equations the coefficients of which are rational functions over rationals in the independent variables. The notion of strong consistency which we introduced earlier for linear systems is extended to nonlinear ones. For orthogonal and uniform grids we describe an algorithmic procedure for the verification of the strong consistency based on the computation of difference standard bases. The concepts and algorithmic methods of the present paper are illustrated by two finite difference approximations to the two-dimensional Navier-Stokes equations. One of these approximations is strongly consistent, while the other is not.

Vladimir P. Gerdt

### The Circuit Model of Quantum Computation and Its Simulation with Mathematica

We consider an application of the

Mathematica

package

QuantumCircuit

to simulation of quantum circuits implementing two of the best known quantum algorithms, namely, the Grover search algorithm and the Shor algorithm for order finding. The algorithms are discussed in detail and concrete examples of their application are demonstrated. The main features of the package

QuantumCircuit

which can be used for the simulation of an arbitrary quantum algorithm are briefly described.

Vladimir P. Gerdt, Alexander N. Prokopenya

### Proteins Studied by Computer Simulations

Computer simulations can complement experiments in molecular biology; and are sometimes the only instrument to probe fundamental processes in the cell. However, their use is hampered by poor convergence. I summarize a number of now widely utilized algorithms that help to alleviate these sampling difficulties, and review recent results that demonstrate the power of these techniques in protein simulations.

Ulrich H. E. Hansmann

### Functional Methods in Stochastic Systems

Field-theoretic construction of functional representations of solutions of stochastic differential equations and master equations is reviewed. A generic expression for the generating function of Green functions of stochastic systems is put forward. Relation of ambiguities in stochastic differential equations and in the functional representations is discussed. Ordinary differential equations for expectation values and correlation functions are inferred with the aid of a variational approach.

Juha Honkonen

### Mathematical Modeling of Finite Quantum Systems

We consider the problem of quantum behavior in the finite background. Introduction of continuum or other infinities into physics leads only to technical complications without any need for them in description of empirical observations. The finite approach makes the problem constructive and more tractable. We argue that quantum behavior is a natural consequence of the dynamical system symmetries. It is a result of fundamental impossibility to trace identity of indistinguishable objects in their evolution — only information about invariant combinations of such objects is available. We demonstrate that any quantum dynamics can be embedded into a simple permutation dynamics. Quantum phenomena, such as interferences, arise in invariant subspaces of permutation representations of the symmetry group of a system. Observable quantities can be expressed in terms of the permutation invariants.

Vladimir V. Kornyak

### Multi-channel Computations in Low-Dimensional Few-Body Physics

In this lecture I give a brief review of low-dimensional few-body problems recently encountered in attempting a quantitative description of ultracold atoms and molecules confined in 2D and 1D optical lattices. Multi-channel nature of these processes has required the development of special computational methods and algorithms which I discuss here as well as the most interesting results obtained with the offered computational technique and future perspectives.

Vladimir S. Melezhik

### Mathematical Models to Predict the Critical Conditions for Bacterial Self-healing of Concrete

Two mathematical models for bacterial self-healing of a crack are considered. The study is embedded within the framework of investigating the potential of bacteria to act as a catalyst of the self-healing process in concrete, that is the ability of concrete to repair occurring cracks autonomously. The first model concerns an analytic formalism to compute the probability that a crack hits an encapsulated particle. Hence, it predicts the probability that the self-healing process starts. The second model of the self-healing process is based on a moving boundary problem. A Galerkin finite-element method is used to solve the diffusion equations. The moving boundaries are tracked using a level set method.

Serguey V. Zemskov, Henk M. Jonkers, Fred J. Vermolen

### A New Discretization Scheme in Field Theory

We propose a new discretization scheme for field theory, in which the space time coordinates are assumed to be operators forming a noncommutative algebra. Working in a discrete representation of that algebra, one obtains naturally a discretization scheme. The original theory should be recovered for representations of large dimensionality. The procedure is illustrated with space-like coordinates that form a Heisenberg algebra. Advantages exist with respect to conventional lattice field theory: fermions can easily be put on a lattice and the continuum limit is recovered without the problems appearing in the conventional formalism; however other types of problems appear.

