Inverse problems are a ubiquitous feature of interdisciplinary mathematics, arising in many physical, biomedical, geological and astrophysical problems. I am particularly interested in inverse problems associated with optics, both classical and quantum, in random media. This involves formulating accurate mathematical models of imaging modalities, developing methods for image reconstruction, and creating computationally efficient algorithms for imaging.
Discrete diffuse optical tomography
I considered a discrete analog of diffuse optical tomography, studying the time-independent diffusion equation on a graph with boundary in the presence of a non-uniform vertex potential. In two papers I analyzed both the forward problem (solving the equation given the potential) and the inverse problem (recovering the potential given boundary data). For the forward problem I developed and analyzed a perturbative approach based on a discrete version of the Born series, formulating necessary conditions for the convergence of my method and using techniques from representation theory, physics and combinatorics to apply it to many families of graphs. For the inverse problem, I inverted the series used to solve the forward problem, obtaining a reconstruction algorithm based on a discrete version of inverse Born series. Using methods from several complex variables and degree theory I obtained sufficient conditions under which the inverse Born series converges to the true vertex potential.
Acousto-optics in random media
The acousto-optic effect refers to the scattering of light from a medium whose optical properties are modulated by an acoustic wave. Previously, I worked on developing a first-principles theory of the acousto-optic effect in random media. I am currently working on solving the corresponding hybrid inverse problem both for radiative transport and the diffusion.
Fast solvers for radiative transfer equations
The propagation of electromagnetic waves in random media over long distances is frequently modelled by radiative transfer equations (RTEs). I developed an algorithm for solving RTEs in piecewise homogeneous media via volume integral equations (VIEs). By discretizing the VIEs pertaining to RTEs using truncated harmonic angular expansions and spatial pulse functions defined on a Cartesian spatial grid, I was able to use the free-space vacuum RTE Green function in conjunction with this basis to obtain Toeplitz operators that can be efficiently generated, stored, and applied. Currently I am developing a surface integral solver for the RTE in piecewise constant media.
Quantum optics in chiral and bi-directional waveguides
We are looking at the problem of single-photon propagation in many-atom waveguide QED systems with disorder. I am interested both in numerical simulations of photons in quantum qubit chains as well as waveguide QED systems in the continuum limit, increasing the density of atoms while simultaneously decreasing their coupling to the electro-magnetic field.