The FANTOM consortium has generated a large gene expression dataset of different cell lines and tissue cultures using the single-molecule sequencing technology of HeliscopeCAGE. This provides a unique opportunity to investigate novel associations between gene expression over time and different cell types. Here, we create a MatLab wrapper for a powerful and computationally intensive set of statistics known as Maximal Information Coefficient, and then calculate this statistic for a large, comprehensive dataset containing gene expression of a variety of differentiating tissues. We then distinguish between linear and nonlinear associations, and then create gene association networks. Following this analysis, we are then able to identify clusters of linear gene associations that then associate nonlinearly with other clusters of linearity, providing insight to much more complex connections between gene expression patterns than previously anticipated.

Phase-field models are emerging as a promising strategy to simulate interfacial phenomena. Rather than tracking interfaces explicitly as done in sharp interface descriptions, these models use a diffuse order parameter to monitor interfaces implicitly. This implicit description, as well as solid physical and mathematical footings, allow phase-field models to overcome problems found by predecessors. Nonetheless, the method has significant drawbacks. The phase-field framework relies on the solution of high-order, nonlinear partial differential equations. Solving these equations entails a considerable computational cost, so finding efficient strategies to handle them is important. Also, standard discretization strategies can many times lead to incorrect solutions. This happens because, for numerical solutions to phase-field equations to be valid, physical conditions such as mass conservation and free energy monotonicity need to be guaranteed. In this work, we focus on the development of thermodynamically consistent algorithms for time integration of phase-field models. The first part of this thesis focuses on an energy-stable numerical strategy developed for the phase-field crystal equation. This model was put forward to model microstructure evolution. The algorithm developed conserves, guarantees energy stability and is second order accurate in time. The second part of the thesis presents two numerical schemes that generalize literature regarding energy-stable methods for conserved and non-conserved phase-field models. The time discretization strategies can conserve mass if needed, are energy-stable, and second order accurate in time. We also develop an adaptive time-stepping strategy, which can be applied to any second-order accurate scheme. This time-adaptive strategy relies on a backward approximation to give an accurate error estimator. The spatial discretization, in both parts, relies on a mixed finite element formulation and isogeometric analysis. The codes are available online and implemented in PetIGA, a high-performance isogeometric analysis framework.

The theory of nonlinear Thomson scattering is presented. A model for the scattered light is developed. The orthogonal polarizations of the second harmonic of the scattered light are examined. The predictions of the model are compared to measurements by our group in collaboration with the Extreme Light Laboratory at the University of Nebraska-Lincoln (UNL). The veracity of the theory and model are confirmed by comparison to the experimental data.

The research of this dissertation is formulated in two fields, i.e., the theoretical and computational studies of circumferential wrinkling on soft nanofibers and the swelling mechanics study of a bi-layered spherical hydrogel containing a hard core. Continuous polymer nanofibers have been massively produced by means of the low-cost, top-down electrospinning technique. As a unique surface instability phenomenon, surface wrinkling in circumferential direction is commonly observed on soft nanofibers in electrospinning. In this study, a theoretical continuum mechanics model is developed to explore the mechanisms of circumferential wrinkling on soft nanofibers under uniaxial stretching. The model is able to examine the effects of elastic properties, surface energy, and fiber radius on the critical axial stretch to trigger circumferential wrinkling and to discover the threshold fiber radius to initiate spontaneous wrinkling. In addition, nonlinear finite element method (FEM) is further adopted to predict the critical mismatch strain to evoke circumferential wrinkling in core-shell polymer nanofibers containing a hard core, as a powerful computational tool to simulate controllable wrinkling on soft nanofibers via co-electrospinning polymer nanofibers incorporated with nanoparticles as the core. The studies provide rational understanding of surface wrinkling in polymer nanofibers and technical approaches to actively tune surface morphologies of polymer nanofibers for particular applications, e.g. high-grade filtration, oil-water separation, polymer nanocomposites, wound dressing, tissue scaffolding, drug delivery, and renewable energy harvesting, conversion, and storage, etc. Furthermore, hydrogels are made of cross-linked polymer chains that can swell significantly when imbibing water and exhibit inhomogeneous deformation, stress, and, water concentration fields when the swelling is constrained. In this study, a continuum mechanics field theory is adopted to study the swelling behavior of a bi-layered spherical hydrogel containing a hard core. The problem is reduced into a two-point boundary value problem of a 2nd-order nonlinear ordinary differential equation (ODE) and solved numerically. Effects of material properties on the deformation, stress, and water concentration fields of the hydrogel are examined. The study offers a rational route to design and regulate hydrogels with tailorable swelling behavior for practical applications in drug delivery, leakage blocking, etc.