Ciprian Sorin Acatrinei

### Two-Loop Calculation of the Anomalous Exponents in the Kazantsev-Kraichnan Model of Magnetic Hydrodynamics

The problem of anomalous scaling in magnetohydrodynamics turbulence is considered within the framework of the kinematic approximation, in the presence of a large-scale background magnetic field. Field theoretic renormalization group methods are applied to the Kazantsev–Kraichnan model of a passive vector advected by the Gaussian velocity field with zero mean and correlation function ∝

δ

(

t

−

t

′)/

k

d

+

ε

. Inertial-range anomalous scaling for the tensor pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain “dangerous” composite operators, whose negative critical dimensions determine the anomalous exponents. The main technical result is the calculation of the anomalous exponents in the order

ε

2

of the

ε

expansion (two-loop approximation).

Nikolay V. Antonov, Nikolay M. Gulitskiy

### Mathematical Modeling of Irregular Integrated Optical Waveguides

The paper presents the formulation of the dispersion relations for regular planar waveguides and smoothly irregular integrated optical waveguides in forms of elementary continuous functions. The stable algorithm for computing the roots of the dispersion relations for regular and irregular waveguides, which is applicable in the case of real and complex coefficients of the phase retardation of waveguide modes, is developed. The theoretical and numerical method for studying the characteristics of inhomogeneous waveguide modes in the overcritical regimes is elaborated.

In contrast to the zero approximation of the adiabatic modes method, and moreover to the method of comparison waveguides, the fields of irregular waveguide modes of an integrated optical waveguide in the first approximation are described by a pair of equations of coupled oscillators allowing resonance solutions.

Edik A. Ayryan, Alexander A. Egorov, Leonid A. Sevastyanov, Konstantin P. Lovetskiy, Anton L. Sevastyanov

### Two Notes on Continuous-Time Neurodynamical Systems

The neurodynamical model of recurrent networks in this paper is approached from an engineering perspective, i.e., to make networks efficient in terms of topology and capture dynamics of time-varying systems. Neural dynamics in that case can be considered from two aspects, convergence of state variables (memory recall) and the number, position, local stability and domains of attraction of equilibrium states (memory capacity). The purpose of this work is to investigate some relationship between Lyapunov exponents and the recurrent neural network model described by the concrete system of delay-differential equations.

Ivan Daňo

### On the Mathematical Modelling of the Annihilation Process

Using field-theoretic approach the reaction process

A

+

A

→ ∅ is studied in the vicinity of space dimension

d

c

= 2 by means of the perturbative renormalization group. Dimensional regularization with the use of minimal subtraction scheme is applied and fixed points with corresponding regions of stability are calculated to the two-loop approximation in double (

ε

,Δ) expansion.

Michal Hnatič, Juha Honkonen, Tomáš Lučivjanský

### The Crossing Numbers of Join of Paths and Cycles with Two Graphs of Order Five

The crossing number cr(

G

) of a graph

G

is the minimal number of crossings over all drawings of

G

in the plane. Only few results concerning crossing numbers of graphs obtained as join product of two graphs are known. There are collected the exact values of crossing numbers for join of all graphs of at most four vertices with paths and cycles. In the paper, we extend these results. For two special graphs

G

on five vertices, we give the crossing numbers of the join products

G

+

D

n

,

G

+

P

n

, and

G

+

C

n

, where

D

n

consists on

n

isolated vertices,

P

n

and

C

n

are the path and cycle on

n

vertices, respectively.

Marián Klešč, Štefan Schrötter

### Exact Solution of a Moisture Drying System with Phase Transition

An exact solution of a linear system of moisture transfer with phase transition is proposed. The system consists of three equations. The first equation is a diffusion equation for liquid moisture concentration

w

l

, the second one is a diffusion equation for saturated vapor concentration

w

v

. Both equations are tied with the rate

I

of change of moisture concentration that arises in the pores due to the evaporation or condensation. The third equation is algebraic one and describes two complementary parts of the pores volume, the part, where the liquid moisture is present and the part, where saturated vapor is present.

