University of Kentucky Applied Math Seminar


The Applied Math seminar has speakers twice a month during the school year. We generally meet in POT 745 from 11-noon. Past and upcoming speakers are listed below. If you would like to be added to the mailing list send an email to with "subscribe UKAPPLIEDMATH-L YourFirstName YourLastName" in the message body (not the subject line!). The email address you send this from is the one that will be subscribed to the list. If you are interested in speaking in the seminar please send an email to or

Academic Year 2018-19

  1. Nov 15, 2018
    CB 211 from 11:00am-12:00 pm
    Title: Mathematical deep learning for drug discovery
    Speaker: Guowei Wei, Michigan State University.

    Designing efficient drugs for curing diseases is of essential importance for the 21st century's life science. Computer-aided drug design and discovery has obtained a significant recognition recently. However, the geometric complexity of protein-drug complexes remains a grand challenge to conventional computational methods, including machine learning algorithms. We assume that the physics of interest of protein-drug complexes lies on low-dimensional manifolds or subspaces embedded in a high-dimensional data space. We devise topological abstraction, differential geometry reduction, graph simplification, and multiscale modeling to construct low-dimensional representations of biomolecules in massive and diverse datasets. These representations are integrated with various deep learning algorithms for the predictions of protein-ligand binding affinity, drug toxicity, drug solubility, drug partition coefficient and mutation induced protein stability change, and for the discrimination of active ligands from decoys. I will briefly discuss the working principle of various techniques and their performance in D3R Grand Challenges, a worldwide competition series in computer-aided drug design and discovery (

  2. Nov 1, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Mathematics for Breast Cancer Research: investigating the role of iron
    Speaker: Luis Sordo-Vieira, The Jackson Laboratory.

    Breast cancer cells are addicted to iron. The mechanisms by which malignant cells acquire and contain high levels of iron are not completely understood. Furthermore, other cell types in a tumor, such as immune cells, can either aid or inhibit cancer cells from acquiring high levels of iron. In order to shed light in the question of how iron affects breast cancer growth, we are applying mathematical tools including polynomial dynamical systems over finite fields and 3D multiscale mathematical modeling. In this talk we will survey how mathematics is aiding in understanding the mechanisms of this addictive iron behavior of malignant cells, and present some preliminary work.

  3. Sept 27, 2018
    POT 745 from 11:00am-12:00 pm
    Title: A Mathematical Model for the Force and Energetics in Competitive Running
    Speaker: Margaret Grogan, University of Kentucky.

    Competitive running has been around for thousands of years and many people have wondered what the optimal form and strategy is for running a race. In his paper, Behncke develops a simple mathematical model that focuses on the relationships and dynamics between the forces and energetics at play in order to find an optimal strategy for racing various distances. In this talk, I will describe the biomechanics, energetics, and optimization of running in Behncke's model and present his findings. Note: you do not have to like running to come to this talk :)

  4. Sept 13, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Preconditioning for Accurate Solutions of the Biharmonic Eigenvalue Problem
    Speaker: Kasey Bray, University of Kentucky.

    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.

Academic Year 2017-18

  1. April 26, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Finding cycles in discrete dynamical systems
    Speaker: Mihai Tohaneanu, University of Kentucky.

    Discrete dynamical systems often exhibit chaotic behavior, and as a result finding cycles can be computationally expensive. I present a new approach to this problem, based on adding a nonlinear feedback that stabilizes the cycles. We are then able to find cycles numerically in polynomial time. The main theoretical new insight is casting the problem in the language of complex analysis, and finding new complex polynomials that generalize work of Ted Suffridge that optimize the number of steps one needs in order to stabilize the system. This is joint work with D. Dmitrishin, A. Khamitova and A. Stokolos.

  2. April 26, 2018
    POT 745 from 1:00pm-2:00 pm
    Title: Complex unitary recurrent neural networks using scaled Cayley transform
    Speaker: K.D.Gayan Maduranga, University of Kentucky.

    Recurrent neural networks (RNNs) have been successfully used in wide range of sequential problems. Despite this success, RNNs suffer from the vanishing or exploding gradients problem. One recent method ''scaled Cayley orthogonal recurrent neural network'' (scoRNN) addresses this issue by maintaining an orthogonal recurrent weight matrix by parametrizing a skew-symmetric matrix through a scaled Cayley transform. The initial implementation of scoRNN used an orthogonal recurrent matrix and we extend the idea to the complex case using unitary matrices. We discuss the advantage the complex scoRNN has over the traditional scoRNN and implementation issues.

  3. April 19, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Effects of Thermoregulation on Human Sleep Patterns: A Mathematical Model of Sleep-Wake Cycles with REM-NREM Subcircuit
    Speaker: Alicia Prieto Langarica, Youngstown State University.

    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.

  4. April 12, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Disease ecology meets economics
    Speaker: Calistus Ngonghala, University of Florida

    Understanding why some human populations remain extremely poor despite current development trends around the world remains a mystery to the natural, social and mathematical sciences. The poor rely on their immediate natural environment for subsistence and suffer from high burdens of infectious diseases. We present a general framework for modeling the ecology of poverty and disease, focusing on infectious diseases and renewable resources. Interactions between these ecological drivers of poverty and economics create reinforcing feedbacks resulting in three possible development regimes: 1) globally stable wealthy/healthy development, 2) globally stable unwealthy/unhealthy development, and 3) bistability. We show that the proportion of parameters leading to poverty is larger than that resulting in healthy/wealthy development; bistability consistently emerges as a general property of generalized disease-economic systems and that the systems under consideration are most sensitive to human disease parameters. The framework highlights feedbacks, processes and parameters that are important to measure in future studies of development, to identify effective and sustainable pathways out of poverty.

  5. April 5, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Epidemiological models examining two susceptible classes
    Speaker: Christina Edholm, University of Tennessee

    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.

  6. March 29, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Investigating the structure of Earth's interior
    Speaker: Keely O'Farrell, University of Kentucky.

    This talk will focus on the fluid dynamics of Earth and planetary mantles (interiors) and their surface manifestations. By necessity, convection in planetary mantles is largely studied using numerical models on supercomputers, though the right parameter range is still often out of reach. In order to solve the equations governing fluid dynamics inside the Earth, we need to know about the velocity, temperature density and general structure (such as viscosity) of the interior. Over the past few decades, much work has been done to constrain the viscosity structure of the Earth's mantle using inverse techniques, viscoelastic modelling and post-glacial rebound data. Variations in the Earth's gravitational potential anomalies (geoid) provide constraints on the density structure in the mantle. Seismic tomography can be used to investigate radial viscosity variations on instantaneous flow models. By specifying a possible viscosity structure and predicting a synthetic geoid, we can compare with the observed geoid to see how well our viscosity structure matches the real Earth. Examining over 50 tomographic models we found 2 possible profiles for the viscosity structure inside the Earth.

  7. March 22, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Simulating Within-Vector Generation of the Malaria Parasite Diversity
    Speaker: Olivia Prosper, University of Kentucky.