Baseband digital communication in electro-magnetic measurement while drilling (EM-MWD) systems is often corrupted by non-white surface noise. The inability to reliably decode the transmitted signals in a noisy environment limits the depth at which EM-MWD systems can operate. Correlation receivers, which are optimal in the presence of additive white Gaussian noise, can be sub-optimal in the presence of various types of field noise at different drilling sites. This thesis investigates the application of artificial neural networks (ANN) as communication receivers in EM-MWD baseband digital communication systems. The performances of various ANN architectures and training algorithms are studied and compared with conventional correlation receivers via computer simulations. Standard symbol error rate (SER) test results show that the NN receiver is able to adapt to site-specific noise and thus outperforms the traditional correlation receiver.

This thesis focuses primarily on generalizations and enhancements of the Wiener path integral (WPI) technique for stochastic response analysis and optimization of diverse nonlinear dynamic systems of engineering interest. Concisely, the WPI technique, which has proven to be a potent mathematical tool in theoretical physics, has been recently extended to address problems in stochastic engineering dynamics. Herein, the WPI technique has been significantly enhanced in terms of computational efficiency and versatility; these results are presented in Chapters 2-5. Specifically, in Chapter 2 a brief introduction to the standard WPI solution approach is outlined. In Chapter 3, a novel methodology is presented, which utilizes theoretical results from calculus of variations to extend the WPI for determining marginalized response PDFs of n-degree-of-freedom (n-DOF) nonlinear systems. The associated computational cost relates to the dimension of the PDF and is essentially independent from the dimension n of the system. In several commonly encountered cases, the aforementioned methodology improves the computational efficiency of the WPI by orders of magnitude, and exhibits a significant advantage over the commonly utilized Monte-Carlo-simulation (MCS). Moreover, in Chapter 4, an extension of the WPI technique is presented for addressing the challenge of determining the stochastic response of nonlinear dynamical systems under the presence of singularities in the diffusion matrix. The key idea behind this approach is to partition the original system into an underdetermined system of SDEs corresponding to a nonsingular diffusion matrix and an underdetermined system of homogeneous differential equations; the latter is treated as a dynamic constraint that allows for employing constrained variational/optimization solution methods. In Chapter 5, this approach is applied for the stochastic response analysis and optimization of electromechanical vibratory energy harvesters. Next, in Chapter 6, a technique from computational algebraic geometry has been developed, which is based on the concept of Gröbner basis and is capable of determining the entire solution set of systems of polynomial equations. This technique has been utilized to address diverse challenging problems in engineering mechanics. First, after formulating the WPI as a minimization problem, it is shown in Chapter 7 that the corresponding objective function is convex, and thus, convergence of numerical schemes to the global optimum is guaranteed. Second, in Chapter 8, the computational algebraic geometry technique has been applied to the challenging problem of determining nonlinear normal modes (NNMs) corresponding to multi-degree-of-freedom dynamical systems as defined in [1], and has been shown to yield improvements in accuracy compared to the standard treatment in the literature.

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This thesis presents analytic and computational advances in the prediction of perturbative nonlinear optical spectroscopies. The contributions of this thesis are packaged together in an open source, freely available piece of software called ultrafast spectroscopy suite (UFSS). It is designed to automatically simulate nonlinear optical spectroscopies for any phase-matching or phase-cycling condition, including finite pulse effects. UFSS includes an algorithm called the diagram generator (DG) that automates the process of writing out all of the Feynman diagrams that contribute to a desired phase-matching or phase-cycling condition, and includes all pulse overlap diagrams when relevant, paving the way toward automation of perturbative calculations. Further, many diagrams can be automatically combined into composite diagrams, giving an exponential decrease in computation time of high-order calculations. Composite diagrams even allow for the efficient study of Rabi oscillations as a function of pulse amplitude, by summing many orders of perturbation theory. The perturbative calculations are done using a novel algorithm presented in this thesis called Ultrafast Ultrafast spectroscopy (UF2). UF2 is an efficient method for determining diagrammatic contributions to spectra including arbitrary (whether analytical or experimentally measured) pulse shapes. It uses the speed of the fast Fourier transform to be as much as 500 times faster than direct propagation techniques for small model Hamiltonians (for Hamiltonian dimension of 100 or less). UF2 outperforms direct propagation techniques for a wide range of model systems, with the speed boost diminishing as the dimension of the model Hamiltonian increases. UF2 can predict spectra for any model system whose relevant Hilbert space that can be described using a finite basis and that can be diagonalized numerically, and users are free to specify their own model. UFSS includes a model generator that generates Hamiltonians and Liouvillians of vibronic systems, allowing users to easily simulate NLOSs for a wide range of model system parameters. UFSS is a fully functional piece of software for simulating any NLOS, to any desired order in perturbation theory.

