Doctor of Philosophy

dissertationThis dissertation consists of two parts that focus on two interrelated areas of Applied Mathematics. The first part explores fundamental properties and applications of functions with values in L-spaces. The second part is connected to Approximation Theory and dives deeper into the analysis of functions with values in specific classes of L-spaces (in particular, L-spaces of sets). In the first project devoted to the theory and numerical methods for the solution of integral equations, we explore linear Volterra and Fredholm integral equations for functions with values in L-spaces (which are generalizations of set-valued and fuzzy-valued functions). In this study, we prove the existence and uniqueness of the solution for such equations, suggest algorithms for finding approximate solutions, and study their convergence. The exploration of these equations is of great importance given the wide variety of their applications in biology (population modeling), physics (heat conduction), and engineering (feedback systems), among others. We extend the aforementioned results of existence and uniqueness to nonlinear equations. In addition, we study the dependence of solutions of such equations on variations in the data. In order to be able to better analyze the convergence of the suggested algorithms for the solutions of integral equations, we develop new results on the approximation of functions with values in L-spaces by adapted linear positive operators (Bernstein, Schoenberg, modified Schoenberg operators, and piecewise linear interpolation). The second project is devoted to problems of interpolation by generalized polynomials and splines for functions whose values lie in a specific L-space, namely a space of sets. Because the structure of such a space is richer than the structure of a general L-space, we have additional tools available (e.g., the support function of a set) which allow us to obtain deeper results for the approximation and interpolation of set-valued functions. We are working on defining various methods of approximation based on the support function of a set. Questions related to error estimates of the approximation of set-valued functions by those novel methods are also investigated

Hägusad teist liiki integraalvõrrandid

Käesolevas doktoritöös on uuritud hägusaid teist liiki integraalvõrrandeid. Need võrrandid sisaldavad hägusaid funktsioone, s.t. funktsioone, mille väärtused on hägusad arvud. Me tõestasime tulemuse sileda tuumaga hägusate Volterra integraalvõrrandite lahendite sileduse kohta. Kui integraalvõrrandi tuum muudab märki, siis integraalvõrrandi lahend pole üldiselt sile. Nende võrrandite lahendamiseks me vaatlesime kollokatsioonimeetodit tükiti lineaarsete ja tükiti konstantsete funktsioonide ruumis. Kasutades lahendi sileduse tulemusi tõestasime meetodite koonduvuskiiruse. Me vaatlesime ka nõrgalt singulaarse tuumaga hägusaid Volterra integraalvõrrandeid. Uurisime lahendi olemasolu, ühesust, siledust ja hägusust. Ülesande ligikaudseks lahendamiseks kasutasime kollokatsioonimeetodit tükiti polünoomide ruumis. Tõestasime meetodite koonduvuskiiruse ning uurisime lähislahendi hägusust. Nii analüüs kui ka numbrilised eksperimendid näitavad, et gradueeritud võrke kasutades saame parema koonduvuskiiruse kui ühtlase võrgu korral. Teist liiki hägusate Fredholmi integraalvõrrandite lahendamiseks pakkusime uue lahendusmeetodi, mis põhineb kõigi võrrandis esinevate funktsioonide lähendamisel Tšebõšovi polünoomidega. Uurisime nii täpse kui ka ligikaudse lahendi olemasolu ja ühesust. Tõestasime meetodi koonduvuse ja lähislahendi hägususe.In this thesis we investigated fuzzy integral equations of the second kind. These equations contain fuzzy functions, i.e. functions whose values are fuzzy numbers. We proved a regularity result for solution of fuzzy Volterra integral equations with smooth kernels. If the kernel changes sign, then the solution is not smooth in general. We proposed collocation method with triangular and rectangular basis functions for solving these equations. Using the regularity result we estimated the order of convergence of these methods. We also investigated fuzzy Volterra integral equations with weakly singular kernels. The existence, regularity and the fuzziness of the exact solution is studied. Collocation methods on discontinuous piecewise polynomial spaces are proposed. A convergence analysis is given. The fuzziness of the approximate solution is investigated. Both the analysis and numerical methods show that graded mesh is better than uniform mesh for these problems. We proposed a new numerical method for solving fuzzy Fredholm integral equations of the second kind. This method is based on approximation of all functions involved by Chebyshev polynomials. We analyzed the existence and uniqueness of both exact and approximate fuzzy solutions. We proved the convergence and fuzziness of the approximate solution.https://www.ester.ee/record=b539569