    Plasmodium falciparum, the malaria parasite causing the most severe disease in humans, undergoes an asexual stage within the human host, and a sexual stage within the vector host, Anopheles mosquitoes. Because mosquitoes may be superinfected with parasites of different genotypes, this sexual stage of the parasite life-cycle presents the opportunity to create genetically novel parasites. To investigate the role that mosquitoes' biology plays on the generation of parasite diversity, which introduces bottlenecks in the parasites' development, we first constructed a stochastic model of parasite development within-mosquito, generating a distribution of parasite densities at five parasite life-cycle stages: gamete, zygote, ookinete, oocyst, and sporozoite, over the lifespan of a mosquito. We then coupled a model of sequence diversity generation via recombination between genotypes to the stochastic parasite population model. Our model framework shows that bottlenecks entering the oocyst stage decrease diversity from the initial gametocyte population in a mosquito's blood meal, but diversity increases with the possibility for recombination and proliferation in the formation of sporozoites. Furthermore, when we begin with only two distinct parasite genotypes in the initial gametocyte population, the probability of transmitting more than two unique genotypes from mosquito to human is over 50% for a wide range of initial gametocyte densities.

  8. March 1, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Model-dependent and model-independent control of biological network models
    Speaker: Jorge G. T. Zanudo, Dana-Farber Cancer Institute and Broad Institute.

    Network models of intracellular signaling and regulation are ubiquitous in systems biology research because of their ability to integrate the current knowledge of a biological process and test new findings and hypotheses. An often asked question is how to control a network model and drive it towards its dynamical attractors (which have been found to be identifiable with phenotypes or stable patterns of activity of the modeled system), and which nodes and interventions are required to do so. In this talk, we will introduce two recently developed network control methods -feedback vertex set control and stable motif control- that use the graph structure of a network model to identify nodes that drive the system towards an attractor of interest (i.e., nodes sufficient for attractor control). Feedback vertex set control makes predictions that apply to all network models with a given graph structure and stable motif control makes predictions for a specific model instance, and this allows us to compare the results of model-independent and model-dependent network control. We illustrate these methods with various examples and discuss the aspects of each method that makes its predictions dependent or independent of the model.

  9. February 22, 2018
    POT 745 from 11:00am-12:00 pm
    Title: The Potential Role of Subclinical Infection in Outbreaks of Emerging Pathogens
    Speaker: Nourridine Siewe, NIMBIOS.

    Many rare or emerging diseases exhibit different epidemioligical behaviors from outbreak to outbreak, leaving it unclear how to best characterize the relevant facets that could be exploited for outbreak mitigation/control. Some studies have already proposed considering the role of active subclinical infections co-emerging and cocirculating as part of the process of emergence of a novel pathogen. However, consideration of the role of subclinical infections in emerging disease dynamics have usually avoided considering the full set of possible influences. Most recently, the Ebola outbreak 2014 seems to fit all the criteria for possible involvement of subclinical circulation. We argue that an understanding of the potential mechanism for diversity in observed epidemiological dynamics may be of considerable importance in understanding and preparing for outbreaks of novel and/or emerging diseases.

  10. February 15, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Ubiquitous Doubling Algorithms, General Theory, and Applications
    Speaker: Ren-Cang Li, University of Texas at Arlington.

    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.

  11. February 1, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Modeling RNA secondary structure with auxiliary information
    Speaker: David Murrugarra, University of Kentucky.

    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.

  12. January 18, 2018
    POT 745 from 11:00am-12:00 pm
    Title: Spatial Dynamics of Vector Borne Diseases
    Speaker: Omar Saucedo, Mathematical Biosciences Institute.

    Vector-borne diseases affects approximately 1 billion people and accounts for 17% of all infectious diseases. With travel becoming more frequent across the global, it is important to understand the spatial dynamics of vector-borne diseases. Host movement plays a key part on how a disease can be distributed as it enables a pathogen to invade a new environment, and helps the persistence of a disease in locations that would otherwise be isolated. In this talk, we will explore how spatial heterogeneity combines with mobility network structure to influence vector-borne disease dynamics.

  13. November 30, 2017
    FPAT 253 from 2:00pm-3:00 pm
    Title: Orthogonal Recurrent Neural Networks with Scaled Cayley Transform
    Speaker: Kyle Helfrich, University of Kentucky

    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.

  14. November 14, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Dynamic Programming in Secondary Structure Inference
    Speaker: Devin Willmott, University of Kentucky

    Given an RNA sequence, secondary structure inference is the problem of predicting that sequence's base pairs. A variety of methods for this problem exist; among the most popular are minimum free energy (MFE) methods, which assign each possible secondary structure an energy based on the presence or absence of various substructures, with negative energy structures being more likely to occur naturally. These methods then use dynamic programming to predict the lowest free energy structure(s) efficiently. We will give an introduction to dynamic programming, talk about why it is necessary for approaching this problem efficiently, and discuss some of the shortcomings of the method. If time permits, we will also talk about connections to machine learning methods for secondary structure prediction.

  15. October 19, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Computational Polypharmacology: A Machine Learning Approach
    Speaker: Sally Ellingson, UK Division of Biomedical Informatics

    Drug discovery is a lengthy, expensive, and sometimes fatal process. It is also an extremely difficult task to perform with a full understanding of experimental results. Drugs are studied in test tubes which lack a realistic in vivo environment and in animal models having limited validity for human conditions. Even when new drugs pass screening experiments with no red flags, they fail during human clinical trials after a great amount of time and money has been invested. Thus, an economic burden is created that eventually must be recuperated with the few drugs that do pass FDA approval. Computational methods that consistently improve predictive accuracy over laboratory and animal testing for the entire human proteome and huge chemical space of potential drugs could revolutionize pharmaceutical research and development. The utilization of such computational tools will increase the return on future investments in health-related research and provide access to new, better understood therapies. The state-of-the-art in many computational methodologies include machine learning approaches. In our digitalized, data-driven world, there is a wealth of knowledge available that is beyond the processing power of an individual researcher or even team of researchers. The goal of my work is to improve the prediction of novel drug safety and efficacy by increasing the accuracy of predicting polypharmacological networks, investigating how drugs interact with the entire proteome. We integrate traditional computational simulations of protein and drug interactions (such as the efficient molecular docking calculation), cheminformatics features of drug-like molecules, and features describing individual proteins to improve the prediction of drug and protein binding. Each component investigated provides some level of predictive utility in isolation. For example, I have seen in my own work that a small number of drug features calculated from current cheminformatics programs can identify active compounds for a given protein with greater than 99% accuracy. These same drug features have been used in machine learning models in combination with docking scores to rescore interactions with one candidate drug to multiple proteins. The individual components of a molecular docking scoring function can be used as features in a machine learning model to greatly improve the accuracy of identifying active compounds in models specific for one protein. From a different perspective, protein features have been used in machine learning models to predict the druggability of a protein. The hypothesis of this work is that the combination of all these components can be used in one model that would vastly improve the accuracy of predicting the effects of new proteins and classes of drugs. Presented here is a first step of showing that it can be done for a class of functionally related proteins (kinases). Kinases have been chosen to study because kinase inhibitors are the largest class of new cancer therapies and selectively inhibiting a kinase is difficult due to their high sequence similarity, making off-target interactions with kinases a common cause of adverse drug reactions.