The system is solved by means of the variables separation method.

Eva Litavcová, Miron Pavluš, Ján Seman, Ibrohim Sarhadov

### Pseudo-Differential Operators in an Operational Model of the Quantum Measurement of Observables

The measurement procedure transforms an isolated (closed) quantum system into an open one. The operators of observables of a rather simple explicit form are converted into pseudo-differential operators of more complicated form. A stable numerical method for studying the discrete spectra of the measured observables on the basis of the observed discrete spectra of the expected observables is developed. Thus, a method for establishing a correspondence between theoretical data in the conventional quantum mechanics of (isolated) quantum objects and the experimental data on their measured values for open quantum objects is proposed.

Leonid Sevastyanov, Alexander Zorin, Alexander Gorbachev

### Exactly Solvable Models for the Generalized Schrödinger Equation

The Darboux transformation operator technique is applied to the generalized Schrödinger equation. The procedure is used for constructing exactly solvable models. The influence of the distance between levels on the form of the potentials is investigated. In particular, symmetric and asymmetric double well and triple well potentials are generated.

Alina Suzko, Elena Velicheva

### Floating Point Degree of Precision in Numerical Quadrature

In the floating point computation of an integral by means of an interpolatory quadrature sum, the algebraic degree of precision

d

, of the quadrature sum is to be abandoned in the favour of its floating point degree of precision,

$d\sb{\mathrm{fp}}$

, the value of which significantly varies with the extent and localization of the integration domain over the real axis. The use of

$d\sb{\mathrm{fp}}$

instead of

d

drastically sharpens the admissible bounds of variation of the integrand in the Bayesian automatic adaptive quadrature.

Sanda Adam, Gheorghe Adam

### Numerical Simulations of Heat and Moisture Transfer Subject to the Phase Transition

A model for the description of the heat and moisture transfer in a porous material is proposed. The density of the saturated vapor and the transfer coefficients of the liquid and vapor moistures depend on the temperature. At the same time, the conductivity coefficient of the porous material depends on the moisture. On the basis of the proposed model, a numerical simulation of the heat and moisture transfer for a drying process has been performed.

Ilkizar V. Amirkhanov, Taisia P. Puzynina, Igor V. Puzynin, Ibrohim Sarhadov, Erika Pavlušová, Miron Pavluš

### Numerical Study of Fluxon Solutions of Sine-Gordon Equation under the Influence of the Boundary Conditions

The dependence on the boundary conditions of the fluxon solutions of a boundary problem for the sine-Gordon equation (SGE) is investigated numerically. Interconnection between fluxon and constant solutions is analyzed. Numerical results are discussed in the context of the long Josephson junction model.

Pavlina Khristova Atanasova, Elena Zemlyanaya, Yury Shukrinov

### Computer Modeling of the Immune System Reconstruction after Peripheral Blood Stem Cell Transplantation

A mathematical model for leukopoiesis has been proposed in literature, consisting of a system of ordinary differential equations with two delays. The aim of this study was to verify the assumption that the model is applicable also for T, B and NK cell recovery after stem cell transplantation and to tune the model parameters on the base of clinical data. The numerical tests illustrate the influence of the model parameters to the behavior of the monitored white blood cell populations. They also show that for certain parameter values the numerically obtained behaviour of B cells is in agreement with the clinical data.

Gergana Bencheva, Lidia Gartcheva, Antoaneta Michova, Margarita Guenova

### Comparison of Some Finite Difference Schemes for Boussinesq Paradigm Equation

The aim of the paper is to propose and study families of finite difference schemes for solving the Boussinesq Paradigm Equation. The nonlinear term of the equation is approximated in three different ways. We obtained a pair of implicit (with respect to the nonlinearity) families of schemes and an explicit one. All schemes have second rate of convergence in space and time. Numerical tests performed confirm our theoretical results regarding accuracy and convergence of all three schemes.

