Mathematical Sciences: Waves and Diffusion in Random Media

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9622854 Papanicolaou The research is in the following areas: Pulse reflection from Random Media and Radiative Transport, the Focusing Singularity of the Nonlinear Schrodinger Equation, Convection Enhanced Diffusion, and Direct and Inverse Problems for High Contrast Media. In Pulse reflection from random media the goal is to extend the previous work for acoustic waves to elastic waves. It is proposed to get deeper into the study of transport approximations for waves and the derivation of effective boundary conditions for the transport equations.In the Nonlinear Schrodinger Equation the goal is to address the very hard problem of whether or not time dispersion prevents the formation of finite time singularities. In convection enhanced diffusion the plans are to continue the past study convection dominated phenomena by considering time dependent and nonlinear flows and using the saddle point variational principles that proved so useful in the previous work by the PI. In the area of High Contrast Media it is proposed to to focus primarily on inverse problems where linearized inversion schemes are not suitable when the parameters to be estimated have large variations over the spatial domain. %%% This research is aimed at understanding the behavior of complex materials and environments using advanced mathematical modeling and solution techniques. Applications include seismic imaging in exploration seismology and in a more global environment, tomographic imaging of geophysical environments, turbulent dispersion of contaminants and related problems. This research is closely related to the applications, and addresses specific issues that arise there, while staying at the cutting edge of what mathematical analysis can do at present and producing new and interesting mathematics.