TAAC'2013: Section "Applied Mathematics"

In this work we describe a class of numerical algorithms for the solution of stiff ODE systems which arise from specific implementation of implicit Runge-Kutta (RK) methods using the principle of steadying. These algorithms do not require Jacobian evaluations and factorizations which is beneficial when the dimension of the ODE is large, and can be naturally parallelized. It is shown that the resulting computational scheme is equivalent to some explicit RK method of first order, but it converges to the solution which corresponds to the base implicit method of high order. Numerical comparison with explicit DOP853 solver is given.

The notion of the bank credit risk and its main components — expected and unexpected losses as well as the methods of their estimation are considered in the article. Some of them are the default probability, the loss given default, exposure at default and expected losses had been investigated by the author and have been realized as the software at the "Ukreximbank". The correspondent decisions are taken on the basis of the obtained results. The further work will concern with the investigation of unexpected losses and constructing the algorithms for CreditVaR estimation as well as programming and testing the obtained results.

This paper presents an approach to predict influenza epidemics based on weather environment. Based on actual statistical data and deterministic models the prediction was made using the multiple linear regression and GMDH tools. Compared the forecasting results got by multiple linear regression and GMDH tools.

The control problem for the singularly perturbed system with delay with indeterminate initial conditions and integral quadratic constraints on the control resources according to the minimax criterion is considered. Iterative procedure of constructing control response that approximates the optimal solution with given accuracy with respect to a small positive parameter is proposed.

The paper deals with optimal control problem for parabolic-hyperbolic equations with quadratic criteria and non-local boundary conditions. We have shown the way of solution of this task and found the general view of zero component of a distributed optimal control, using the Fredholm integral equation of the second type. Also we got formulas for some formulation of problem. Results are going to be used for expanding the solution to a non-zero components of the optimal control.

The method of detonation processes modeling and distribution of a detonation wave based on a discrete-event modeling method is proposed. Detonation experiments of a carbon dust in the oxygen environment are carried out at various concentration parameters. Statistical results of energy and impulse profile changing throughout a settlement area are compiled.

The article discusses issue of stability of dynamic macroeconomic model taking into account the monetary policy of the Central Bank. Comparison of equilibrium states of static and dynamic models was held. The correlations that were obtained economically justify values of the real interest rate.

The paper deals with the differential-algebraic systems with impulse impact under the assumption that systems under the consideration can be reduced to the central canonical form. We find necessary and sufficient conditions for the stability of such systems and generalize the Floquet-Lyapunov theory for systems of this type with periodic coefficients.

The article deals with the Black-Scholes model where parameters depend on the time and environmental state, conditions under which the fair price of an option before and after averaging coincide are considered.

The purpose of the work is organized, analyzed and a mathematical formulation of the optimal distribution of resources in the GRID-system. The needs of applications require the creation of distributed computing systems and their optimal use is an important issue. Therefore, the construction and the study of organizational models of functioning of GRID-systems is defined mission work. In the work was constructed mathematical models of the four major organizational problems functioning GRID-systems: the problem of redistribution of resources, two-criteria and minimax optimization problems and planning problem. Was defined class of problems solved these problems, the mathematical statement is made and formulated conditions of construction of efficient computational schemes.

This paper described approach of organization distributed computing for solving scientific problems. Iterative distributed algorithms are reduced to the MapReduce model and their performance is measured and analyzed. We implement iterative computation using Hadoop MapReduce technology in three parts: first, through controlling iterations of MapReduce Job (class executing call of run() method for Mapper and Reducer); second, implementing initialization for the first iteration step; and, finally, implement separate class for updating data on each step of iteration. Research result is the comparison of the result with different cluster settings and MPI model.

We consider the problem of the computation of $\inf_p \theta p$ over the set of exponent pairs $P \ni p$ under linear constrains for a certain class of objective functions $\theta$. An effective backtracking algorithm is presented, which runs in a linear number of steps for typical inputs. The output of the algorithm leads to the improvement and establishing new estimates in the various divisor problems in the analytical number theory.

The paper summarizes the authors' recent work on developing and proving an exponentially convergent numerical-analytical method (the FD-method) for solving Sturm-Liouville problems with a singular Legendre operator and a singular potential. It gives a concise general overview of the FD-method, outlines the proof of its convergence and exponential convergence rate when applied to the particular problem at hand and talks briefly about its software implementation.