  16. October 5, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Application of Orthogonal Polynomials and the Euclidean Algorithm to Interpolation and Cubature
    Speaker: Larry Harris, University of Kentucky

    Numbers \(h_0 > h_1 > \cdots > h_m\) are alternation points for corresponding orthogonal polynomials \(p_0, p_1,\ldots, p_m\) if \[ p_{m-j}(h_n) = (-1)^n p_j(h_n),\quad 0\leq n,j\leq m. \] For example, the Chebyshev points \(h_n = \cos(n\pi/m)\), \(0 \leq n \leq m\) are alternation points for the Chebyshev polynomials \(T_0,\ldots, T_m\). We show that any decreasing numbers are alternation points for some corresponding orthogonal polynomials. This is applied to produce Lagrange polynomials and cubature formulas for nodes in \(R^2\) whose coordinates are even and odd pairs of points from a finite decreasing sequence.

  17. September 14, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Radiative transport and optical tomography
    Speaker: Francis Chung, University of Kentucky

    Optical tomography is the process of reconstructing the optical parameters of the inside of an object from measurements taken on the boundary. This problem is hard if light inside the object is scattered -- if it bounces around a lot and refuses to travel in straight lines. To solve optical tomography problems, we need a mathematical model for light propagation inside a scattering medium. In this talk I'll give a brief introduction to one such model -- the radiative transport model -- and talk a little bit about its behavior and its implications for optical tomography.

  18. August 31, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Preconditioning for Accurate Solutions of Linear Systems and Eigenvalue Problems
    Speaker: Qiang Ye, University of Kentucky

    This paper develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditioner M for an ill-conditioned linear system Ax=b, we show that, if the inverse of the preconditioner can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse-equivalent accuracy is introduced to describe higher accuracy that is achieved and an error analysis will be presented. As an application, we use the preconditioning approach to accurately compute a few smallest eigenvalues of certain ill-conditioned matrices. Numerical examples are presented to illustrate the error analysis and the performance of the methods.

Academic Year 2016-17

  1. April 20, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Two-Dimensional PCA with F-Norm Minimization
    Speaker: Jing Wei, University of Kentucky

    Master's Talk.
    Two-dimensional principle component analysis (2DPCA) has been widely used for face image representation and recognition. But it is sensitive to the presence of outliers. To alleviate this problem, we propose a novel robust 2DPCA, namely 2DPCA with F-norm minimization (F-2DPCA), which is intuitive and directly derived from 2DPCA. In F-2DPCA, distance in spatial dimensions (attribute dimensions) is measured in F-norm, while the summation over different data points uses 1-norm. Thus it is robust to outliers and rotational invariant as well. To solve F-2DPCA, we propose a fast iterative algorithm, which has a closed-form solution in each iteration, and prove its convergence. Experimental results on face image databases illustrate its effectiveness and advantages.

  2. April 13, 2017
    POT 245 from 11:00am-12:00 pm
    Title: Theory and Application of a Direct Solution Algorithm for Large Dense Matrices of Boundary Element Methods
    Speaker: Robert John Thomas, University of Kentucky

    Master's Talk
    Subject Paper: Martinsson, and Rokhlin. "A Fast Direct Solver for Boundary Integral Equations in Two Dimensions." Journal of Computational Physics 205.1 (2005): 1-23. Web. ISSN: 0021-9991 ; DOI: 10.1016/ In computational science and engineering, the numerical solution of partial differential equations is effected through the solution of extremely large linear systems. Finite element and finite difference methods give rise to sparse matrices that admit iterative solution techniques. Acoustic and electromagnetic scattering problems, however, are often better approached via boundary element methods. These result in huge dense matrices that would be prohibitively expensive to solve conventionally. The subject paper details a method to construct the matrix inverse directly. The nature of the boundary integrals causes the system matrix to exhibit rank deficiency of blocks further removed from the diagonal. A modified QR algorithm from the literature both reveals the rank and approximates the nullspace basis of such blocks. An algorithm based on the Schur complement is then applied iteratively, inverting selected pivot blocks. The approach is extended to a hierarchical application reminiscent of Greengard and Rokhlin's Fast Multipole Method. This Master's Degree examination talk will present the theory of the key elements of the method, as well as the performance metrics of the derived algorithms. A sample implementation with numerical results will also be described.

  3. March 30, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Master's talk
    Speaker: Kehelwala Dewage Maduranga, University of Kentucky

    This master's talk will present the following paper: Theory of Inexact {Krylov} Subspace Methods and Applications to Scientific Computing Valeria Simoncini and Daniel B. Szyld SIAM Journal on Scientific Computing, 25, 454-477, 2003.

  4. March 9, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Algebraic Statistics Applications in Epidemiology
    Speaker: Luis Garcia Puente, Sam Houston State University

    Interactions between single nucleotide polymorphisms (SNPs) and complex diseases have been an important topic throughout epidemiological studies. Previous studies have mostly focused on gene variables at a single locus. In this talk, I will discuss a focused candidate gene study to test the interaction of multiple SNPs with the risk of different types of cancer. We will exemplify the fact that traditional asympotic results in statistical analysis do not apply in our setting. This is due mainly to the fact that we have a relatively small fixed data set. In our work we develop a new statistical approach using techniques from the field of algebraic statistics. Algebraic statistics focuses on mathematical aspects of statistical models, where algebraic, geometric and combinatorial insights can be useful to study behavior of statistical procedures. Using the R package algstat, developed by Kahle, Garcia Puente, and Yoshida, we implemented an algebraic statistics method that can test for independence between several variables and the desease. We applied our methods to the study of gene-gene interaction on cancer data obtained from the European case-control study Gen-Air extending previous work by Ricceri, Fassino, Matullo, Roggero, Torrente, Vineis, and Terracini.

  5. March 2, 2017
    POT 745 from 11:00am-12:00 pm
    Title: Tallgrass Prairie Ecosystem Restoration: Modeling the Impact of the Conservation Reserve Program
    Speaker: Anna Mummert, Marshall University

    The tallgrass prairie ecosystem has been reduced to a fraction of its original extent, due to rapid conversion to other land use types, especially agricultural and urban. Restoration is a relatively new process to convert agricultural land back to communities dominated by native vegetation, including prairies. The most notable restoration project for prairies is the Conservation Reserve Program (CRP) administered by the USDA Farm Service Agency. We develop a compartmental model for the Midwestern tallgrass prairie ecosystem, incorporating the impact of human population on land use changes. Restoration via participation in CRP is included. Historical data is used to determine model parameter ranges. Local and global sensitivity analyses are performed. Our findings emphasize the importance of increasing incentives for CRP enrollment as a means to restoring the tallgrass prairie ecoregion.

