I am an applied mathematician, interested in the field of numerical analysis. What is numerical analysis? See the essay by L. N. Trefethen. I was never interested in mathematics until high school algebra, where I had a wonderful teacher, Andrew Blount. Initially intimated by mathematics, I did not enter college as a math major, but soon realized it would continue to be my favorite subject. I loved my linear algebra class and continued studying matrices in graduate school. My graduate advisor, Charlie Van Loan, has amazing creativity in solving numerical problems and I hope I can advise students as well as he does!
Now to the details. My research area is numerical linear algebra. Specifically, the theme of my research includes finding efficient algorithms for computation with Kronecker products. As computing power increases, large scale problems become more tractable in engineering and data analysis. Computations with tensors are naturally occurring in such areas. The Kronecker product is used extensively and allows for fast algorithms because of its regular structure. The specific problems I have tackled in my research are examples of tensoring lower-dimensional objects for higher-dimensional data analysis.
Below, I briefly describe specific problems I have worked on. Also see my publications list.
My current interests involve computational and theoretical analysis of tensor decompositions. We define an order-p tensor to be a structure, A, indexed as A = (ai1,i2,...,ip) in Rn1xn2x...xnP. An order-1 tensor is a vector and an order-2 tensor is a matrix. Third-order tensors can be visualized as a cube of data.
The type of tensor decompositions I have concentrated on are types of extensions of the Singular Value Decomposition (SVD) for matrices. While the matrix SVD is well-understood mathematically, numerically, and com- putationally, finding an appropriate decomposition for higher-order tensors is nontrivial; even familiar matrix concepts such as diagonalization and rank become ambiguous and complicated. My dissertation work involved designing algorithms to compute tensor decompositions as well as analyzing the theoretical notion of rank as it applies to higher-order tensors.
With Misha Kilmer, I have been working on new ways to define products of tensors, so that multiplication for tensors is a closed operation. This has led to new ways to generalize certain matrix factorizations such as QR, SVD, and eigenvalue decompositions. Based on this new SVD extension, we can develop new algorithms for compressing multi-way data.
In Summer 2008, I had two REU Students, Scott Ladenheim and Emily Miller, work with me on such extensions. One of the applications they worked on was video compression. Click here for their work on video compression.
Another problem I have worked on involving Kronecker products involves shifted linear systems. That is, solving (A-lambda*I)x= b, where A is a Kronecker product of matrices.
Our algorithm invokes the complex Schur decomposition when an Ai has complex eigenvalues. We also proved a perturbation theorem that shows when a real linear system is subjected to complex perturbations, the real part of the solution to the perturbed system solves a nearby real linear system. The result is that is it "numerically safe" to obtain real solutions via the complex Schur decomposition. This result is critical because for this type of problem, one cannot work with the real Schur form without greatly increasing the volume of work.
We have also published results on a related problem, where A is a product of matrices.
As a musician, I am always fasinated by melody lines and the patterns of such. When I have time, I occasionally do some mathematical analysis of certain melody lines. See my talk on Linear Recurrence Relations in Music.