Non-ideal dynamical system "pendulum-electric motor" with taking into account the delay effect of electric motor impulse on the pendulum and the delay of the medium reaction on the dynamical state of the pendulum is considered. The approaches that reduce the mathematical model of the system to a system of differential equations without delay are proposed. The maps of dynamical regimes are constructed and analysed. The influence of various factors of delay on steady-state regimes of the system "pendulum-electric motor" is investigated.

Studied the fluctuations of stock returns based on the analysis of data obtained as a result of trading Public Joint Stock Company "Raiffeisen Bank Aval". Built graphs the cumulative distribution of the normalized stock returns for different time intervals and determined that the resulting curves are well described by power law. Established that Ukrainian stocks exhibit behavior similar as the shares of the leading foreign countries.

A well-known example of global optimization that provides solutions within fixed error limits is optimization of functions with a known Lipschitz constant. We propose a novel method called Pareto Lipschitzian Optimization (PLO) that provides solutions within fixed error limits for functions with unknown Lipschitz constants. We use Pareto Optimality (OP) for comparing. The criteria list is formulated out of set of all unknown Lipschitz constants. All they are regarded as multiple criteria in a search for Potentially Optimal (PO) solution. We introduce experimental implementation of PLO in high process computing (HPC) system. PLO algorithm, its realization and parallel solution is made for Vilnius University Institute of Mathematics and Informatics cluster. HPC implementation of PLO is a complex solution is not realized with straight forward methods. We compare counting and optimization speed between simple and HPC solution. The profit of usage of HPC in the PLO is described in experimental state.

A weakly nonlinear impulsive differential system with center or saddle phase portrait type is considered. System has variable moments of impulses. Points of discontinuity form two parallel lines. Possible types of cycles and conditions of their existence are found for corresponding linear system. Based on these results conditions of existence of cycles of the nonlinear system in vicinities of linear system cycles are found.

Particle Swarm Optimization (PSO) is an optimization technique originally described for continuous problems. However, PSO modified to apply to discrete problems appears to produce optimal or nearly optimal solutions in reasonable time. Such results are often unreachable for deterministic approaches. The paper introduces the detailed PSO algorithm adapted to solve the group of discrete problems that can be represented in a certain form.

We study a certain class of two-dimensional systems of differential equations that undergo an impulsive action at the moment when phase point passes through a given straight line in plane. Necessary and sufficient conditions for asymptotic stability (non-stability) of solutions of such system and the existence of periodic solutions that have one (two) impulsive perturbations per period are stated. That is the solutions that generate so-called one- (two-) impulsive discontinuous cycles. The result of research may be used in the study of oscillatory processes that undergo a short-term perturbations.

Conversion the problem of analyzing a continuous-time random process to that of analyzing a random sequence, which is much simpler, is useful in various applications. In the paper we work with the wavelet based expansions which have numerous advantages over Fourier series, and often lead to stable computations. This approach is widely used in statistics to estimate a curve given observations of the curve plus some noise. So, conditions of uniform convergence for Battle-Lemarie wavelets expansions of Gaussian random processes are presented. Another useful in various computational applications thing is the rate of convergence, especially if we are interested in the optimality of the stochastic approximation or the simulations. An explicit estimate of the rate of uniform convergence for Battle-Lemarie wavelets expansions of Gaussian random processes is obtained.

I present a new risk process where premiums are modeled by a normal inverse Gaussian (NIG) L'evy process, and claims by a Gamma L'evy process. I illustrate proposed model with simulated graph and derive a Cramer-Lundberg upperbound for ruin probability.

Axisymmetric problem for an elastic semistrip under the action of an external loading is solved. On the lateral faces the conditions of coupling are executed, at the end the conditions of the elasticity first main problem are given. With the help of the integral sin-, cos-Fourier transformations the initial problem is reduced to the one-dimensional one, which is formulated as a vector semi-homogeneous boundary value problem. Its solving with the help of the Green's matrix leads to the solving of the integral-differential equation with regard to the unknown derivative of the displacement at the strip's end. The equation is solved approximately by the orthogonal polynomials' method with taking into consideration the real singularities of the unknown function at the ends of the integration's interval. The use of the inverse integral transformations allow to obtain expressions for the displacements' and stress' estimation. The values of the normal stress within the strip are investigated.

By example of concrete system with one degree of freedom, that firstly was investigated analytically, specificity of numerical integration of ordinary second-order differential equations for systems with Coulomb and viscous friction was considered. A modification of Runge-Kutta formulas for mentioned systems was developed. Efficiency of modified Runge-Kutta formulas was shown for particular case by comparison of numerical and analytical solutions. A mentioned modification of Runge-Kutta formulas can be easily extended for systems with dry friction and any given finite number of degrees of freedom.