  6. February 23, 2017
    POT 745 from 11:00am-12:00 pm
    Title: The Inverse q-Numerical Range Problem and Connections to the Davis-Wielandt Shell and the Pseudospectra of a Matrix
    Speaker: Russell Carden, University of Kentucky

    Numerical ranges and related sets provide insights into the behavior of iterative algorithms for solving systems of equations and computing eigenvalues. Inverse numerical range problems attempt to enhance these insights. We generalize the inverse numerical range problem, as proposed by Uhlig, to the inverse $q$-numerical range problem, and propose an algorithm for solving the problem that relies on convexity. To determine an approximation to the boundary of the $q$-numerical range, as needed by our algorithm, we must approximate the top of the Davis-Wielandt shell, a generalization of the numerical range. We found that the Davis-Wielandt shell is in a sense conjugate to the the extreme singular values of the resolvent of a matrix. Knowing the Davis-Wielandt shell allows for the approximation of the $q$-numerical range, the pseudospectra and the Davis-Wielandt shell for any allowed M\"{o}bius transformation of a matrix. We provide some examples illustrating these connections, as well as how to solve the inverse $q$-numerical range problem.

  7. February 16, 2017
    POT 745 from 11:00am-12:00 pm
    Title: RNA Secondary Structure Inference with Recurrent Neural Networks
    Speaker: Devin Willmott, University of Kentucky

    RNA secondary structure inference is the problem of taking an RNA sequence and predicting which elements of the sequence are paired together. We will begin by converting the problem into a mathematically palatable form, and then look at some currently popular methods for inferring RNA secondary structure. Our work centers around the comparison of two methods that work with sequential data: hidden Markov models (HMMs) and recurrent neural networks (RNNs). We will discuss some of the particular strengths and weaknesses of each in the context of RNA secondary structure inference, see some preliminary results of each method's application to the problem, and (if time permits) talk about future research directions that exploit the combinatorial structure of RNA.

  8. February 9, 2017
    POT 745 from 11:00am-12:00 pm
    Title: A quantitative comparison of quarantine and symptom monitoring
    Speaker: Lauren Childs, Virginia Tech

    Quarantine and symptom monitoring of contacts with suspected exposure to an infectious disease are key interventions for the control of emerging epidemics; however, there does not yet exist a quantitative framework for comparing the control performance of each. Here, we use an agent-based branching model of seven case study diseases to show how the choice of intervention is influenced by the natural history of the infectious disease, its inherent transmissibility, and the intervention feasibility in the particular healthcare setting. We use this information to identify the most important characteristics of the disease and setting that need to be characterized for an emerging pathogen in order to make an informed decision between quarantine and symptom monitoring.

  9. December 8, 2016
    POT 745 from 10:30am-11:30 am
    Title: Accurately Computing Eigenvalues of Extremely Ill-conditioned Matrices, with an Application to the Biharmonic Operator
    Speaker: Kasey Bray, University of Kentucky

    We are primarily concerned with computing smaller eigenvalues of large, extremely ill-conditioned matrices. After discussing where the standard algorithms fail to compute such eigenvalues with any accuracy, we offer a solution to the problem for diagonally dominant matrices. We will then apply this solution to accurately compute an eigenvalue of the biharmonic operator on the unit circle.

  10. December 1, 2016
    POT 745 from 11am-noon
    Title: Optical tomography on graphs
    Speaker: Jeremy Hoskins, University of Michigan

    Diffuse optical tomography is an imaging modality frequently used in imaging biomedical systems. Here we discuss a discrete analog defined on graphs, which we call discrete diffuse optical tomography (DDOT). The goal of DDOT is to recover a vertex potential from boundary measurements. In this talk, we present a novel method for solving the inverse problem associated with DDOT, proving necessary conditions for recovery. Finally, we show how to modify our method to incorporate additional information on the structure of the potential and multifrequency measurements.

  11. November 17, 2016
    POT 745 from 11am-noon
    Title: Applications of Singular Value Decomposition to cryptography and privacy
    Speaker: Luis Sordo Vieira, University of Kentucky

    There have been recent attempts to encrypt images and text using the singular Value decomposition of a matrix. We talk about some of these protocols and results and possible benefits. We also mention some protocols to preserve privacy in data mining. We will quickly overview SVD in the beginning.

  12. November 3, 2016
    POT 745 from 11am-noon
    Title: Structural and Functional Characterization of Expected and Aberrant Metal Ion Coordination in Proteins
    Speaker: Hunter Moseley, University of Kentucky

    Metalloproteins bind and utilize metal ions for a variety of biological purposes. Due to the ubiquity of metalloprotein involvement throughout these processes across all domains of life, how proteins coordinate metal ions for different biochemical functions is of great relevance to understanding the implementation of these biological processes. Towards these ends, we have improved our methodology for structurally and functionally characterizing metal binding sites in metalloproteins. Our new ligand detection method is statistically much more robust, producing estimated false positive and false negative rates of ~0.11% and ~1.2%, respectively. Additional improvements expand both the range of metal ions and their coordination number that can be effectively analyzed. Also, the inclusion of many additional quality control filters has significantly improved structure-function Spearman correlations as demonstrated by rho values greater than 0.90 for several metal coordination analyses and even one rho value above 0.95. Also, improvements in bond-length distributions have revealed bond-length modes specific to chemical functional groups involved in multidentation. Using these improved methods, we analyzed all single metal ion binding sites with Zn, Mg, Ca, Fe, and Na ions in wwPDB, producing statistically rigorous results supporting the existence of both a significant number of unexpected compressed angles and subsequent aberrant metal ion coordination geometries (CGs) within structurally known metalloproteins. By recognizing these aberrant CGs in our clustering analyses, high correlations are achieved between structural and functional descriptions of metal ion coordination. Moreover, distinct biochemical functions are associated with aberrant CGs versus non-aberrant CGs.

  13. October 27, 2016
    POT 745 from 11am-noon
    Title: Spatial heterogeneity, host movement, and the transmission of mosquito-borne disease
    Speaker: Olivia Prosper, University of Kentucky

    The Ross-Macdonald framework, a suite of mathematical models for the transmission of mosquito-borne disease, made numerous simplifying assumptions including that transmission occurs in a homogeneous environment. Despite these assumptions, this modeling framework has been invaluable to the study of vector-borne disease and to informing public health policy. In recent years, more attention has been paid to the role of human movement in regions with spatially heterogeneous disease transmission. In this talk, I will introduce a metapopulation framework for vector-borne disease, based on the Ross-Maconald model, in which human movement connects discrete populations with different levels of malaria transmission. I will discuss properties of this model, compare these properties to the homogeneous case, and will discuss the implications for malaria control. Next, I will present some of the challenges that arise when linking this theoretical framework to a real-world problem. Finally, I will discuss an approach developed to address one of these challenges, namely identifying the appropriate network structure for the metapopulation model, using either mobile phone or geographical data.

  14. October 20, 2016
    POT 745 from 11am-noon
    Title: Synchrony in a Boolean network model of the L-arabinose operon
    Speaker: Matthew Macauley, Clemson University

    In genetics, an operon is a segment of DNA that contains several co-transcribed genes, which together form a functional regulatory unit. Operons have primarily been studied in prokaryotes, with both the lactose and tryptophan operons in E. Coli having been classically modeled with differential equations and more recently, with Boolean networks. The L-arabinose operon in E. coli encodes proteins that function in the catabolism of arabinose. This operon has several complex features, such as a protein that acts both as an activator, a DNA looping repressing mechanism, and the lack of inducer exclusion by glucose. In this talk, I will propose a Boolean network model of the ara operon, and then show how computational algebra in Sage establishes that for 11 of the 12 choices of initial conditions, the state space contains a single fixed point that correctly predicts the biology. The final initial condition describes the case where there are medium levels of arabinose and no glucose, and it successfully predicts bistability of the system. Finally, I will compare the state space under synchronous and asynchronous update, and show how the former has several artificial cycles that go away under a general asynchronous update.

