Mathematical Sciences: Waves and Diffusion in Random Media

主要研究方向为：随机介质的脉冲反射与辐射输输、非线性薛定谔方程的聚焦奇点、对流增强扩散、高对比度介质的正反问题。在随机介质的脉冲反射中，目标是将以前的声波工作扩展到弹性波。建议深入研究波的输运近似和输运方程的有效边界条件的推导。在非线性薛定谔方程中，目标是解决时间色散是否阻止有限时间奇点的形成这一非常困难的问题。在对流增强扩散中，计划是通过考虑时间相关和非线性流动，并使用在PI以前的工作中证明非常有用的鞍点变分原理，继续过去对对流主导现象的研究。在高对比度介质领域，建议主要关注当待估计参数在空间域上有很大变化时，线性化反演方案不适合的反演问题。这项研究旨在利用先进的数学建模和求解技术来理解复杂材料和环境的行为。应用包括地震成像在勘探地震学和更全球化的环境中，地球物理环境的层析成像，污染物的湍流分散和相关问题。这项研究与应用密切相关，并解决了其中出现的具体问题，同时保持了数学分析目前可以做的最前沿，并产生了新的和有趣的数学。

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.