Graduate Students Seminar

Speaker 1: Ellie Gurvich

Title

Weak Solutions for an Implicit, Degenerate Poroelastic Plate System

Abstract

The mathematical theory of poroelasticity was developed for geoscience applications (e.g., petroleum engineering). More recently, it has been incorporated into biological models, owing to the poroelastic nature of biological tissues. We consider a recent biologically-motivated plate model obtained as a scaled limit of the three-dimensional quasi-static Biot system of poroelasticity. We allow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Existence and uniqueness of weak solutions are obtained using the theory of implicit, degenerate evolution equations.

Speaker 2: Vahid Andalib

Title

High-Dimensional Data Classification with Applications to DNA Microarrays

Abstract

Several classification algorithms have been proposed in low-dimensional settings where the number of features p is less than observations n, and each observation belongs to one of the target variable’s classes, such as linear discriminant analysis (LDA) and logistic regression. However, in the high-dimensional setting where p ≫ n, classical methods face both theoretical and computational challenges, e.g., classical LDA cannot be applied since the sample covariance matrix becomes singular. We review two of the modern high-dimensional classifiers, Nearest Shrunken Centroid (NSC) and Shrunken Centroids Regularized Discriminant Analysis (SCRDA). These algorithms are developed especially for classification problems in high-dimension low-sample size situations, for example, class prediction in DNA microarray data to classify and predict the diagnostic category of a sample based on its gene expression profile. Finally, we apply these two classifiers to a real microarray data.