  15. October 13, 2016
    POT 745 from 11am-noon
    Title: The role of networks on disease spread and intervention strategies
    Speaker: Michael Kelly, Transylvania University

    The interconnectedness of communities has played a major role in disease spread within a population. This has become especially true in the case of waterborne diseases such as cholera, where multiple transmission pathways exist. Understanding the role of networks on disease outbreaks has become crucial when considering where intervention strategies should be focused. We investigate questions of optimal vaccination distributions on heterogeneous community networks in the case of cholera outbreaks; both in response to and preemptively before an outbreak. For responsive strategies, optimal control on a system of ordinary differential equations is developed to minimize the number of infected individuals in the population. For preemptive strategies, a constrained optimization problem is used that seeks to minimize the risk of outbreak on the network while incorporating uncertainty in disease transmissibility. Both also focus on minimizing the associated cost of implementation. The two methods will be discussed, simulations are shown for varying scenarios and networks, and results provide guidance on where to prioritize vaccination in light of outbreaks.

  16. September 29, 2016
    POT 745 from 11am-noon
    Title: Long Short Term Memory
    Speaker: John A. Hirdt, Department of Mathematics, University of Kentucky

    Long Short Term Memory or LSTMs as they are more commonly known, are the most popular type of Recurrent Neural Network used in Machine Learning. LSTMs popularity comes from their ability to capture long-term dependencies in sequential data sets. LSTMs often outperform other RNNs and many Hidden Markov Models when applied to various applications. One popular example of LSTM use is the Netflix user rating example. Users watch a movie, rate it and then watch another movie, and continue with this pattern creating a sequence of reviews. Using LSTMs we can model this sequence and make predictions about a users favorite genre of movie as well as make predictions about future movies a user may want to watch. Finally, we look at how LSTMs can be applied to a variety of problems, including those that are non-sequential.

  17. September 22, 2016
    POT 745 from 11am-noon
    Title: An Efficient Ascending Auction for Assignment Problems
    with De Liu, Carlson School of Management- University of Minnesota
    Speaker: Adib Bagh, Department of Mathematics, University of Kentucky

    We review basic concepts in the theory of auctions. We then introduce a simple ascending auction that allocates heterogeneous objects among bidders with purely private unit demands. Our auction design differs from existing dynamic auctions in a number of ways: it solicits a single new bid from selected bidders at a time, thus minimizing bidder information revelation; it uses a simple and intuitive price adjustment procedure; the seller can set starting prices above his valuations. Despite these new features, (i) the auction stops in a finite time, (ii) sincere bidding at every stage of the auction is an ex-post Nash equilibrium, and (iii) for given valuations, the auction ending prices and revenue depend only on starting prices. We establish sincere bidding and path-independent ending prices using combinatorial arguments. We demonstrate via simulations that our proposed auctions is better than existing auctions in preserving the privacy of the bidders.

  18. September 8, 2016
    POT 745 from 11am-noon
    Title: Insight into Molecular through Subcellular Calcium Signaling via Multi-Scale Simulation
    Speaker: Peter Kekenes-Huskey, Department of Chemistry, University of Kentucky

    Calcium is critical to a wide range of physiological processes, including neurological function, immune responses, and muscle contraction. Calcium-dependent signaling pathways enlist a variety of proteins and channels that must rapidly and selectively bind calcium against thousand-fold higher cationic concentrations. Frequently these pathways further require the co-localization of these proteins within specialized subcellular structures to function properly. Our lab has developed multi-scale simulation tools to elucidate how protein structure and co-localization facilitate intracellular calcium signaling. Developments include combining molecular simulations with a statistical mechanical model of ion binding, a homogenization theory to upscale molecular interactions into micron-scale diffusion models, and reaction-diffusion simulations that leverage sub-micron microscopy data. In this seminar, I will describe these tools and their applications toward molecular mechanisms of calcium-selective recognition and cross-talk between co-localized calcium binding proteins inside the cell.

  19. September 1, 2016
    POT 745 from 11am-noon
    Title: Hidden Markov Models with Applications to RNA Folding
    Speaker: David Murrugarra, Department of Mathematics, University of Kentucky

    This talk will give an introduction to RNA secondary structure prediction using the Nearest Neighborhood Thermodynamic Model (NNTM) and then will present Hidden Markov Models (HMMs) and potential applications for the problem of RNA folding.

Academic Year 2015-16

  1. April 28, 2016
    POT 745 from 11am-noon
    Title: Qualitative Assesment of the Role of Temperature Variations on Malaria Transmission Dynamics
    Speaker: Folashade B. Agusto, Department of Ecology and Evolutionary Biology, University of Kansas

    A new mechanistic deterministic model for assessing the impact of temperature variability on malaria transmission dynamics is developed. The effects of sensitivity and uncertainty in estimates of the parameter values used in numerical simulations of the model are analyzed. These analyses reveal that, for temperatures in the range [16-34]°C, the parameters of the model that have the dominant influence on the disease dynamics are the mosquito carrying capacity, transmission probability per contact for susceptible mosquitoes, human recruitment rate, mosquito maturation rate, biting rate, transmission probability per contact for susceptible humans, and recovery rate from first-time infections. This study emphasize the combined use of mosquito-reduction strategy and personal protection against mosquito bite during the periods when the mean monthly temperatures are in the range [16.7, 25]°C. For higher daily mean temperatures in the range [26, 34]°C, mosquito-reduction strategy should be emphasized ahead of personal protection. Numerical simulations of the model reveal that mosquito maturation rate has a minimum sensitivity (to the associated reproduction threshold of the model) at T = 24°C and maximum at T = 30°C. The mosquito biting rate has maximum sensitivity at T = 26°C, while the minimum value for the transmission probability per bite for susceptible mosquitoes occurs at T = 24°C. Furthermore, disease burden increases for temperatures between 16°C and 25°C and decreases beyond 25°C. This finding, which supports a recent study by other authors, suggests the importance of the role of global warming on future malaria transmission trends.

  2. April 21, 2016
    POT 745 from 11am-noon
    Title: Generative Neural Networks in Semi-Supervised Learning
    Speaker: Devin Willmott

    Semi-supervised learning is a relatively new machine learning concept that seeks to use both labeled and unlabeled data to perform supervised learning tasks. We will look at two network types with some promising applications to semi-supervised learning: ladder networks and adversarial networks. For each, we will discuss the motivations behind their architectures & training methods, and derive some favorable theoretical properties about their capabilities.

  3. April 20, 2016
    POT 110 from 2-3pm
    Title: Matrix Factorization Techniques for Recommender Systems
    Speaker: Zhen Luo

    Recommendation Systems apply Information Retrieval techniques to select the online information relevant to a given user. Collaborative Filtering (CF) is currently most widely used approach to build Recommendation System. To address this issue, the collaborative filtering recommendation algorithm is based on singular value decomposition (SVD) . How the SVD works to make recommendations is presented in this master talk.

