In the field of uncertainty quantification, the effects of parameter uncertainties on scientific simulations may be studied by integrating or approximating a quantity of interest as a function over the parameter space. If this is done numerically, using regular grids with a fixed resolution, the required computational work increases exponentially with respect to the number of uncertain parameters – a phenomenon known as the curse of dimensionality. We study two methods that can help break this curse: discrete least squares polynomial approximation and kernel-based approximation. For the former, we adaptively determine sparse polynomial bases and use evaluations in random, quasi-optimally distributed evaluation nodes; for the latter, we use evaluations in sparse grids, as introduced by Smolyak. To mitigate the additional cost of solving differential equations at each evaluation node, we extend multilevel methods to the approximation of response surfaces. For this purpose, we provide a general analysis that exhibits multilevel algorithms as special cases of an abstract version of Smolyak’s algorithm. In financial mathematics, high-dimensional approximation problems occur in the pricing of derivatives with multiple underlying assets. The value function of American options can theoretically be determined backwards in time using the dynamic programming principle. Numerical implementations, however, face the curse of dimensionality because each asset corresponds to a dimension in the domain of the value function. Lack of regularity of the value function at the optimal exercise boundary further increases the computational complexity. As an alternative, we propose a novel method that determines an optimal exercise strategy as the solution of a stochastic optimization problem and subsequently computes the option value by simple Monte Carlo simulation. For this purpose, we represent the American option price as the supremum of the expected payoff over a set of randomized exercise strategies. Unlike the corresponding classical representation over subsets of Euclidean space, this relax- ation gives rise to a well-behaved objective function that can be globally optimized using standard optimization routines.

The theme of this masters thesis are driven chaotic oscillators. The aim of this project is show the various types of driven chaotic oscillator and propose mathematical model solutions using numerical methods. In the first part of this thesis are shown theory of chaos, history of chaos theory, chaotic systems and chaos quantifiers. Next is numerical analysis of differential equations second order by Runge-Kutta fourth order method. Next part contains circuit blocks and synthesis of oscillators. In next part are defined all types of oscillators. Parameters of analysis, equations, circuits and simulations are defined for each type of driven chaotic oscillators. In each subchapter is design electrical circuit. This circuit is simulated and some of them realized.

"The purpose of this paper will be to develop a semi-automatic process for numerical solution of ordinary differential equations, associated commonly with the names of Runge and Kutta, which by its essential features can be characterized as an iterative 'method of successive substitutions'"--Introduction. / Typescript. / "August, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: H. C. Griffith, Professor Directing Paper. / Includes bibliographical references (leaf 48).

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In this dissertation, we investigate the Riccati diagonal stability and explore some extensions of this notion. Riccati diagonal stability plays an important role in the stability analysis of linear time-delay systems. It is known that if a linear time-delay system is Riccati diagonally stable then it admits a diagonal Lyapunov-Krasovskii functional. The existence of such a functional implies the asymptotic stability of the linear time-delay system. This diagonal stability problem has other applications in applied areas such as physical sciences and population dynamics. We also study the Lyapunov diagonal stability, which has a clear connection to the Riccati diagonal stability. Using a separation theorem, we first provide new proofs for some existing results on the Lyapunov-type diagonal stability. We also construct a new, shorter, and more transparent proof for a well-known result by Kraaijevanger that gives explicit conditions for the Lyapunov diagonal stability on matrices in $\mathbb{R}^{3 \times 3}$. In addition, we give several necessary and sufficient conditions for matrices in $\mathbb{R}^{3 \times 3}$ to be Lyapunov diagonally stable. Furthermore, we present an extension of the so-called Riccati diagonal stability to the Riccati $\alpha$-scalar stability. We derive two new characterizations regarding the Riccati $\alpha$-scalar solution of the Riccati matrix inequality so as to expand and broaden the relevant existing results. We also generalize this notion to consider a common $\alpha$-scalar solution for a family of Riccati matrix inequalities. We shall refer to this new generalization as common Riccati $\alpha$-scalar stability. As an application for the main results, we further explore families of block triangular matrices. Finally, motivated by recent developments, we formulate the problem of Riccati $\alpha$-stability. We present a necessary and sufficient condition for this type of stability and study the connection between Riccati $\alpha$-stability of a pair of $\alpha$-block matrices and Riccati stability of the diagonal block pairs. Moreover, we generalize the Riccati $\alpha$-stability by considering a family of pairs of $\alpha$-block matrices and give a new characterization for this new case.

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