Numerical Solution of Some Uncertain Diffusion Problems

Diffusion is an important phenomenon in various fields of science and engineering. These problems depend on various parameters viz. diffusion coefficients, geometry, material properties, initial and boundary conditions etc. Governing differential equations with deterministic parameters have been well studied. But, in real practice these parameters may not be crisp (exact) rather it involves vague, imprecise and incomplete information about the system variables and parameters. Uncertainties occur due to error in measurements, observations, experiments, applying different operating conditions or it may be due to maintenance induced errors, etc. As such, it is an important concern to model these type of uncertainties. Traditionally uncertain problems are modelled through probabilistic approach. But probabilistic methods may not able to deliver reliable results at the required precision without sufficient data. In this context, interval and fuzzy theory may be used to manage such uncertainties. Accordingly, the system parameters and variables are represented here as interval and fuzzy numbers. Generally, we get interval or fuzzy system of equations for uncertain steady state problems with interval or fuzzy parameters whereas interval or fuzzy eigenvalue problems may be obtained for unsteady state. This thesis redefined interval or fuzzy arithmetic in order to handle the uncertain problems. The proposed arithmetic has been used to solve fuzzy and interval system of equations and eigenvalue problems. Various numerical methods viz. Finite Element Method (FEM), Wavelet Method (WM), Euler Maruyama and Milstein Methods are studied by introducing interval or fuzzy theory. The proposed arithmetic has been combined with FEM and WM to develop Interval or Fuzzy Finite Element Method (I/FFEM) and Interval or Fuzzy Wavelet Method (I/FWM). Further, it may be pointed out that sometimes systems may possess uncertainties due to randomness and fuzziness of the parameters. As such, here we have hybridized the concept of fuzziness as well as stochasticity to develop numerical fuzzy stochastic methods viz. interval or Fuzzy Euler Maruyama and Interval/Fuzzy Milstein. These methods are also been used to solve various diffusion problems. Numerical examples and different application problems are solved to demonstrate the efficiency and capabilities of the developed methods. In this respect, imprecisely defined diffusion problems such as heat conduction and conjugate heat transfer in rod, homogeneous and non-homogeneous fin and plate, along with one group, multi group and point kinetic neutron diffusion with interval or fuzzy uncertainties have been investigated. The convergence of the field variables have been investigated with respect to the number of element discretization of the domain in case of I/FEM. Accordingly, convergence of the proposed interval or fuzzy FEM has been studied for unsteady heat conduction in a cylindrical rod. For conjugate heat transfer problems, the convergence of uncertain temperature distributions with respect to the number of element discretizations has also been studied. Further, various combinations of uncertain parameters are considered and the sensitivity of these parameters has been reported. Next, one group and two group problems have been solved and the sensitivity of the uncertain parameters in the context of fast and thermal neutrons are presented. The hybrid fuzzy stochastic methods have also been used to investigate uncertain stochastic point kinetic neutron diffusion problem. Uncertain variation of neutron populations are analysed by considering two random samples. Developed interval or fuzzy WM has also been used to solve uncertain differential equation. Finally obtained results for the said problems are compared in special cases for the validation of proposed methods

Fuzzy Systems

This book presents some recent specialized works of theoretical study in the domain of fuzzy systems. Over eight sections and fifteen chapters, the volume addresses fuzzy systems concepts and promotes them in practical applications in the following thematic areas: fuzzy mathematics, decision making, clustering, adaptive neural fuzzy inference systems, control systems, process monitoring, green infrastructure, and medicine. The studies published in the book develop new theoretical concepts that improve the properties and performances of fuzzy systems. This book is a useful resource for specialists, engineers, professors, and students

Neutrosophic SuperHyperAlgebra and New Types of Topologies

In general, a system S (that may be a company, association, institution, society, country, etc.) is formed by sub-systems Si { or P(S), the powerset of S }, and each sub-system Si is formed by sub-sub-systems Sij { or P(P(S)) = P2(S) } and so on. That’s why the n-th PowerSet of a Set S { defined recursively and denoted by Pn(S) = P(Pn-1(S) } was introduced, to better describes the organization of people, beings, objects etc. in our real world. The n-th PowerSet was used in defining the SuperHyperOperation, SuperHyperAxiom, and their corresponding Neutrosophic SuperHyperOperation, Neutrosophic SuperHyperAxiom in order to build the SuperHyperAlgebra and Neutrosophic SuperHyperAlgebra. In general, in any field of knowledge, one in fact encounters SuperHyperStructures. Also, six new types of topologies have been introduced in the last years (2019-2022), such as: Refined Neutrosophic Topology, Refined Neutrosophic Crisp Topology, NeutroTopology, AntiTopology, SuperHyperTopology, and Neutrosophic SuperHyperTopology

