This project studies the propagation of high frequency waves in highly heterogeneous media. Radiative transport theory provides an accurate macroscopic description of the microscopic interactions between the propagating waves and the rapidly-fluctuating underlying medium. The objectives of the research are threefold. First, extend radiative transport theory to include boundary conditions and surface and volume wave interaction. Second, solve the volume and surface equations numerically using a Monte Carlo method based on a probabilistic representation of radiative transfer with polarization. Finally, assess the domain of validity of radiative transfer by setting up suitable inverse problems that allow determination of statistical parameters characterizing the underlying medium from boundary measurements.
A major recent success of radiative transport theory is the modeling of the propagation of seismic waves in the earth's crust. Numerical study of seismic wave propagation over hundreds of kilometers remains prohibitive with a microscopic model, but simulations are now accessible using radiative transfer theory and suitable statistical methods. Results promise to yield better understanding and prediction of earthquakes. Another application of radiative transport and its diffusion approximation is near-infrared spectroscopy. This novel method is increasingly used in medical imaging for monitoring properties of human tissues. Numerical simulation of forward and inverse transport problems remains an active field of research. The project provides the mathematical and numerical analysis to address these issues.
