Research

Research Interests

Current Research

(Fall 2019 - present) My PhD research has focused on trimmed serendipity elements. Trimmed serendipity finite elements are a relatively new family of finite elements that can be used to solve PDEs on quad and cube meshes. I have focused on the implementation and application of these elements inside of an open source FEM package called Firedrake. This has allowed us to do a thorough computational comparison between trimmed serendipity elements and more traditional tensor product elements, resulting in the paper here. These elements have the advantage of requiring less computational cost than traditional elements do, and so we are progressing toward applying these elements to applications where being able to compute faster is a necessity.

I enjoy using code to try and solve interesting application problems, whether that is in a finite element setting, a machine learning setting, or on graphs.

Past Research

(Summer 2021) Supervised by Andrew Gillette in conjunction with Brendan Keith and Socratis Petrides at Lawrence Livermore National Laboratory. Using MFEM to find numerical solutions to PDEs on a variety of meshes, we then applied reinforcement learning techniques to teach an agent how to do adaptively refine meshes.

(Summer 2020) Supervised by Andrew Gillette at Lawrence Livermore National Laboratory. During this summer, I worked on two different topics. The first was solving lid-driven cavity flows with the eventual goal of being able to use the data generated by simulations in MFEM to predict the number of vortices present in the cavity as time went on. The second project involved using gradient boosted decision trees to make predictions about chemical compounds.

(Summer 2019) Supervised by Vitaliy Gyrya at Los Alamos National Laboratory. When solving large eddy simulations for turbulent models, direct numerical simulation (DNS) is one method for doing so. However, DNS is a computationally expensive method that can be very slow. Our goal was to leverage machine learning to be able to learn parameters in the large eddy simulation and predict flows in these models similar to what the large eddy simulations would give.

(Spring 2017 - Spring 2019) Supervised by Andrew Gillette in conjunction with Josh Levine. The goal of this research was to find a way to solve fractional PDEs on complex geometries. We chose to adapt discrete exterior calculus (DEC) -- a method for computing solutions to PDEs on simplicial meshes -- to a fractional PDE setting. This required defining a fractional discrete exterior derivative and illustrating some convergence results to give a proof of the concept, and resulted in this paper (arxiv preprint).

(Fall 2017--RTG) Supervised by Andrew Gillette. We worked on understanding theoretical numerical convergence results for curved finite elements. The focus was on elements that came from rational or polynomial transformations of the unit square.

(Spring 2017--Spring Term Paper) Supervised by Rob Indik. We read and worked through The Matrix Eigenvalue Problem by David S. Watkins. During this, I implemented various structured eigenvalue solvers in Matlab and tested them on random matrices with the structure of interest.

Presentations

(June 2018--Poster Presentation) Fractional Laplacian Smoothing. I presented preliminary results for fractional Laplacian smoothing at the poster session for the ICERM workshop Fractional PDEs: Theory, Algorithms, and Applications.

(December 2017--Talk) Curved Finite Elements. I presented theoretical convergence results from the third semester research project with Andrew Gillette at the Applied Mathematics Second Year Research Conference.

(May 2017--Talk) The Eigenvalue Problem. I presented results and algorithms that were implemented during the second semester research project at the First Year Applied Math Research Workshop.