Mathematics Research

My advisor for my PhD thesis at MIT was Professor Laurent Demanet of MIT's Imaging and Computing Group.

For my PhD thesis, I worked on absorbing boundary conditions for the Helmholtz equation. This is in the field of in numerical analysis and scientific computing, more precisely wave propagation problems and fast algorithms.

Scattering off an object far away.

More specifically, I looked at variable media, where typically an absorbing layer will need to be very thick in order to absorb the waves sufficiently for practical purposes. In situations where one needs to solve the Helmholtz equation many times, as is the case in some geophysics and optics applications, compressing this wide absorbing layer as a precomputation leads to speed-ups for each subsequent solve.

The figure above is a part of the Dirichlet-to-Neumann map for a medium with a discontinuity, where we have removed the oscillations caused by the first-arrival travel time. The Dirichlet-to-Neumann map is a crucial operator in the study of absorbing boundary conditions.

The image below depicts the low-rank hierarchical matrix decomposition of a DtN map that was first compressed using Matrix Probing.

Low-rank hierarchical decomposition of a DtN map.

I collaborated with Christophe Geuzaine and Alexandre Vion on incorporating this compressed DtN map in their Domain Decomposition Method (DDM) solver for the Helmholtz equation.