Professor Ion Grama, Universite  Bretagne Sud, Lorient (UBS)

Title: Conditioned limit theorems for products of random matrices and Markov chains with applications to branching processes.

Abstract: Consider a random walk defined by the consecutive action of independent identically distributed random matrices on a starting point outside the unit ball in the d dimensional Euclidean space. We study the first moment when the walk enters the unit ball. We study the exact behaviour of this time and prove conditioned limit theorems for the associated Markov walk. This extends to the case of walks on group GL(d,R) the well known results by Spitzer. The existence of the harmonic function related to the Markov walk turns out to be crucial point of the proof. We have extended these results to general Markov chains and applied them to study the branching processes in Markov environment.    

Speaker: Prof. Zhezhen Jin, Columbia University (Hosted by Jiancheng Jiang)

Title:   Variance estimation in semiparametric regression models

Abstract: In semiparametric regression analysis, objective functions and estimating functions for regression parameters are often nonsmooth and non-monotone, which results in difficulty in the corresponding variance estimation. I will discuss the issues and present available and newly developed  methods with general theory, implementation and demonstrate the methods with examples.

Prof. Yanqing Wang, Georgia State University (Hosted by Yinghao Pan)

Title:   Learning-Based Biomarker-Assisted Rules for Optimized Clinical Benefit under A Risk-Constraint

Abstract: Novel biomarkers, in combination with currently available clinical information, have been sought to improve clinical decision making in many branches of medicine, including screening, surveillance, and prognosis. Statistical methods are needed to integrate such diverse information to develop targeted interventions that balance benefit and harm. In the specific setting of disease detection, we propose novel approaches to construct a multiple-marker-based decision rule by directly optimizing a benefit function, while controlling harm at a maximally tolerable level. The new approaches include plug-in and direct-optimization-based algorithms, and they allow for the construction of both nonparametric and parametric rules. A study of asymptotic properties of the proposed estimators is provided. Simulation results demonstrate good clinical utilities for the resulting decision rules under various scenarios. The methods are applied to a biomarker study in prostate cancer surveillance.

Professor Yang Yang, Michigan Tech University

Title: A Discrete Fracture Model for Single-phase Flow on Non-conforming Meshes
Abstract: The discrete fracture model (DFM) has been widely used in the simulation of fluid flow in fractured porous media. Traditional DFM use the so-called hybrid-dimensional approach to treat fractures explicitly as low-dimensional entries (e.g. line entries in 2D media and face entries in 3D media) on the interfaces of matrix cells to avoid local grid refinements in fractured region and then couple the matrix and fracture flow systems together based on the principle of superposition with the fracture thickness used as the dimensional homogeneity factor. Because of this methodology, DFM is considered to be limited on conforming meshes and thus may raise difficulties in generating high qualified unstructured meshes due to the complexity of fracture’s geometrical morphology. In this paper, we clarify that the discrete fracture model actually can be extended to non-conforming meshes without any essential changes. To show it clearly, we provide another perspective for DFM based on hybrid-dimensional representation of permeability tensor modified from the comb model to describe fractures as one-dimensional line Dirac delta functions contained in permeability tensor. A finite element DFM scheme for single-phase flow on non-conforming meshes is then derived by applying Galerkin finite element method to it. Analytical analysis and numerical experiments show that our DFM scheme automatically degenerates to the classical finite element DFM when the mesh is conforming with fractures. Moreover, the accuracy and efficiency of the model on non-conforming meshes is demonstrated by testing several benchmark problems. This model is also applicable to curved fracture with variable thickness.

Professor Shanshan Zhao, Biostatistics and Computational Biology Branch, National Institute of Environmental Health Sciences

Title:   Accommodating limit-of-detection in environmental mixture analysis

Abstract: Humans are exposed to a multitude of environmental toxicants daily, and there is a great interest in developing statistical methods for assessing the effects of environmental mixtures on various health outcomes. One difficulty is that multiple chemicals in the mixture can be subject to left-censoring due to varying limits of detection (LOD). Conventional approaches either ignore these measures, dichotomize them at the limits, or replace them with arbitrary values such as LOD/Ö2. Methods have been proposed to handle a single biomarker with limit of detection in such setting, by joint modeling the left-censored biomarker measure with an AFT model and the disease outcome with a generalized linear model. We extend this method to handle multiple correlated biomarkers subject to LOD, through a newly proposed nonparametric estimator of the multivariate survival function and innovative computational approaches. We apply the proposed method to the LIFECODES birth cohort to elucidate the relationship between maternal urinary trace metals and oxidative stress markers. 

Dr. Michael DiPasquale,  Colorado State University.

Title: Extending Wilf’s Conjecture

Abstract: Suppose $a_1,a_2,\ldots,a_n$ is a set of positive integers which are relatively prime.  The set of all integers which can be written as a non-negative integer combination of $a_1,\ldots,a_n$ is called a numerical semigroup.  The non-negative integers which are not in the numerical semigroup is called the set of holes of the semigroup.  It is known that the set of holes of a numerical semigroup is finite, and the largest hole is called the conductor.  In a four-page note in the American Mathematical Monthly in 1978, Herbert Wilf asked a question about the density of the set of holes of a numerical semigroup in the interval from zero through the conductor.  This question is still widely open and has become known as Wilf’s conjecture.  We will discuss this conjecture and a recent extension of it to higher dimensions which is joint work with C. Cisto, G. Failla, Z. Flores, C. Peterson, and R. Utano.

Prof. Haizhao Yang, Purdue University

Title: $O(N \log^\alpha N)$ matvec and preconditioners for highly oscillatory integral transforms

Abstract: One of the key problems in scientific computing is the acceleration of matrix computation for large problem sizes. This talk introduces several $O(N \log^\alpha N)$ algorithms for dense matrix multiplications and solving linear systems from highly oscillatory phenomena, e.g, evaluating oscillatory integral transform and special function transforms, performing their inverse transforms, solving boundary integral equations in the high-frequency regime, etc. Based on recent advances of randomized numerical linear algebra and matrix recovery, we are able to approximate these dense matrices and their inverse in $O(N \log^\alpha N)$ operations, leading to efficient matvec and preconditioners for highly oscillatory integral transforms.

Prof. Zhihua Su, University of Florida

Title:  Envelope Models and Methods

Abstract:  This talk presents a new statistical concept called an envelope. An envelope has the potential to achieve substantial efficiency gains in multivariate analysis by identifying and cleaning up immaterial information in the data. The efficiency gains will be demonstrated both by theory and example. Some recent developments in this area, including partial envelopes and heteroscedastic envelope models, will also be discussed. They refine and extend the enveloping idea, adapting it to more data types and increasing the potential to achieve efficiency gains. Applications of envelopes and their connection to other fields will also be mentioned. (Hosted by Dr. Yanqing Sun, UNC Charlotte)