  4. April 14, 2016
    POT 745 from 11am-noon
    Master's Talk
    Speaker: Jonathan Proctor, University of Kentucky

    Jonathan will be presenting the paper

    Numerical Methods for Electronic Structure Calculations of Materials

  5. April 7, 2016
    POT 745 from 11am-noon
    Learning About When and Where from Imagery
    Speaker: Nathan Jacobs, University of Kentucky

    Every day billions of images are uploaded to the Internet. Together they provide many high-resolution pictures of the world, from panoramic views of natural landscapes to detailed views of what someone had for dinner. Many are tagged with when and where the picture was taken, thus providing an opportunity to better understand how the appearance of objects and scenes varies with respect to location and time. This talk describes my work in using learning-based methods to extract geo-spatial properties from imagery. In particular, I will focus on two recent research thrusts: using deep convolutional neural networks to geo-calibrate social network imagery and using such imagery to build geo-temporal models of human appearance.

  6. March 31, 2016
    POT 745 from 11am-noon
    The benefits of elliptic curve cryptography
    Speaker: Luis Sordo Vieira, University of Kentucky

    We will introduce the basis of elliptic curve cryptography. Roughly speaking ECC is based on the group structure of the points defined on an elliptic curve over a finite field and the difficulty of solving the discrete log problem. The applications are many, such as signature verification and pseudo random generators. No knowledge of algebraic geometry is required.

  7. March 10, 2016
    POT 745 from 11am-noon
    Computing Exponentials of Essentially Non-negative Matrices with Entry-wise Accuracy
    Speaker: Qiang Ye, University of Kentucky

    A real square matrix is said to be essentially non-negative if all of its off-diagonal entries are non-negative. In this talk, I will present new perturbation results and algorithms that demonstrate that the exponential of an essentially non-negative matrix can be computed with entrywise relative accuracy.

  8. March 3, 2016
    POT 745 from 11am-noon
    Learning Algorithms for Restricted Boltzmann Machines
    Speaker: Devin Willmott, University of Kentucky

    Restricted Boltzmann machines (RBMs) have played a central role in the development of deep learning. In this talk, we will introduce the theoretical framework behind stochastic binary RBMs, give motivation and a derivation for the most commonly used RBM learning algorithm (contrastive divergence), and prove some analytic results related to its convergence properties.

  9. February 4, 2016
    POT 745 from 11am-noon
    Algebraic methods in computational biology
    Speaker: Reinhard Laubenbacher, Director, Center for Quantitative Medicine, UConn Health Center
    Abstract: As biology has become a data-rich science, more biological phenomena have become amenable to modeling and analysis using mathematical and statistical methods. At the same time, more mathematical areas have developed applications in the biosciences, in particular algebra, discrete mathematics, topology, and geometry. This talk will present some case studies from algebra and discrete mathematics applied to the construction and analysis of dynamic models of biological networks. Some emerging themes will be highlighted, outlining a broader research agenda at the interface of biology and algebra and discrete mathematics. No special knowledge in any of these fields is required to follow the presentation.
  10. January 28, 2016
    POT 745 from 11am-noon
    Estimating Propensity Parameters using Google PageRank and Genetic Algorithms
    Speaker: David Murrugarra, University of Kentucky
    Abstract: Stochastic Boolean networks, or more generally stochastic discrete networks, are an important class of computational models for molecular interaction networks. The stochasticity stems from the updating schedule. The standard updating schedules include the synchronous update, where all the nodes are updated at the same time and gives a deterministic dynamic, and the asynchronous update, where a random node is updated at each time step that gives a stochastic dynamics. A more general stochastic setting considers propensity parameters for updating each node. SDDS is a modeling framework that considers two propensity values for updating each node, one when the update has a positive impact on the variable, that is, when the update causes the variable to increase its value, and the other when the update is negative, that is, when the update causes it to decrease its value. This extension adds a complexity in parameter estimation of the propensity parameters. This talk presents a method for estimating the propensity parameters for SDDS. The method is based on adding noise to the system using the Google PageRank approach to make the system ergodic and thus guaranteeing the existence of a stationary distribution and then with the use of a genetic algorithm the propensity parameters are estimated.
  11. November 12, 2015
    POT 745 from 11am-noon
    Fast algorithms for large scale eigenvalue and singular value calculations
    Speaker: Yunkai Zhou, Southern Methodist University
    Abstract: The first part of this talk is on accelerating a block Davidson method for computing large scale eigenvalue decomposition (EVD) and singular value decomposition (SVD). We use two type of filters for the acceleration, one based on polynomial filters, the other based on rational filters. Our method uses the least amount of memory comparing with other state-of-the-art algorithms, but can achieve similar or better computational speed.
    The second part of the talk is on a recently developed spectrum partition methods based on ARPACK (or the eigs() in Matlab). It can be used to conveniently compute several thousands of eigenpairs for matrices with large dimensions. In comparison, eigs() without partition applied to the same problems would either take very long to converge or run out of memory. Our partitioned method is designed to be intrinsically-parallel, suitable for solving very large eigenproblems on supercomputers.
  12. November 5, 2015
    POT 745 from 11am-1pm
    The Krylov Subspace Methods for the Computation of Matrix Exponentials
    Speaker: Hao Wang, University of Kentucky
    Abstract: The problem of computing the matrix exponential \(e^{tA}\) arises in many theoretical and practical problems. Many methods have been developed to accurately and efficiently compute this matrix function or its product with a vector, i.e., \(e^{tA}v\). In the past few decades, with the increasing need of the computation for large sparse matrices, iterative methods such as the Krylov subspace methods have proved to be a powerful class of methods in dealing with many linear algebra problems. The Krylov subspace methods have been introduced for computing matrix exponentials by Gallopoulos and Saad, and the corresponding error bounds that aim at explaining the convergence properties have been extensively studied. Many of those bounds show that the speed of convergence depends on the norm of the matrix, while some others emphasize the important role played by the spectral distribution for some special matrices. For example, it is shown in a recent work by Ye that the speed of convergence is closely related to the condition number, namely the convergence is fast for a well-conditioned matrix no matter how large the norm is.
    In this dissertation, we derive new error bounds for computing \(e^{tA}v\) with non-symmetric \(A\), using the spectral information of \(A\). Our result is based on the assumption that the field of values of \(A\) lies entirely in the left half of the complex plane, such that the underlying dynamic system is stable. The new bounds show that the speed of convergence is related to the size and shape of the rectangle containing the field of values, and they agree with the existing results when \(A\) is nearly symmetric. Furthermore, we also derive a simpler error bound for the special case when \(A\) is skew-Hermitian. This bound explains an observed convergence behavior where the approximation error initially stagnates for certain iterations before it starts to converge. In deriving our new error bounds, we use sharper estimates of the decay property of exponentials of Hessenberg matrices, by constructing polynomial approximations of the exponential function in the region containing the field of values. The Jacobi elliptic functions are used to construct the conformal mappings and generate the Faber polynomials. We also present numerical tests to demonstrate the behavior of the new error bounds.
  13. October 22, 2015
    POT 745 from 11am-noon
    On the perfect reconstruction of the topology of dynamic networks
    Speaker: Alan Veliz-Cuba, University of Dayton Ohio
    Abstract: The network inference problem consists in reconstructing the topology or wiring diagram of a dynamic network from time-series data. Even though this problem has been studied in the past, there is no algorithm that guarantees perfect reconstruction of the topology of a dynamic network. In this talk I will present a framework and algorithm to solve the network inference problem for discrete-time networks that, given enough data, is guaranteed to reconstruct the topology of a dynamic network perfectly. The framework uses tools from algebraic geometry.
  14. October 8, 2015
    POT 745 from 11am-noon
    An Introduction to Wavelets Speaker: David Roach, Western Kentucky University
    Abstract: In this talk, I will introduce the concept of a wavelet from a theoretical perspective as well as how the wavelet can used to approximate data which contains high frequency data at multiple resolutions.
  15. September 24, 2015
    POT 745 from 11am-noon
    Multivariate Decomposition Method for \(\infty\)-Variate Integration Speaker: Grzegorz W. Wasilkowski, University of Kentucky
    Abstract: We present a Multivariate Decomposition Method (MDM) for approximating integrals of functions with countably many variables. We assume that the integrands have mixed first order partial derivatives bounded in a \(\gamma=\{\gamma_u\}_{u\subset \mathbb{N}_+}\)-weighted \(L_p\) norm. We also assume that the integrands can be evaluated only at points with finitely many \((d)\) coordinates different than zero and that the cost of such a sampling is equal to \(\$(d)\) for a given cost function \(\$\). We show that MDM can approximate the integrals with the worst case error bounded by \(\varepsilon\) at cost proportional \[\varepsilon^{-1+|O(\ln(1/\varepsilon)/\ln(\ln(1/\varepsilon)))|}\] even if the cost function is exponential in \(d\) , i.e., \(\$(d)=e^{O(d)}\). This is an almost optimal method since all algorithms for univariate functions \((d=1)\) from this space have the cost bounded from below by \(\Omega(1/\varepsilon)\).
  16. September 10, 2015
    No Seminar.
    We will meet for lunch around noon to discuss future activities of the seminar.
    We encourage you to attend the Math Biology journal club that will bee meeting at 2pm in POT 945.
  17. September 1, 2015
    POT 745 from 1-2pm
    Singular Value Computation and Subspace Clustering
    Speaker: Qiao Liang, University of Kentucky

