Research interests

My research interests include

Inverse Problems and Imaging
Numerical Linear Algebra
Applications to environmental science, medical imaging, etc.

The relevant papers can be found here.

Software

Computing Karhunen-Loeve Decomposition on irregular grids link
Interpolatory tensor decomposition link
Hybrid iterative methods for Inverse Problems link

To get a sense of my research interests, here are some of my past projects.

Randomized Numerical Linear Algebra

Randomized matrix methods are emerging as a powerful tool for dimensionality reduction in large-scale applications arising in Scientific Computing. I am developing numerical algorithms and analysis to efficiently compute low-rank factorizations, approximate matrix trace, determinants, and compute interpolatory decompositions. These approximations are useful in applications such as optimal experimental design, matrix biclustering, etc.

Collaborators: Ilse Ipsen, Alen Alexanderian

Tensor decompositions

Many datasets can be naturally viewed as tensors. For example, a handwriting dataset contains images of handritten digits from multiple people. This can be viewed as a three-way tensor (pixels x people x digits). Efficient representation of theses datasets is important, as is the computational cost of handling these datasets, particularly in large-dimensions. I develop algorithms for computing tensor decompositions, with applications to facial recognition, handwriting digit classification.

Collaborators: Misha Kilmer

Optical imaging for biomedical applications

Diffuse optical tomography (DOT) is an imaging technique in which the region of interest is illuminated with near infrared light at multiple wavelengths and observations of the resulting scattered diffuse fields at a number of locations surrounding the medium are used to image tumors. I have worked on making hyperspectral DOT more computationally feasible, which can enable it to be a viable non-invasive, cheap option for detecting breast cancer. Photoacoustic Tomography is another interest of mine, where both optical and acoustic measurements are used in image reconstruction.

Collaborators: Eric Miller, Misha Kilmer, Sergio Fantini, Sarah Vallelian.

Subsurface imaging

Accurate characterization of the subsurface properties such as hydraulic conductivity is necessary to manage underground sites, contaminant remediation (see cool video here), and locating natural resources. I have worked on fast algorithms for estimating hydraulic conductivity from head measurements, quantifying uncertainty associated with reconstructions and linear solvers for frequency and time-dependent groundwater simulations.

Collaborators: Peter Kitanidis, Tania Bakhos, Jonghyun Lee.

CO monitoring

To reduce CO emissions from power plants and industries, CO is captured and injected in super-critical form into deep geological formations such as oil and natural gas fields, and saline aquifers. Accurate real-time monitoring of CO plumes is necessary for several reasons: first, it serves as an early warning of leaks, to serve as a tool for public policy and second, optimization of the storage process will require detailed images of CO distribution in space and time. A good introduction is available here.

Collaborators: Eric Miller and Peter Kitanidis

Krylov subspace solvers for shifted systems

Several frequency and time-dependent problems boil down to the solution of shifted systems of equations of the form

(K + sigma_j M) x_j = b qquad j=1,dots,N_f

We have developed efficient solvers for systems of these kinds by exploiting shift-invariant property of Krylov subspaces.

Collaborators: Tania Bakhos, Peter Kitanidis, Daniel Szyld and Scott Ladenheim