Milena Dimova, Natalia Kolkovska

### Simulation of Shapiro Steps in Current-Voltage Characteristics of Intrinsic Josephson Junctions in High Temperature Superconductors

The phase dynamics of intrinsic Josephson junctions (IJJ) in high temperature superconductors is investigated. We calculate numerically the current-voltage characteristics (CVC) of IJJ in the framework of the capacitively coupled Josephson junctions model with diffusion current. The effect of electromagnetic irradiation on the CVC of coupled Josephson junctions is demonstrated. Variation of the Shapiro steps and of the critical current with the radiation amplitude is systematically studied for a single junction. These results are generalized for a stack of coupled Josephson junctions. The effects of coupling between junctions on the Shapiro steps in the CVC and on the electric charge oscillations in the superconducting layers at the parametric resonance are shown.

Mahmoud Gaafar, Yury Shukrinov, Hussein El Samman, Sanaa Maize

### Second Order Scheme for Maxwell’s Equations with Discontinuous Electromagnetic Properties

A second order finite volume scheme for numerical solution of non-stationary Maxwell’s equations with discontinuous dielectric permittivity and magnetic permeability on unstructured meshes is suggested. The scheme is based on Godunov, Lax-Wendroff, and Van Leer approaches. The distinctive feature of the considered scheme is calculation of derivatives that ensures approximation even near electromagnetic properties discontinuity. Numerical tests confirm the second order of approximation of the proposed scheme for cases of linear and curvilinear discontinuities.

Timur Z. Ismagilov

### Simulation of Current Voltage Characteristics of Intrinsic Josephson Junctions in HTSC

The current-voltage characteristics (CVC) of capacitively coupled Josephson junctions are numerically calculated. The methods of branch structure simulation in the CVC and charge dynamics of the coupled Josephson junctions are described in detail. We discuss the features of the parametric resonance and its manifestation in the CVC and in the charge-charge diagrams.

Yury Shukrinov, Ilhom Rahmonov, Mohammad Hamdipour

### Numerical Study of Stationary, Time-Periodic, and Quasiperiodic Two-Soliton Complexes in the Damped-Driven Nonlinear Schrödinger Equation

We compile a chart of stationary and oscillatory two-soliton attractors on a plane of two parameters of the damped-driven nonlinear Schrödinger equation. Stable stationary and time-periodic complexes are shown to coexist.

Elena Zemlyanaya, Nora Alexeeva

### Algorithms and Software for Event Reconstruction in the RICH, TRD and MUCH Detectors of the CBM Experiment

The Compressed Baryonic Matter (CBM) experiment at the future FAIR facility at Darmstadt will measure dileptons emitted from the hot and dense phase in heavy-ion collisions. Very fast event reconstruction is extremely important for CBM because of the huge amount of data which has to be handled. In this contribution the parallel event reconstruction algorithms in the Ring Imaging CHerenkov detector, Transition Radiation Detector and muon system are presented. Modern CPUs have two features, which enable parallel programming. First, the SSE technology allows using the SIMD execution model. Second, multi core CPUs enable to use multithreading. Both features were implemented in the reconstruction software. Simulation results show a significant speed up factor.

Semen Lebedev, Claudia Höhne, Ivan Kisel, Andrey Lebedev, Gennady Ososkov

### On-Line Data Processing in the Dubna Gas Filled Recoil Separator Experiments

The on-line data processing for PC based integrated systems for the protection and the determination and monitoring of parameters at the Dubna Gas Filled Recoil Separator is considered [2]. These systems are developed for long-term experiments at the U400 FLNR cyclotron and are aimed at the synthesis of super heavy nuclei in heavy ion induced complete fusion reactions. Parameters and events related to: a) beam and cyclotron; b) separator itself, c) detection system, d) target and entrance window, are measured and stored in the protocol file of the experiment [3]. Special attention is paid to generating alarm signals and implementing the appropriate procedures. The method of active correlations is considered in detail. Just this technique made it possible to search for pointers on a potential multi-chain event and to provide a deep suppression of background products coming from the cyclotron in real-time mode.

