Projects

Structured Covariance Matrix Estimation

In joint work with Anil Damle and Sam Otto at Cornell, I am working to develop ultra-low sample complexity methods for estimating covariance matrices associated with forecasts or climatologies of high dimensional dynamical systems. This work is in early stages; stay tuned for more!

Numerical Methods for Data Assimilation

With Ian Grooms (CU Boulder Dept. of Applied Mathematics), I developed an ensemble Kalman filtering algorithm which uses techniques from numerical quadrature and Krylov subspace iteration to update the ensemble from prior to posterior under a localized forecast covariance matrix. Compared to most existing methods, this algorithm achieves a more favorable trade-off between accuracy and cost when the system being modeled is extremely high-dimensional. With Chris Snyder (NSF NCAR Mesoscale and Microscale Meteorology), we have also been working to implement and test this algorithm on a full-scale forecasting problem using MPAS and JEDI.

For more information, please read our preprint on arXiv, or this set of slides.

Efficient Matrix Factorization

With Anil Damle at Cornell, I developed an algorithm for efficiently computing column-pivoted QR (CPQR) factorizations of extremely large matrices. It is challenging to compute a CPQR factorization in a manner that takes advantage of BLAS-3 optimizations in core matrix-matrix products routines. Whereas most existing solutions to this problem target matrices with far more rows than columns, our algorithm targets the opposite situation: matrices with far more columns than rows. Matrices of this sort appear in scientific applications related to data clustering, model reduction, and computational chemistry. For a wide class of matrices arising in these domains, our algorithm runs significantly faster than the standard GEQP3.

For more information, please read our preprint on arXiv, or these slides. This work has also been submitted to the SIAM Journal on Scientific Computing.

Fine-Grained Analysis of Interpolative Decompositions

Working with Anil Damle at Cornell and Alex Buzali, now at Harvard, I analyzed how the accuracy of interpolative decomposition algorithms for low-rank approximation are affected by the structure of the matrix being approximated. This work resulted in a set of theorems which describe the effects of subspace coherence and residual stable rank on interpolative decomposition approximation error, along with a comprehensive set of numerical experiments that compare several low-rank approximation algorithms across variations in relevant problem structures.

This work will soon appear in the SIAM Journal of Scientific Computing. You may also read our preprint on arXiv or this set of slides.