I am a PhD candidate in the Department of Computational and Applied Mathematics at Rice University (CAAM) under the supervision of Mark Embree. Much of my research branches from topics covered in the Physics of Strings Seminar. My master's thesis analyzed an understudied model of magnetic damping in elastic conductors. Currently, I am preparing a paper on a data analysis that arose from laboratory experiments. This paper will describe a fast and efficient algorithm to decompose a vector into a sum of complex exponentials. This problem goes by many names: exponential interpolation, cisoid parameter estimation, modal analysis, and exponential analysis to name a few. Simply put, given some 'data' ${\bf y} \in \mathbb{C}^n$ we seek to find parameters ${\bf a} \in \mathbb{C}^m$ and ${\bf \lambda} \in \mathbb{C}^m$ such that $y_j \approx \sum_{k=1}^m a_k e^{j\lambda_k} \quad \text{for} \quad 0 \le j \le n-1.$ We encountered this problem when trying to find the eigenvalues of a violin string. There $m\approx 200$ and $n\approx 10^6$; as we found out, this is far larger than existing algorithms can handle. As a result, I have developed an algorithm which is dramatically faster than existing subspace or algebraic approaches (HSVD, Matrix Pencil Method, Kung's method, etc) and maximum likelihood methods. These existing algorithms both cost $\mathcal{O}(nm^2)$, whereas I can solve the problem in $\mathcal{O}(n\log n + m^3)$.

### Rubens' tube

The header image is a picture of a Rubens' Tube I constructed after completing my qualifying exams. Here the tube is being driven at the frequency corresponding to the third eigenvalue. The shorter flames at the end are due to the longer channel through which the propane must escape.

### Code Snippets

• Often times, thousands of numerical experiments are needed to generate statistics. This short Matlab template displays the current iteration on a single line and estimates when all the iterations will be complete.

### Contact

jeffreyh@rice.edu
Duncan Hall 2108

### curriculum vitae

PDF as of November 2011

Git

Screen

TeX Install