Yury Tsyganov, Alexander Polyakov, Alexander Sukhov, Victor Zlokazov

### Automatic Calibration of Multi-strip Position-Sensitive Detector

Calibration is the transformation of the output channels of a measuring device into the physical values (energies, times, angles etc.) If dealt with manually, it is a labor- and time-consuming procedure even if only few detectors are used. However, the situation worsens appreciably if the calibration of multi-detector systems is required, where the number of registering devices extends to hundreds [1]. The calibration is aggravated by the fact that needed pivotal channel numbers should be determined from peak-like distributions. But peak distribution is an informal pattern so that a procedure of image recognition should be employed to discard the operator interference. The report discusses an algorithm for an automatic calibration and describes instances of its verification with use of both simulated data and real alpha spectra.

Victor Zlokazov, Vladimir Utyonkov, Yury Tsyganov

### Numerical Simulation of Heat Conductivity in Composite Object with Cylindrical Symmetry

A parallel algorithm for numerical solution of the mixed problem for heat transport with discontinuous coefficients is presented. The problem is motivated by simulation of heat conductivity in a composite object, when it is heated by the electric current passing through one relatively thin layer. The object is considered to be a cryogenic cell pulse (in the millisecond range) feeding the working gases into some source of highly charged ions. Results are reported for a common configuration of the cell.

Alexander Ayriyan, Edik A. Ayryan, Eugeny Donets, Ján Pribiš

### Simulation of Holography Using Multiprocessor Systems

This paper describes a software package developed for simulations in the field of holography. As the main computational algorithm takes considerable amount of time and has

N

4

time complexity, the software should be implemented on HPC-computers. A software package (BinNet) with the required functionality was written in C++ programming language. Computations were performed on the MVS-100K JSCC RAS supercomputer (140 TFLOPS) and on the MIIT T4700 supercomputer (4.7 TFLOPS). The developed software package showed good scalability on these clusters. Calculation algorithms, the software package structure, speedup and efficiency benchmarks are described. Some critical time-consuming parts of the algorithm were ported to NVIDIA CUDA. Results of tests on TESLA C1060 are also presented.

Dmitry Knyazkov

### GPU Computing in Biomolecular Modeling and Nanodesign

In addition to the intended use of graphics processing units (GPU) to accelerate computer games, their potential has become apparent for scientific computations in the recent years. Molecular modeling and molecular design are only few examples of numerous research areas that are significantly benefiting from novel developments of hardware and software platforms. For example, the impact of high computational power of GPUs has been demonstrated in molecular dynamics (MD) simulations or quantum chemical (QC) calculations. Thus far, several MD programs have been adapted to GPU computing, including NAMD/VMD, GROMACS, AMBER,

etc

. In addition, modeling tools intended for molecular design based on receptor-ligand interactions, such as molecular docking or core hopping protocols have recently been updated for GPU environment. The tremendous increase in the computing power facilitated by the integration of GPUs and the availability of GPU-based systems could accelerate material research on nanoscale. The price/performance ratio of GPU-based systems supports the development of custom-made protocols for efficient modeling of biomolecular systems and nanostructures. GPU-related molecular modeling tools will also accelerate the combined quantum chemical/molecular mechanics (QC/MM) methodologies. An overview of the performance of NVIDIA Tesla GPU-based system built for high-performance and high-throughput computing aimed for biomolecular modeling and nanodesign is presented.

Tibor Kožár

### Numerical Modeling of Nanoparticles Tracking in the Blood Stream

The 3D computer simulation of magnetic nanoparticles transport in the blood flow under the influence of a magnetic field was developed in the diffuse programming environment ROOT. This simulation enabled to calculate the force acting on these particles and to find the conditions under which it is possible to capture the transmitted material in the desired area and to keep it, so that the medicinal product could satisfy its therapeutic role at its release.

Lucia Val’ová, Ján Jadlovský, Oxana Streltsova, Peter Kopčanský, Milan Timko, Martina Kubovčíková, Martina Koneracká, Vlasta Závišová

### Parallel Numerical Calculations of Quantum Trimer Systems

In this report, total angular momentum representation is used in order to write the Schrödinger equation for the trimer as a finite system of coupled three-dimensional differential equations. The complex scaling method is used for studying resonances of trimers. To get a numerical approximation of the problem, a combination of the high-order finite element method with the spectral method is used. This approach results in a generalized eigenvalue problem. Its spectrum describes bound states and resonances of the trimers. Different parallelization techniques (OpenMP, MPI, GPGPU) are considered for the calculation of the matrix elements of the problem. Their efficiency and scalability are discussed and compared.

