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Browsing Theses and Dissertations by Author "Tourassis, Vassilios D."
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Item Open Access Convergence Rate of Fourier Neural Networks(Nazarbayev University School of Science and Technology, 2019-04-26) Zhumekenov, Abylay; Assylbekov, Zhenisbek; Tourassis, Vassilios D.The paper investigates a convergence rate for 2-layer feedforward Fourier Neural Network (FNN). Such networks are motivated by the approximation properties of wellknown Fourier series. Several implementations of FNNs were proposed since 1990’s: by Gallant and White; A. Silvescu; Tan, Zuo and Cai; Liu. The main focus of this research is Silvescu’s FNN, because such activation function does not fit into the category of networks, where the linearly transformed input is exposed to activation. The latter ones were extensively described by Hornik in 1989. In regard to non-trivial Silvescu’s FNN, its convergence rate is proven to be of order 𝑂(1/𝑛). The paper continues investigating classes of functions approximated by Silvescu FNN, which appeared to be from Schwartz space and space of positive definite functions.Item Open Access Experimental study of Pac-Man conditions for learn-ability of discrete linear dynamical systems(Nazarbayev University School of Science and Technology, 2019-05-01) Damiyev, Zhaksybek; Takhanov, Rustem; Tourassis, Vassilios D.In this work, we are going to reconstruct parameters of a discrete dynamical system with a hidden layer, given by a quadruple of matrices (𝐴,𝐵,𝐶,𝐷), from system’s past behaviour. First, we reproduced experimentally the well-known result of Hardt et al. that the reconstruction can be made under some conditions, called Pac-Man conditions. Then we demonstrated experimentally that the system approaches the global minimum even if an input 𝑥 is a sequence of i.i.d. random variables with a nongaussian distribution. We also formulated hypotheses beyond Pac-Man conditions that Gradient Descent solves the problem if the operator norm (or alternatively, the spectral radius) of transition matrix 𝐴 is bounded by 1 and obtained the negative result, i.e. a counterexample to those conjectures.Item Open Access Explorations on chaotic behaviors of Recurrent Neural Networks(Nazarbayev University School of Science and Technology, 2019-04-29) Myrzakhmetov, Bagdat; Assylbekov, Zhenisbek; Takhanov, Rustem; Tourassis, Vassilios D.In this thesis work we analyzed the dynamics of the Recurrent Neural Network architectures. We explored the chaotic nature of state-of-the-art Recurrent Neural Networks: Vanilla Recurrent Network, Recurrent Highway Networks and Structurally Constrained Recurrent Network. Our experiments showed that they exhibit chaotic behavior in the absence of input data. We also proposed a way of removing chaos chaos from Recurrent Neural Networks. Our findings show that initialization of the weight matrices during the training plays an important role, as initialization with the matrices whose norm is smaller than one will lead to the non-chaotic behavior of the Recurrent Neural Networks. The advantage of the non-chaotic cells is stable dynamics. At the end, we tested our chaos-free version of the Recurrent Highway Networks (RHN) in a real-world application. In a sequence-to-sequence modeling experiments, particularly in the language modeling task, chaos-free version of RHN perform on par with the original version by using the same hyperparameters.Item Open Access Global stability analysis for a tick-borne model(Nazarbayev University School of Science and Technology, 2019-05-06) Koptleuova, Daiana; Kashkynbayev, Ardak; Tourassis, Vassilios D.This thesis consider three type of epidemiological models: SIR, SIS and SIRS with nonlinear incidence rate and piecewise constant delay of generalized type. In this paper the total population size is varied with time elapse. We study the global asymptotic stability of the disease-free and endemic equilibrium states of models by constructing suitable Lyapunov functions and Lyapunov–LaSalle technique. The main contribution of this master thesis is to develop more realistic compartmental models by extending the literature of models with piecewise constant delay. The theoretical findings are illustrated through numerical simulations.Item Open Access The lumped model parameters approach for static and dynamic power-law beam problems(Nazarbayev University School of Science and Technology, 2019-04-29) Begzhigitov, Madi; Skrzypacz, Piotr; Tourassis, Vassilios D.It is important to estimate the natural frequencies of the structural elements in the design of mechanical or electromechanical structures. There is a wide use of single lumped-parameter spring-mass models in the industry for materials . Their behaviour is linear by Hooke’s law within the geometric and loading conditions. In this work, the lumped-parameter theory is generalized for Hollomon’s power-law materials and the