    Abstract: In this dissertation we discuss two problems. In the First part, we consider the problem of computing a few extreme singular values of a symmetric defnite generalized eigenvalue problem or a large and sparse matrix C. Most existing numerical methods are based on reformulating the singular value problem as an equivalent symmetric eigenvalue problem. The standard method of choice of computing a few extreme eigenvalues of a large symmetric matrix is the Lanczos or the implicitly restarted Lanczos method. These methods usually employ a shift-and-invert transformation to accelerate the speed of convergence, which is not practical for truly large problems. With this in mind, Golub and Ye proposes an inverse-free preconditioned Krylov subspace method, which uses preconditioning instead of shift-and-invert to accelerate the convergence. The inverse-free Krylov subspace method focuses on the computation of one extreme eigenvalue and a deflation technique is needed to compute additional eigenvalues. The Wielandt deflation has been considered and can be used in a straightforward manner. However, the Wielandt deflation alters the structure of the problem and may cause some difficulties in certain applications such as the singular value computations. So we First propose to consider a deformation by restriction method for the inverse-free Krylov subspace method. We generalize the original convergence theory for the inverse-free preconditioned Krylov subspace method to justify this deflation scheme. We next extend the inverse-free Krylov subspace method with deflation by restriction to the singular value problem. We consider preconditioning based on robust incomplete factorization to accelerate the convergence. Numerical examples are provided to demonstrate effciency and robustness of the new algorithm. In the second part of this thesis, we consider the so-called subspace clustering problem, which aims for extracting a multi-subspace structure from a collection of points lying in a high-dimensional space. Recently, methods based on Self Expressive Property(SEP) such as Sparse Subspace Clustering(SSC) and Low Rank Representations( LRR) have been shown to enjoy superior performances than other methods. Self Expressive Property means the points can be expressed as linear combinations of themselves. However, methods with SEP may result in representations that are not amenable to clustering through graph partitioning. We propose a method where the points are expressed in terms of an orthonormal basis. The orthonormal basis is optimally chosen in the sense that the representation of all points is sparsest. Nnumerical results are given to illustrate the effectiveness and effciency of this method.

Academic Year 2014-15

  1. April 23, 2015
    POT 945 from 11-noon
    Making Do with Less: An Introduction to Compressed Sensing
    Master's Presentation
    Speaker: Fouche Smith

  2. April 16, 2015
    POT 745 from 2:15-3:30pm
    A Matrix Analysis of Centrality Measures
    Master's Presentation
    Speaker: Sarach Orchard

    Abstract: When analyzing a network, one of the most basic concerns is identifying the "important" nodes in the network. What defines "important" can vary from network to network, depending on what one is trying to analyze about the network. In this paper by Benzi and Klymko several different centrality measures, methods of computing node importance, are introduced and compared. We will see that some centrality measures give more information about the network on a local scale, while others help to analyze on a more global scale. In particular, the paper analyzes the behavior of these measures as we let the parameters defining them approach certain limits that appear to be problematic.

  3. April 9, 2015
    CP 222 from 5-6pm (refreshemnts at 4:30pm)
    The Problem of Bus-Bunching and What to Do About It
    SIAM Talk
    Speaker: Dr. John Bartholdi of Georgia Institute of Technology

    Abstract: The main challenge for an urban bus system is to maintain constant headways between successive buses. Most bus systems try to adhere to a schedule, but the natural dynamics of the system tends to collapse headways so that buses travel in bunches. What can be done about it? We discuss some models of the phenomenon and show some ways to coordinating buses. In addition, we introduce a new idea that abandons the idea of a schedule and any a priori headway and enables equal headways to emerge spontaneously. We also report on the implementation for a public bus route in Atlanta.

    (joint work with Donald D. Eisenstein, University of Chicago)

  4. April 2, 2015
    POT 245 from 3:30-4:30pm
    Optimality of the Neighbor Joining Algorithm and Faces of the Balanced Minimum Evolution Polytope
    Speaker: Dr. Ruriko Yoshida of the University of the University of Kentucky Department of Statistics

    Abstract: Balanced minimum evolution (BME) is a statistically consistent distance-based method to reconstruct a phylogenetic tree from an alignment of molecular data. In 2008, Eickmeyer, Huggins, Pachter, and myself developed a notion of the BME polytope, the convex hull of the BME vectors obtained from Pauplin's formula applied to all binary trees. We also showed that the BME can be formulated as a linear programming problem over the BME polytope. The BME is related to the Neighbor Joining (NJ) algorithm, now known to be a greedy optimization of the BME principle. Further, the NJ and BME algorithms have been studied previously to understand when the NJ algorithm returns a BME tree for small numbers of taxa. In this talk we aim to elucidate the structure of the BME polytope and strengthen knowledge of the connection between the BME method and NJ algorithm. We first show that any subtree-prune-regraft move from a binary tree to another binary tree corresponds to an edge of the BME polytope. Moreover, we describe an entire family of faces parametrized by disjoint clades. We show that these clade-faces are smaller-dimensional BME polytopes themselves. Finally, we show that for any order of joining nodes to form a tree, there exists an associated distance matrix (i.e., dissimilarity map) for which the NJ algorithm returns the BME tree. More strongly, we show that the BME cone and every NJ cone associated to a tree T have an intersection of positive measure. We end this talk with the current and future projects on phylogenomics with biologists in University of Kentucky and Eastern Kentucky University. This work is supported by NIH.