Evgeny Yarevsky

### A Graph Annihilation Problem

In this paper a new minimal cost graph flow problem is defined. Such kind of problem appears by optimal solving of certain financial tasks connected to the currency exchange at the cash currency markets. The initial problem containing absolute values in the objective function which provides the total costs of the financial transaction is transformed to a linear programming problem in the standard form. An example of the solution of the simple graph annihilation problem is given.

Ján Buša

### Piecewise Scaling in a Model of Neural Network Dynamics

Realistic neural network (RNN) model was proposed in 1981 by Kropotov and Pakhomov [4] for description of most important neuro-physiological dynamical mechanisms. In the modified RNN (MRNN) model [1,3] the defined by the Bogdanov-Hebb principle dynamics of interneuron interactions and a dissipation were introduced. In results the dynamics of the system appeared to be more stable and also a possibility arose to investigate the structure processes. The stable regimes of the MRNN model can be classified as periodical and non-periodical ones. A special case of non-periodical regime is the critical dynamics. It is characterized by consequences of quasi-periodical patterns of neuron activity with mean value of one equal 1/2. The distribution of durations of the patterns of such a kind is presented by a piecewise potential function.

German Chernykh, Yury Pis’mak

### Prediction of Financial Markets Using Agent-Based Modeling with Optimization Driven by Statistical Evaluation of Historical Data

This paper introduces agent-based model for simple prediction of financial markets, where each agent predicts development of selected subset of assets pairs in time by separately examining the similarities between ask and bid assets histories. Agent’s fitness is proportional to the wealth accumulated by exercising long and short trading positions, with regards to predicted development of assets. Although the model is iterative and operates on equidistant price data, agents are encouraged to optimize their trading frequency to maximize simulated wealth (fitness). The model evolves by enforcing competitive behavior through optimization processes.

Jana Kočišová, Denis Horváth, Tomáš Kasanický, Ján Buša

### Multi-agent Based Analysis of Financial Data

In this work the system of agents is applied to establish a model of the nonlinear distributed signal processing. The evolution of the system of the agents – by the prediction time scale diversified trend followers, has been studied for the stochastic time-varying environments represented by the real currency-exchange time series. The time varying population and its statistical characteristics have been analyzed in the non-interacting and interacting cases. The outputs of our analysis are presented in the form of the mean life-times, mean utilities and corresponding distributions. They show that populations are susceptible to the strength and form of inter-agent interaction. We believe that our results will be useful for the development of the robust adaptive prediction systems.

Tomáš Tokár, Denis Horváth, Michal Hnatič

### Constraints on Control Parameters of Asynchronous Differential Evolution

The efficiency of an algorithm to find the global minimum depends on its ability to keep population diversity during evolutionary iterations. Statistical variance can serve as a measure of population diversity. We analyse the expected population variance after mutation and crossover for

best/1/bin

strategy of Classical Differential Evolution and for new strategies of a novel Asynchronous Differential Evolution. Relations between the control parameters (

N

p

,

F

,

C

r

) of algorithms and the extension factor of population variance are derived. Constraints on control parameters to prevent premature convergence of the algorithm are suggested and compared with phase portraits (convergence domains) for several benchmark functions.

Evgeniya Zhabitskaya

### Asynchronous Differential Evolution

Differential Evolution (DE) is an algorithm to solve possibly nonlinear and non-differentiable global optimization problems. Classical Differential Evolution (CDE) employs a synchronous generation-based evolution strategy. We propose a modification of the CDE algorithm by incorporating mutation, crossover and selection operations into an asynchronous strategy. A novel Asynchronous Differential Evolution (ADE) is well suited for parallel optimization. Moreover even in the sequential mode its rate of convergence is competitive to CDE. The performance of the Asynchronous Differential Evolution is reported on a set of benchmark functions.

Evgeniya Zhabitskaya, Mikhail Zhabitsky

### Backmatter

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