2020-2021

We present a divergence-free (hybrid) discontinuous Galerkin scheme for incompressible flow problems, including incompressible Euler and Naver-Stokes equations, incompressible MHD, and the phase-field model of incompressible two phase flow.

Main features of the scheme includes globally divergence-free velocity approximation/exact mass conservation, inherent (minimal amount) numerical dissipation (via DG upwinding) for convection terms which makes the scheme stable in the convection-dominated regime without using extra residual-based stabilizations, efficient linear system solvers via hybridization

We develops a fixed-accuracy confidence interval for P(X > c) when X follows a gamma distribution, Γ(α, β), and c is a preassigned positive constant through: 1) a purely sequential procedure with known shape parameter α and unknown rate parameter β; and 2) a nonparametric purely sequential procedure with both shape and rate parameters unknown. Both procedures enjoy appealing asymptotic first-order efficiency and asymptotic consistency properties. Extensive simulations validate the theoretical findings. Real-life data examples are included to illustrate the practical applicability of both procedures.

For a regular local ring R of dimension at least 2, the successive localized blow-ups of the maximal ideal form a tree of regular local rings which are in one-to-one correspondence with the divisorial valuation rings birationally dominating R. The 2-dimensional case is especially well-behaved in many ways: for instance, a classical result of Zariski is that integrally closed ideals have unique factorization, and a result of Abhyankar is that every infinite directed union of such blow-ups is a valuation ring. In this colloquium, we examine rings between R and its field of fractions that can be constructed from blow-ups and from divisorial valuations in the 2-dimensional case and in higher dimensions. In particular, we’ll discuss ascending unions of blow-ups, the convergence of valuation rings associated to them in the patch topology on valuations, and generalizations of these notions to more settings. In addition, we’ll discuss families of rings that can be obtained by intersecting collections of divisorial valuation rings.

A fundamental issue in the study of geometric objects defined by algebraic equations is that of efficacy of representation: we wish to encode only the properties that interest us, and to do so we introduce toy models. Tropical geometry yields a fascinating class polyhedral toy models that have offered new insights across a variety of scientific disciplines. In this talk I will introduce the basic tropical paradigm and give some applications to the study of algebraic curves and differential equations.

The field of online algorithms considers computational settings in which algorithms make permanent real-time decisions while receiving input over time. The challenge lies in making decisions that will be nearly optimal for a range of possible future scenarios. In this talk, I will discuss two directions within online algorithms. The first is the Convex Body Chasing problem, a geometric form of convex optimization in an online setting. I will show the resolution of a long-standing question. Secondly, I will discuss a new model of Secretary problems aimed at bridging between worst-case and average-case models, and make connections to problems in online learning.

Contextual bandit problems are important for sequential learning in various practical settings that require balancing the exploration-exploitation trade-off to maximize total rewards. Motivated by applications in health care, we consider a contextual bandit framework with delays in observing the rewards. Although there has been substantial work on handling delays in standard multi-armed bandit problems, the field of contextual bandits with delayed feedback, especially with nonparametric estimation tools remains largely unexplored. Thus, in this work we develop randomized allocation strategies that incorporate delayed rewards using nonparametric regression methods for estimating the mean reward functions. In the talk, we discuss these strategies and establish their theoretical properties such as strong consistency and finite-time regret bounds along with illustrating the performance on simulated/real datasets. Towards the end of the talk, we will briefly introduce other applications of interest that we study using nonparametric estimation techniques.

Single-index models are practical, useful tools for modeling and analyzing many clinical and psychological studies with complex non-linear covariate effects on the response. We propose frequentist and Bayesian methods for monotone single-index models using the Bernstein polynomial basis to represent the link function. The monotonicity of the unknown link function creates a clinically interpretable index, along with the relative importance of the covariates on the index. We develop a computationally-simple, iterative, profile likelihood-based method for the frequentist analysis. To ease the computational complexity of the Bayesian analysis, we also develop a novel, and efficient Metropolis-Hastings step to sample from the conditional posterior distribution of the index parameters. These methodologies and their advantages over existing methods are illustrated via simulation studies. These methods are also used to analyze depression based measures among adolescent girls.

One-shot devices are products that can only be used once. Typical one-shot devices include air-bags, fire-extinguishers, fireworks etc. The observations from those devices are either success and failure at the time of test/use. So, there is usually a considerable loss of information, and hence, the estimation of life characteristics becomes a difficult problem. In this case, the estimation problem has been discussed by many authors, mostly in a parametric setting. In this talk, we will focus on the following aspects of one-shot devices test data. First, we will discuss the Bayesian estimation and a semi-parametric estimation method for simple one-shot devices. Since most one-shot devices contain many components and that failure of any one of them may lead to the device’s failure, a competing risk model will be discussed next in a one-shot device testing context. The second section will discuss the maximum likelihood estimation of model parameters using the EM algorithm, the Bayesian estimation, and the semi-parametric estimation for such a competing risk scenario. Finally, we will conclude the presentation by mentioning some open problems. This talk is based on the following publications:

1. Ling, M.H., So, H.Y., and Balakrishnan, N., Likelihood inference under proportional hazards model for one-shot device testing, IEEE Transactions on Reliability, vol. 65(1), pp. 446-458, 2016.

2. Balakrishnan, N., So, H.Y., and Ling, M.H., A Bayesian approach for one-shot device testing with Weibull lifetimes, (Currently under review).

