New numerical algorithms are devised for high-order, efficient quadrature in domains arising from the intersection of a hyperrectangle and a manifold implicitly defined by level sets. By casting the manifold locally as the graph of a function (implicitly evaluated through a recurrence relation for the zero level set), a recursion stack is set up in which the interface and integrand information of a single dimension after another will be treated. Efficient means for the resulting dimension reduction process are developed, including maps for identifying lower-dimensional hyperrectangle facets, algorithms for minimal coordinate-flip vertex traversal, which, together with our multilinear-form-based derivative approximation algorithms, are used for checking a proposed integration direction on a facet, as well as algorithms for detecting interface-free sub-hyperrectangles. The multidimensional quadrature nodes generated by this method are inside their respective domains (hence, the method does not require any extension of the integrand) and the quadrature weights inherit any positivity of the underlying single-dimensional quadrature method, if present. The accuracy and efficiency of the method are demonstrated through convergence and timing studies for test cases in spaces of up to seven dimensions. The strengths and weaknesses of the method in high dimensional spaces are discussed. / A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Summer Semester 2017. / July 17, 2017. / approximate integration, C, C++ implementation, cubature quadrature, implicit manifold, level set, recursive dimension reduction algorithm / Includes bibliographical references. / Mark Sussman, Professor Directing Dissertation; Tomasz Plewa, University Representative; Nick Moore, Committee Member; Giray Okten, Committee Member.

We present a new algorithm for computing the Perron root of a nonnegative irreducible matrix. The algorithm is formulated by combining a reciprocal of the well known Collatz's formula with a special inverse iteration algorithm discussed in [10, Linear Algebra Appl., 15 (1976), pp 235-242]. Numerical experiments demonstrate that our algorithm is able to compute the Perron root accurately and faster than other well known algorithms; in particular, when the size of the matrix is large. The proof of convergence of our algorithm is also presented.

Time reversal is a general technique in wave propagation in inhomogeneous media when a signal is recorded at points of a device called time reversal mirror, gets time reversed and radiated back in the medium. The resulting field has a property of refocusing. Time reversal in acoustics has been extensively studied both experimentally and theoretically. In this thesis we consider the problem of time reversal of electromagnetic waves in inhomogeneous layered media. We use Markov process model for the medium parameters which allows us to exploit diffusion approximation theorem. We show that the field generated by the time reversal mirror focuses at a point of initial source inside of the medium. The size of the focusing spot is of the kind that it is smaller than the one that would be obtained if the medium were homogeneous meaning that the super resolution phenomenon is observed.

This study analyzes two robust failure detection algorithms and applies the algorithms to three power system models. An optimal test signal to distinguish between a failure model and a normal model is calculated using the two algorithms. Advantages and disadvantages of each algorithm, Direct Optimization (DO) and Constrained Control (CC), are discussed. DO uses complex software (Sparse Optimal Control Software by The Boeing Corporation) to solve the necessary and boundary conditions of the optimization problem directly. CC utilizes free software (SciLab by Inria, Enpc.) to solve a two-point boundary value problem based on the necessary and boundary conditions of the optimization problem. Both algorithms yield similar signals, but DO is faster and more accurate yet requires expensive software. CC is not as robust, but can be run on free software and does not need as much fine tuning as the DO algorithm. Examples presented are two DC motor models and a linearized gas turbine model.

In recent years people have seen a rapidly growing market for credit derivatives. Among these traded credit derivatives, a growing interest has been shown on multi-name credit derivatives, whose underlying assets are a pool of defaultable securities. For a multi-name credit derivative, the key is the default dependency structure among the underlying portfolio of reference entities, instead of the individual term structure of default probabilities for each single reference entity as in the case of single-name derivative. So far, however, default dependency modeling is still the most demanding open problem in the pricing of credit derivatives. The research in this dissertation is trying to model the default dependency with aid of perturbation method, which was first proposed by Fouque, Papanicolaou and Sircar (2000) as a powerful tool to pricing options under stochastic volatility. Specifically, after a theoretic result regarding the approximation accuracy of the perturbation method and an application of this method to pricing American options under stochastic volatility by Monte Carlo approach, a multi-dimensional Merton model under stochastic volatility is studied first, and then the multi-dimensional generalization of the first-passage model under stochastic volatility comes next, which is then followed by a copula perturbed from the standard Gaussian copula.