Doctor of Philosophy

dissertationWith modern computational resources rapidly advancing towards exascale, large-scale simulations useful for understanding natural and man-made phenomena are becoming in- creasingly accessible. As a result, the size and complexity of data representing such phenom- ena are also increasing, making the role of data analysis to propel science even more integral. This dissertation presents research on addressing some of the contemporary challenges in the analysis of vector fields--an important type of scientific data useful for representing a multitude of physical phenomena, such as wind flow and ocean currents. In particular, new theories and computational frameworks to enable consistent feature extraction from vector fields are presented. One of the most fundamental challenges in the analysis of vector fields is that their features are defined with respect to reference frames. Unfortunately, there is no single ""correct"" reference frame for analysis, and an unsuitable frame may cause features of interest to remain undetected, thus creating serious physical consequences. This work develops new reference frames that enable extraction of localized features that other techniques and frames fail to detect. As a result, these reference frames objectify the notion of ""correctness"" of features for certain goals by revealing the phenomena of importance from the underlying data. An important consequence of using these local frames is that the analysis of unsteady (time-varying) vector fields can be reduced to the analysis of sequences of steady (time- independent) vector fields, which can be performed using simpler and scalable techniques that allow better data management by accessing the data on a per-time-step basis. Nevertheless, the state-of-the-art analysis of steady vector fields is not robust, as most techniques are numerical in nature. The residing numerical errors can violate consistency with the underlying theory by breaching important fundamental laws, which may lead to serious physical consequences. This dissertation considers consistency as the most fundamental characteristic of computational analysis that must always be preserved, and presents a new discrete theory that uses combinatorial representations and algorithms to provide consistency guarantees during vector field analysis along with the uncertainty visualization of unavoidable discretization errors. Together, the two main contributions of this dissertation address two important concerns regarding feature extraction from scientific data: correctness and precision. The work presented here also opens new avenues for further research by exploring more-general reference frames and more-sophisticated domain discretizations

Efficient frequency averaging techniques for noise and vibration simulations

In the latest decades, noise and vibration characteristics of products have been growing in importance, driven by market expectations and tightening regulations. Accordingly, CAE tools have become irreplaceable in assisting acousticians through the design process, and their accuracy and efficiency are essential to model the behavior of complex engineering systems.This dissertation aims at increasing the computational efficiency of deterministic simulation techniques for steady-state noise and vibration problems. In particular, the focus is on the efficient evaluation of weighted frequency integrals. Classic approaches make use of numerical quadrature to evaluate a frequency integral. However, the response of a vibrating system is commonly a highly oscillating function of frequency, and a large number of sampling points might be required to achieve an accurate integration. As an alternative, the residue theorem is proposed to compute the weighted integrals. The refined integration over real frequencies is replaced by a few computations at complex frequencies, with a consequent increased accuracy and computational efficiency.A weighted integral is first evaluated to compute the band-averaged input power into a vibrating system. The ideal rectangular weighting function is approximated by using the square magnitude of a Butterworth filter. Applying the residue theorem, the integral can be evaluated by computing the system response at a few points in the complex frequency plane. These points are some of the filter poles, equal in number to the order of the filter. This allows for an efficient integration, regardless of the bandwidth of analysis. Such a result is successively generalized. The band integral is computed by moving the path of integration to the complex frequency plane and applying efficient quadrature schemes. Due to the smoothness of the integrand in the complex frequency plane, the accuracy and efficiency of the technique are further increased. Moreover, it is shown that using numerical quadrature in the complex plane indirectly leads to the definition of a novel family of weighting functions over the real frequency domain.The proposed techniques for the evaluation of the band-averaged input power are accurate, efficient, easy to implement and can be employed within any classic deterministic framework. Applications allow assessing the effectiveness of the methodology for complex geometries and frequency dependent properties, and in combination with optimization schemes.When the order of the Butterworth filter is one, its square magnitude corresponds to the Lorentzian function, which allows computing the weighted average over a wide frequency range by evaluating the response at a single complex frequency. However, due to its bell-shape, the Lorentzian is not suitable to evaluate band values, and its use as a weighting function is investigated in three different ways. The ensemble mean input power is estimated by using the Lorentzian-weighted frequency averaging. The same procedure is used to evaluate the direct field dynamic stiffness of a component. Finally, the Lorentzian is used as a mass-frequency density function within the Fuzzy Structure Theory.status: publishe