3. Balakrishnan, N., So, H.Y., and Ling, M.H., EM algorithm for oneshot device testing with competing risks under exponential distribution, Reliability Engineering &System Safety, vol. 137, pp. 129-140, 2015.

4. Balakrishnan, N., So, H.Y., and Ling, M.H., A Bayesian approach for one-shot device testing with exponential lifetimes under competing risks, IEEE Transactions on Reliability, vol. 65(1), pp. 469-485, 2016.

5. Balakrishnan, N., So, H.Y., and Ling, M.H., EM algorithm for one-shot device testing with competing risks under Weibull distribution, IEEE Transactions on Reliability, vol. 65(2), pp. 973-991, 2016.

Small sample sequential multiple assignment randomized trials (snSMARTs) are multistage trial designs to identify the best overall treatment. In snSMARTs, binary response/nonresponse outcomes are measured at the end of the first and second stages. If the patient is responding at the end of the first stage, they continue on the same treatment. Otherwise, they are re-randomized to one of the remaining treatments. In contrast to the binary outcome snSMART design, we propose a novel snSMART design allowing for continuous outcomes. The probability of staying on the same treatment is proportional to the first stage outcome eliminating the need for a categorical tailoring variable defining response/nonresponse. This re-randomization scheme allows for trials to continue without requiring a dichotomous variable. Additionally, this increases the probability of observing patients in all treatment regimens which is critical in small sample studies. I will present simulation results from continuous snSMART trials and estimated treatment effects using Bayesian analysis. Bias and efficiency of treatment effect estimates as well as patient outcomes are examined in this design setting.

Influenza associated cardiac events have been linked to multiple biological pathways in a human host. To study the contribution of influenza virus infection to cardiovascular thrombotic events, we develop a dynamic model which incorporates some key elements of the host immune response, inflammatory response, and blood coagulation. We formulate these biological systems and integrate them into a cohesive modeling framework to show how blood clotting may be connected to influenza virus infection. With blood clot formation inside an artery resulting from influenza virus infection as the primary outcome of this integrated model, we demonstrate how blood clot severity may depend on circulating prothrombin levels. We also utilize our model to leverage clinical data to inform the threshold level of the inflammatory cytokine TNFα which initiates tissue factor induction and subsequent blood clotting. Our model provides a tool to explore how individual biological components contribute to blood clotting events in the presence of influenza infection, to identify individuals at risk of clotting based on their circulating prothrombin levels, and to guide the development of future vaccines to optimally interact with the immune system.

Infectious diseases are one of the leading causes of many deaths around the world. As a result, health officials and the World Health Organization have devoted several resources to educate populations on safety measures which prevent the spread of infectious diseases. Hence restricting population’s movement has been widely used in an effort to limit the outbreak of an infectious disease. In this talk, we will study a multiple-strains PDE infectious disease epidemic model and discuss how population movement can affect the dynamics of the disease.

Pullback dynamics describe the long time behavior of solutions to non-autonomous partial differential equation. Pullback attractors are generalizations of global attractor of a general process. In this work, we give a rather complete study of the tempered pullback dynamics of non-autonomous Navier-Stokes equation with a nonhomogeneous boundary condition on 2D Lipschitz-like domains: we establish not only the existence of a minimal family of pullback attractors, but also estimates on the finite fractal dimension and upper-semicontinuity (by assuming that the external force allows to be of a special form). Some of the technical difficulties that we have to overcome include the non-smoothness of the underlying domain and a somewhat relaxed assumption on the external force.

The mathematical model of Hindmarsh-Rose equations describes biological neuron spiking-bursting of the intracellular membrane potential observed in experiments. It exhibits rich and interesting temporal bursting patterns, especially the 3D complex bifurcations leading to chaotic bursting and dynamics. In this talk, I will briefly present the main results of the global and longtime dynamics of the diffusive and partly diffusive Hindmarsh-Rose equations as well as the random dynamics of the stochastic Hindmarsh-Rose equations with the multiplicative white noise and additive noise, respectively. As a significant extension of these topics on single neuron dynamics, I proposed two new and meaningful mathematical models respectively for two coupled neurons and for a boundary coupled Hindmarsh-Rose neuronal network. The main result shows that the neuronal network is asymptotically synchronized at a uniform exponential rate provided that the combined boundary coupling strength and the stimulating signals exceed a quantified threshold explicitly in terms of the parameters. This talk is based on the joint works with Professors Yuncheng You and Jianzhong Su.

In this talk we focus on the etiology of Alzheimer's Disease and the dynamics of the homeless population as described through partial differential equation models. For Alzheimer's, we build a model coupling the neurotoxic effect of "oligomers" (proteins continuously produced in the brain) with the development of Alzheimer's Disease. The model takes the form of a nonlinear system of reaction-diffusion equations with age-dependent rate constants and, through formal asymptotics, can be compared with clinical data. In studying homelessness, we develop a nonlinear, nonlocal parabolic PDE describing the homeless population density. We state and prove some important results concerning the behavior of the model and, through numerical simulations, show the model can be qualitatively consistent with homeless count data.

In this talk, we will present a few mathematical models that describe different biological phenomenon, like the approach that honey bees take to selecting new nesting sites and cross-contamination in industrial scale food produce. The talk will also outline some of the motifs that show up in seemingly different biological problems, and present challenges that may occur in obtaining precise quantitative models.