  5. March 26, 2015
    POT 245 from 11-noon
    Convexity, star-shapedness, and multiplicity of numerical range and its generalizations
    Speaker: Tin-Yau Tam of the Auburn University Department of Mathematics and Statistics

    Abstract: Given an n×nn\times n complex matrix AA, the classical numerical range (field of values) of A is the following set associated with the quadratic
    W(A)={x*Ax:x*x=1,x is a complex n-tuple} W(A) = \{x^*Ax: x*x=1, x\,\text{ is a complex }\, n\text{-tuple}\}We will start with the celebrated Toeplitz-Hausdorff (1918, 1919) convexity theorem for the classical numerical range. Then we will move on to introduce various generalizations and we will focus on those in the framework of semisimple Lie algebras and compact Lie groups. In our discussions, results on convexity, star-shapedness, and multiplicity will be reviewed, for example, the results of Embry (1970), Westwick (1975), Au-Yeung-Tsing (1983, 84), Cheung-Tsing (1996), Li-Tam (2000), Tam (2002), Dokovic-Tam (2003), Cheung-Tam (2008, 2011), Carden (2009), Cheung-Liu-Tam (2011) and Markus-Tam (2011). We will mention some unsolved problems.

  6. March 12, 2015
    DH 135 from 11am-noon
    Text as Data
    Speaker: J.P. Wedeking of the University of Kentucky Department of Political Science

    Abstract: Professor Wedeking will give a summary of three projects that he has been involved in using text as data (1 is published, 1 is under review, and 1 is ongoing). Specifically, for each of the 3 projects, He will: (1) describe the method he's using, what it generally is used for; (2) the motivation for the project-e.g., the substantive research question and relevant background information; (3) a brief description of the data; and (4) the results of the method and the substantive conclusions. The three projects are: (1) measuring how legal issues are framed (e.g., free speech vs. right to privacy, etc) and how that helps parties win; (2) uncovering the clarity of texts using readability formulas; and (3) scaling justices with texts- uncovering their ideological positions (how liberal or conservative they are) using their words.

  7. December 4, 2014
    POT 145 from 3:00-4:30pm
    Hubs and Authorities Master's Presentation
    Speaker: Nicholas Benthem of the University of Kentucky Department of Mathematics

    Abstract: We introduce the idea of Hub and Authority rankings inside large scale networks with appropriate historical context, and introduce a new form for calculating Hubs and Authorities by turning a directed network into a bipartite network, along with efficient computational tools to evaluate these rankings in large scale networks.

  8. November 6, 2014
    POT 145 from 3:30-4:30pm
    Modeling Foot and Mouth Disease in cattle in northern Cameroon
    Speaker: Matt Orelam of the Ohio State Universsity Mathematical Biosciences Institute

    Abstract: Foot and Mouth Disease (FMD) is endemic in cattle in the Far North Region of Cameroon. While many cattle herds remain in a fixed location throughout the year, there are a small number of mobile herds that migrate depending on the season. These mobile herds share grazing space with many other cattle throughout the year, leading to increased disease transmission. In this talk I will present a multi-scale agent-based simulation model of FMD in northern Cameroon, focusing on the mathematical SIRS epidemic model running both inter- and intra-herd. Various parameters are determined by data from researchers on the ground while others are determined via in silico experimentation. The goal of the first phase of the project is to determine how each herd type contributes to the overall number of secondary infections. This model is a work in progress and the talk is meant to stimulate discussion about means of incorporating epidemic models in a multi-scale setting.

  9. October 16, 2014
    POT 745 from 4-5pm
    Efficient Solutions of Large Saddle-Point Systems
    Speaker: Lola Davidson of the Unviersity of Kentucky Department of Mathematics

    Abstract: Linear systems of saddle-point type arise in a range of applications including optimization, mixed finite-element methods for mechanics and fluid dynamics, economics, and finance. Due to their indefiniteness and generally unfavorable spectral properties, such systems are difficult to solve, particularly when their dimension is very large. In some applications - for example, when simulating fluid flow over large periods of time - such systems have to be solved many times over the course of a single run, and the linear solver rapidly becomes a major bottleneck. For this reason, finding an efficient and scalable solver is of the utmost importance. In this talk, we examined various solution strategies for saddle-point systems.

  10. October 1, 2014
    POT 745 from 3-4pm
    Network Analysis with Matrix Functions
    Speaker: Lothar Reichel of Kent State University

    Abstract: Networks arise in many applications. It is often of interest to be able to identify the most important nodes of a network or to determine the ease of traveling between them. We are interested in carrying out these tasks for large undirected and directed networks. Many quantities of interest can be determined by computing certain matrix functionals. We discuss how for directed and undirected graphs a few steps of the Lanczos method in combination with Gauss-type quadrature rules can be applied to determine upper and lower bounds for quantities of interest.

  11. September 25, 2014
    POT 145 from 3:30-4pm
    Accurate Computations of Matrix Eigenvalues with Applications to Differential Operators
    Speaker: Qiang Ye of the University of Kentucky Department of Mathematics

    Abstract: In this talk, we present our recent works on high relative accuracy algorithms for computing eigenvalues of diagonally dominant matrices. We present an algorithm that computes all eigenvalues of a symmetric diagonally dominant matrix to high relative accuracy. We further consider using the algorithm in an iterative method for a large scale eigenvalue problem and we show how smaller eigenvalues of finite difference discretizations of differential operators can be computed accurately. Numerical examples are presented to demonstrate the high accuracy achieved by the new algorithm.

Corrections to:

plications. It is often of interest to be able to identify the most important nodes of a network or to determine the ease of traveling between them. We are interested in carrying out these tasks for large undirected and directed networks. Many quantities of interest can be determined by computing certain matrix functionals. We discuss how for directed and undirected graphs a few steps of the Lanczos method in combination with Gauss-type quadrature rules can be applied to determine upper and lower bounds for quantities of interest.

  • September 25, 2014
    POT 145 from 3:30-4pm
    Accurate Computations of Matrix Eigenvalues with Applications to Differential Operators
    Speaker: Qiang Ye of the University of Kentucky Department of Mathematics

    Abstract: In this talk, we present our recent works on high relative accuracy algorithms for computing eigenvalues of diagonally dominant matrices. We present an algorithm that computes all eigenvalues of a symmetric diagonally dominant matrix to high relative accuracy. We further consider using the algorithm in an iterative method for a large scale eigenvalue problem and we show how smaller eigenvalues of finite difference discretizations of differential operators can be computed accurately. Numerical examples are presented to demonstrate the high accuracy achieved by the new algorithm.

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