Aleksandras Krylovas

Parallel algorithms for solution of nonlinear diffusion problems in image smoothing 1

Abstract In this work we consider parallel algorithms for solution of nonlinear parabolic PDEs. First mathematical models describing nonlinear diffusion filters are presented. The finite‐volume method is used to approximate differential equations. Parallel algorithms are based on the domain decomposition method. The algorithms are implemented by using ParSol parallelization tool and a brief description of this tool is also presented. The efficiency of proposed parallel algorithms is investigated and results of the scalability analysis are given. Theoretical predictions are compared with results of computational experiments. Application of nonlinear diffusion filters for analysis of computer tomography images is discussed in the last section of the paper.

Mathematical modelling of forecasting the results of knowledge testing

Abstract In this paper a mathematical model for obtaining probability distribution of the knowledge testing results is proposed. Differences and similarities of this model and Item Response Theory (IRT) logistic model are discussed. Probability distributions of 10 items test results for low, middle and high ability populations selecting characteristic functions of the various difficulty items combinations are obtained. Entropy function values for these items combinations are counted. These results enable to formulate recomendations for test items selection for various testing groups according to their attainment level. Method of selection of a suitable item characteristic function based on the Kolmogorov compatibility test, is proposed. This method is illustrated by applying it to a discreet mathematics test item.

Asymptotic method for approximation of resonant interaction of nonlinear multidimensional hyperbolic waves

Abstract A method of averaging along characteristics of weakly nonlinear hyperbolic systems, which was presented in earlier works of the author for one dimensional waves, is generalized for some cases of multidimensional wave problems. In this work we consider such systems and discuss a way to use the internal averaging along characteristics for new problems of asymptotical integration.

Multiple criteria decision-making KEMIRA-M method for solution of location alternatives

Abstract Choice of location in many cases is a key factor setting up a new business object. In this article the KEMIRA-M method is proposed to establish priority of criteria and determine criteria weights. Weighted sum of criteria values was applied for ranking the alternatives. This technique is useful if the evaluation criteria naturally consist of several logically explained groups of criteria. Method requires much less initial information and is based upon searching the solution of optimisation problem. KEMIRA-M is applied for the case study of construction site for non-hazardous waste incineration plant in Vilnius City.

ON THE INTERACTION OF ELASTIC WAVES

Abstract The non-linear mathematical model of the interaction of elastic waves is presented. The conditions of possible resonant interaction of periodic waves are described. The method of internal averaging for getting uniformly valid asymptotic expansions is used in both, ie resonant and non-resonant, cases. Results of numerical experiments are presented for the resonant interaction of the elastic waves.

Entropy–KEMIRA Approach for MCDM Problem Solution in Human Resources Selection Task

Entropy–KEMIRA approach is proposed for criteria ranking and weights determining when solving Multiple Criteria Decision-Making (MCDM) problem in human resources selection task. For the first time the method is applied in the case of three groups of criteria. Weights are calculated by solving optimization problem of maximizing the number of elements, which are “best” according to all three criteria, and minimizing the number of “doubtful” elements. The algorithm of problem solution is presented in the paper. The numerical experiment with three groups of evaluation criteria describing 11 life goals was accomplished.

Expert Estimates Averaging by Constructing Intuitionistic Fuzzy Triangles

AbstractThe problem of ranking (sorting) of m alternatives is considered when experts evaluate each alternative according to k criteria. Functions of arithmetic and geometric averages are constructed for decision making. We present a generalization of this scheme when there are evaluation matrices of several experts and this information is aggregated in the form of triangular intuitionistic fuzzy numbers. Fuzzy triangles were constructed with different uncertainty levels, experts decision matrices and the number of experts varied from 2 to 5. Moreover, method for construction of experts decision probability matrices is proposed in the paper. Ranking results obtained by performing Monte Carlo simulations. Probabilities of errors are compared for arithmetic, geometric, fuzzy arithmetic and fuzzy geometric averages.

Asymptotical Analysis of Some Coupled Nonlinear Wave Equations

Abstract We consider coupled nonlinear equations modelling a family of travelling wave solutions. The goal of our work is to show that the method of internal averaging along characteristics can be used for wide classes of coupled non-linear wave equations such as Korteweg-de Vries, Klein – Gordon, Hirota – Satsuma, etc. The asymptotical analysis reduces a system of coupled non-linear equations to a system of integro – differential averaged equations. The averaged system with the periodical initial conditions disintegrates into independent equations in non-resonance case. These equations describe simple weakly non-linear travelling waves in the non-resonance case. In the resonance case the integro – differential averaged systems describe interaction of waves and give a good asymptotical approximation for exact solutions.

Uniformly Valid Asymptotics for Carrier's Mathematical Model of String Oscillations

AbstractIn the paper, an asymptotic analysis of G.F. Carriers mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the systems solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.

Examples of Asymptotical Analysis of Hyperbolic Equations

We consider a system of weakly nonlinear equations with a small positive parameter e: