# Numerical Analysis

### Structure-Preserving Numerical Methods and Their Applications

Directed by James Adler and Xiaozhe Hu

Under this project, undergraduate students will conduct research related to structure-preserving numerical methods and their applications. Given a “real-world” application, the students will work to develop and analyze numerical simulation techniques that respect and preserve the fundamental physical and mathematical properties of the application’s governing model. In doing so, the students will help design, implement, and test benchmark problems using our in-house software package and identify the best approaches for different parameter settings using scientific machine learning ideas. Specific possible projects are:

- Stable numerical schemes for modeling the flow in porous media with applications in biomechanics and reservoir engineering.
- Data-driven scalable graph algorithms with applications in bioinformatics and electrical engineering.

Eligible students should have some prior background in numerical analysis and numerical linear algebra. We also expect some familiarity with coding, as the students will be given the opportunity to use our computational software package, written in C. During the summer, the students will work on deeper aspects of a project, and be mentored on how to work on both general algorithm development and on specific modeling for practical applications. In the end, they will have first-hand experience in scientific computing research.

#### About the Professors

James Adler is an Associate Professor in the Department of Mathematics at Tufts University. Adler’s research interests are in the area of scientific computing, particularly in the area of computational mathematics and physics. His research focuses on the numerical computation of partial differential equations that are used to model multi-scale physical systems.

Xiaozhe Hu is an Associate Professor in the Department of Mathematics at Tufts University. Hu’s primary research interests are in numerical analysis and scientific computing, with an emphasis on the development, analysis, and implementation of numerical algorithms for solving partial differential equations and graph problems arising from different applications.