Inhibin is one of several hormones which collectively regulate the human female reproductive endocrine system. In recent years, physiologists have been able to separately assay two forms of inhibin. We begin by discussing the physiology and endocrinology that underlie the menstrual cycle. Then, we fit an existing delay differential equation model of the human menstrual cycle to new data describing average hormone levels of young women throughout the menstrual cycle. Next, we consider the existence and stability of equilibrium and periodic solutions, analyze bifurcations, and perform a sensitivity analysis. We compare and contrast these results to previously published results based on the same model but a different data set. Because the introduction of exogenous hormones, whether pharmacological or environmental, can have significant effects on the menstrual cycle, we model the effects of these external hormones. Next, we introduce an expanded model that more fully accounts for the actions of both forms of inhibin. We optimize the parameters to fit the data, then discuss the equilibrium and periodic solutions, bifurcations, and sensitivity for this new model and parameter set. Next, we use the model to consider the effects of exogenous pharmacological hormones. Finally, we discuss the effects on the menstrual cycle of age-related hormone production changes that occur during the peri-menopausal years. This leads to a second parameter set, with only a few variations from the first one, for which the model approximates the monthly hormonal fluctuations for a woman around age 40. In closing, we discuss future possibilities for this line of research.

The purpose of the research has been to develop a class of new finite-element methods, called immersed-interface finite-element methods, to solve elliptic and elasticity interface problems with homogeneous and non-homogeneous jump conditions. Simple non-body-fitted meshes are used. Single functions that satisfy the same non-homogeneous jump conditions are constructed using a level-set representation of the interface. With such functions, the discontinuities across the interface in the solution and flux are removed; and equivalent elliptic and elasticity interface problems with homogeneous jump conditions are formulated. Special finite-element basis functions are constructed for nodal points near the interface to satisfy the homogeneous jump conditions. Error analysis and numerical tests are presented to demonstrate that such methods have an optimal convergence rate. These methods are designed as an efficient component of the finite-element level-set methodology for fast simulation of interface dynamics that does not require re-meshing. Such simulation has been a powerful numerical approach in understanding material properties, biological processes, and many other important phenomena in science and engineering.

Smart material transducers are utilized in wide range of applications, including nanopositioning, fluid pumps, high accuracy, high speed milling, objects, vibration control and/or suppression, and artificial muscles. They are attractive because the resulting devices are solid-state and often very compact. However, the coupling of field or temperature tomechanical deformation, which makes these materials effective transducers, also introduces hysteresis and time-dependent behaviors that must be accommodated in device designs and models before the full potential of compounds can be realized. In this dissertation, we present highly efficient modeling techniques to characterize hysteresis and constitutive nonlinearities in ferroelectric, ferromagnetic, and shape memory alloy compounds and model inversion techniques which permit subsequent linear control designs.

We focus on the problem of clustering large textual data sets. We present 3 well-known clustering algorithms and suggest enhancements involving dimension reduction. We propose a novel method of algorithm aggregation that allows us to use many clustering algorithms at once to arrive on a single solution. This method helps stave off the inconsistency inherent in most clustering algorithms as they are applied to various data sets. We implement our algorithms on several large benchmark data sets.

Let A be an algebraic system with product a*b between elements a and b in A. It is of interest to compare the solvable length t with other invariants, for instance size, order, or dimension of A. Thus we ask, for a given t what is the smallest n such that there is an A of length t and invariant n. It is this problem that we consider for associative algebras, matrix groups, and Lie algebras. We consider A in each case to be subsets of (strictly) upper triangular n by n matrices. Then the invariant is n. We do these for the associative (Lie) algebras of all strictly upper triangular n by n matrices and for the full n by n upper triangular unipotent groups. The answer for n is the same in all cases. Then we restrict the problem to a fixed number of generators. In particular, using only 3 generators and we get the same results for matrix groups and Lie algebras as for the earlier problem. For associative algebras with 1 generator we also get the same result as the general associative algebra case. Finally we consider Lie algebras with 2 generators and here n is larger than in the general case. We also consider the problem of finding the dimension in the associative algebra, the general, and 3 generator Lie algebra cases.