Nonlinear optics is the study of interactions of materials with intense light beams made possible by the invention of laser. Arguably the most trivial but technologically most important nonlinear optical effect is the intensity-dependent nonlinear refraction: an intense light beam can temporarily and reversibly change the refractive index of a material. However, the changes to the refractive index of a material due to the presence of a strong laser beam are very weak---maximum on the order of $10^{-3}$---and tend to be a small fraction of the linear refractive index. It must be noted that at optical frequencies vacuum has a refractive index of 1 and glass has a refractive index of 1.5. Thus, one of the foundational assumptions of nonlinear optics is that the nonlinear optical changes to material properties are always a small perturbation to the linear response. In the 58-year history of nonlinear optics, one of the overarching themes of research has been to find ways to increase the efficiency of nonlinear interactions. This thesis is a collection of six manuscripts motivated by our experimental finding that at least in a certain class of materials the above long-standing view of nonlinear optics does not necessarily hold true. We have found that in a material with low refractive index, known as an epsilon-near-zero material or ENZ material, the nonlinear changes to the refractive index can be a few times larger than the linear refractive index, i.e. the nonlinear response becomes the dominant response of the material in the presence of an intense optical beam. We believe that the results presented in this thesis collectively make a convincing case that ENZ materials are a promising platform for nonlinear nano-optics.

Les programmes TV possède une structure qui, en général, est perdue quand les programmes sont diffusés. Les programmes qui ont été enregistrés via un enregistreur vidéo personnel ou disponibles via des services comme la TV à la demande, ne peuvent être visionnés que d’une façon linéaire. La navigation y est réalisée en utilisant les fonctions basiques d’avance/retour rapide. Dans ce contexte, la structuration automatique de programmes TV apporte une solution originale. En retrouvant la structure d’origine du programme, elle permet d’offrir aux utilisateurs des outils de navigation originaux. Elle peut également servir pour d’autres applications comme la construction des résumés vidéo, l’indexation et la recherche…Cette thèse s’intéresse ainsi à la structuration automatique des programmes TV. L’objectif est de retrouver automatiquement la structure d’origine d’un programme en déterminant le début et la fin de chaque partie qui le compose. L’approche proposée est complètement non-supervisée et adresse une large catégorie de programmes TV comme les jeux, les magazines, les journaux TV… Cette approche exploite les « séparateurs » qui sont de séquences courtes insérées dans les programmes pour en délimiter les différentes parties. Pour cela, une détection des récurrences audio et visuelles est réalisée sur un ensemble d’épisodes du même programme. Ces récurrences sont ensuite classées à l’aide d’arbres de décision pour en extraire les séparateurs. Les attributs utilisés pour la construction des arbres de décision porte sur la détection des applaudissements, la segmentation en scènes, la détection et le clustering des visages et des locuteurs. / TV programs have an underlying structure that is lost when these are broadcasted. The linear mode is the only available reading mode when viewing programs recorded using a Personal Video Recorder or through a TV-on-Demand service. The fast-forward/backward functions are the only available tools for browsing. In this context, program structuring becomes important in order to provide users with novel and useful browsing features. In addition to advanced browsing features, TV program structuring can also be used for summarization, indexing and querying, archiving, etc. This thesis addresses the problem of unsupervised TV program structuring. The idea is to automatically recover the original structure of the program by finding the start time of each part composing it. The proposed approach is completely unsupervised and addresses a large category of programs like TV games, magazines, news…. It is based on the detection of “separators” which are short audio/visual sequences that delimit the different parts of a program. To do so, audio and visual recurrences are first detected from a set of episodes of a same program. In order to extract the separators, the recurrences are then classified using decision trees. These are built based on attributes issued from techniques like applause detection, scenes segmentation, face and speaker detection and clustering.