Title: Preconditioning for Accurate Solutions of the Biharmonic Eigenvalue Problem

Abstract: Solving ill-conditioned systems poses two basic problems: convergence and accuracy. Preconditioning can overcome slow convergence, but this is only practical if the preconditioned system can be formed sufficiently accurately. In fact, for a fourth order operator, existing eigenvalue algorithms may compute smaller eigenvalues with little or no accuracy in standard double precision. In this talk, we combine standard matrix eigenvalue solvers with an accurate preconditioning scheme in order to compute the smallest eigenvalue of the biharmonic operator to machine precision in spite of ill-conditioning. The results on various domains are compared with the best known computations from the literature to demonstrate the accuracy and applicability of the method.

Title: Modeling RNA secondary structure with auxiliary information

Abstract: The secondary structure of an RNA sequence plays an important role in determining its function, but directly observing RNA secondary structure is costly and difficult. Therefore, researchers have developed computational tools to predict the secondary structure of RNAs. One of the most popular methods is the Nearest Neighbor Thermodynamic Model (NNTM). More recently, high-throughput data that correlates with the state of a nucleotide being paired or unpaired has been developed. This data, called SHAPE for `selective 2'-hydroxyl acylation analyzed by primer extension', has been incorporated as auxiliary information into the objective function of NNTM with the goal of improving the accuracy of the predictions. This type of prediction is referred to as SHAPE-directed RNA secondary structure modeling. The addition of auxiliary information usually improves the accuracy of the predictions of NNTM. This talk will discuss challenges in RNA secondary structure modeling using NNTM and will provide ideas for developing synthetic auxiliary information that can be incorporated into NNTM to improve the accuracy of the predictions.

Title:  Effects of Thermoregulation on Human Sleep Patterns: A Mathematical Model of Sleep–Wake Cycles with REM–NREM Subcircuit 

Abstract: In this paper we construct a mathematical model of human sleep–wake regulation with thermoregulation and temperature effects. Simulations of this model show features previously presented in experimental data such as elongation of duration and number of REM bouts across the night as well as the appearance of awakenings due to deviations in body temperature from thermoneutrality. This model helps to demonstrate the importance of temperature in the sleep cycle. Further modifications of the model to include more temperature effects on other aspects of sleep regulation such as sleep and REM latency are discussed.

Title: Epidemiological models examining two susceptible classes

Abstract: Be it the Ebola or Buruli ulcers, we are constantly informed about infectious diseases and the ramifications. We can combat infectious diseases using mathematics to gain insight into diseases dynamics and outbreaks. We will explore using two susceptible classes in epidemiological models. I concentrate on a model for Buruli Ulcers and briefly discuss two other disease models.

Buruli Ulcers is a debilitating disease induced by Mycobacterium ulcerans. The transmission mechanism is not known at this time, but the bacteria is known to live in natural water environments. To understand the role of human contact with water environments in the spread of this disease, we formulate a model to emphasize the interaction between humans and the pathogen in a water environment. Therefore, we included two susceptible classes with one having more exposure to the water environment than the other in our system of differential equations. This work gives insight into the importance of various components of the mechanisms for transmission dynamics.

Title: The Potential Role of Subclinical Infection in Outbreaks of Emerging Pathogens

Abstract: Many rare or emerging diseases exhibit different epidemioligical behaviors from

Title: Ubiquitous Doubling Algorithms, General Theory, and Applications

Abstract: Iterative methods are widely and indispensably used in numerical approximations. Basically, any iterative method is a rule that produces a sequence of approximations and with a reasonable expectation that newer approximations in the sequence are better. The goal of a doubling algorithm is to significantly speed up the approximation process by seeking ways to skip computing most of the approximations in the sequence but sporadically few, in fact, extremely very few: only the $2^i$-th approximations in the sequence, kind of like computing $\alpha^{2^i}$ via repeatedly squaring. However, this idea is only worthwhile if there is a much cheaper way to directly obtain the $2^i$-th approximation from the $2^{i-1}$-st one  than simply following the rule to generate every approximation between the $2^{i-1}$-st and $2^i$-th approximations in order to obtain the $2^i$-th approximation. Anderson (1978) had sought the idea to speed up the simple fixed point iteration for solving the discrete-time algebraic Riccati equation via repeatedly compositions of the fixed point iterative function. As can be imagined, under repeatedly compositions, even a simple function can usually and quickly turn into nonetheless a complicated and unworkable one. In the last 20 years or so in large part due to an extremely elegant way of formulation and analysis, the research in doubling algorithms thrived and continues to be very active, leading to numerical effective and robust algorithms not only for the continuous-time and discrete-time algebraic Riccati equations from optimal control that motivated the research  in the first place but also for $M$-matrix algebraic Riccati equations (MARE), structured eigenvalue problems, and other nonlinear matrix equations. But the resulting theory is somewhat fragmented and sometimes ad hoc. In this talk, we will seek to provide a general and coherent theory, discuss new highly accurate doubling algorithm for MARE, and look at several important applications.

Title:  Orthogonal Recurrent Neural Networks with Scaled Cayley Transform

Abstract: Recurrent Neural Networks (RNNs) are designed to handle sequential data but suffer from vanishing or exploding gradients.  Recent work on Unitary Recurrent Neural Networks (uRNNs) have been used to address this issue and in some cases, exceed the capabilities of Long Short-Term Memory networks (LSTMs).  We propose a simpler and novel update scheme to maintain orthogonal recurrent weight matrices without using complex valued matrices. This is done by parametrizing with a skew-symmetric matrix using the Cayley transform. Such a parametrization is unable to represent matrices with negative one eigenvalues, but this limitation is overcome by scaling the recurrent weight matrix by a diagonal matrix consisting of ones and negative ones.  The proposed training scheme involves a straightforward gradient calculation and update step. In several experiments, the proposed scaled Cayley orthogonal recurrent neural network (scoRNN) achieves superior results with fewer trainable parameters than other unitary RNNs.