lumped-parameters for the corresponding nonlinear restoring force in the spring-like model for the standard geometric and loading conditions of the power-law Euler beams are provided. For each case in the given lumped-parameter model the corresponding effective mass is also calculated. Then, the resulting spring-mass system is solved to validate the solutions as approximations to the corresponding beam system. Numerical validations of the proposed lumped models for the cantilever beam with circular and rectangular cross-sections are presented.Item Open Access Multigrid method for Mild-Slope equation in Coastal Wave Modelling(Nazarbayev University School of Science and Technology, 2019-04-19) Tabarek, Rysbergen; Erlangga, Yogi; Tourassis, Vassilios D.In this thesis we propose and study an efficient iterative multigrid method for the time independent modified mild slope equation with and without energy dissipation term. The algorithm relies on a multigrid method preconditioned with shifted-Laplacian preconditioner and solved by Bi-CGSTAB algorithm. Multigrid analysis results are shown by numerical experiments. Numerical experiments are conducted in depth sloped elliptic shoal introduced by Berkhoff et. alItem Open Access Numerical computations of complexification of Legendrian knots(Nazarbayev University School of Science and Technology, 2019-04-29) Yerzhigit, Bauyrzhan; Lawrence, Mark; Tourassis, Vassilios D.With the recent interest in knots, it is interesting to study their complexification. We have chosen to study Legendrian knots as they have the property that we can reconstruct the original knot from its projection. This property is especially useful in the case of the complexification of a knot as in this case the diagram of the projection of the knot is no longer real. In this paper we show a way to compute complex rational functions that have a Legendrian knot as an image under unit circle.Item Open Access Pulse vaccination of a time-delayed SIRS epidemic model with nonlinear incidence rate(Nazarbayev University School of Science and Technology, 2019-05-03) Yeleussinova, Meruyert; Kashkynbayev, Ardak; Tourassis, Vassilios D.This work deals with an application of pulse vaccination for a varying size of the population of time-delayed 𝑆𝐼𝑅𝑆 epidemic model. The dynamics of the infectious disease depends on the threshold value, 𝑅0, known as the basic reproduction number. In the classical epidemic models, this value is evaluated by means of the next generation matrix. However, this method does not work for non-autonomous systems. Since we consider the pulse vaccination strategy for epidemic models our system is naturally non-autonomous. We follow the general approach to derive 𝑅0 in terms of spectral radii of Poincare maps. Further, we show the existence of an infectious-free periodic solution and its global attractiveness for 𝑅0 < 1 and the persistence of infectious disease for 𝑅0 > 1.Item Open Access Synchronization of Coupled Nonlinear Oscillators with Applications to Photonic Arrays(Nazarbayev University School of Science and Technology, 2019-05-01) Zharas, Banu; Bountis, Anastasios; Tourassis, Vassilios D.In recent years, the study of synchronization of coupled oscillators have been the subject of intense research interest, leading to many new and unexpected phenomena. Our research is first focused on the analysis of a network of coupled nonlinear oscillators exhibiting the breakdown of synchronization into fascinating “chimera states” exhibiting the coexistence of synchronized and unsynchronized parts. We then apply these ideas to laser arrays of photonic “oscillators”, which have numerous applications in optical communications, sensing and imaging. First of all, we demonstrate the occurrence of synchronization and chimera states in a simpler problem, consisting of a ring of coupled 4D simplified Lorenz systems, in which each oscillator is described by a Li-Sprott oscillator [1]. An interesting feature of each oscillator is the coexistence of a limit cycle and two symmetric strange attractors for some specific range of parameters, which influences the global synchronization dynamics and leads to the formation of chimera states. Inspired by this model, we study some fascinating oscillatory phenomena of coupled photonic oscillators consisting of dimers of semiconductor lasers, each of which is capable of performing limit cycle oscillations. Coupling in an appropriate way a large number of dimers in long arrays we find that they can exhibit combinations of oscillatory patterns involving long amplitude oscillations (LAO) and also localized oscillations of very small amplitude close to the fixed points (LOCFP). As preliminary results of this investigation, we show the coexistence of LOA and LOCFP patterns reminiscent of “chimera–like” states and LOCFP “breather– like” phenomena. Both of these behaviors are shown to be spatially robust, when we calculate the Discrete Laplacian of their amplitudes for long times.