Graduate Students Seminar

Speaker 1: Theodore Weinberg

Title

Fast Implementation of Mixed Finite Elements in MATLAB

Abstract

Understanding and modeling flow in porous media is important in many areas including managing groundwater reserves, maintaining CO2 storage facilities, and simulating petroleum reservoirs. This has created a growing need to efficiently describe flow in porous media. Models are typically described by partial differential equations (PDEs). We have developed an efficient implementation of the mixed finite element method for the lowest order Raviart-Thomas elements (RT0), which can be used to discretize problems related to flow in porous media. This implementation was created in MATLAB.  As MATLAB is inefficient with iterative operations, the code had to be vectorized, replacing loops with array operations. In other words, instead of interacting with one element at a time the code interacts with all the elements simultaneously.  The code supports two-dimensional and three-dimensional domains, and the finite elements can be triangles, quadrilaterals, tetrahedrons, or blocks. Based on numerical experiments, we have shown that our implementation is significantly more efficient than the standard approach.

Speaker 2: Yewon Kim

Title

Functional Classification Using Path Length

Abstract

Functional logistic regression model (James 2002) is a useful technique to investigate a relationship between a binary response and its corresponding functional covariate. Although the model in general captures different functional patterns between two classes, it may be problematic when two classes have similar mean functions but, their trajectories are different, in particular in terms of the magnitude of fluctuations. To address this issue, we introduce a path length of functional covariate in order to reflect information on the fluctuation of trajectory in a model. With real data set in Lee et al. (2007), we demonstrate that a new functional logistic model using path length not only leads to improve classification rate but also helps us to identify informative time intervals in which our proposed model plays a significant role to distinguish two classes more sharply, which is achieved by incorporating fused lasso penalty.