波传播反问题的数学分析、计算方法及应用

We propose to study direct and inverse scattering problems in electromagnetic, acoustic and elastic wave propagation. Our focus will be on the development of novel and basic algorithms for solving the related inverse problems. The proposed topics include: (1) computational modelling for multi-scale scattering problems, (2) inverse medium and source problems with multi-frequencies scattered data, (3) convergence and stability analysis with uncertainty principle, (4) algorithms application in Geophysics. The methodology developed will be general and applicable to inverse problems in many other engineering applications. A distinctive feature of the proposed project is to combine the basic algorithm developments and computational modeling techniques to develop novel computational methods for solving many challenging inverse problems. The developed methods are expected to make a breakthrough in the modelling, analysis, and computation of inverse problems and Geophysics.

本项目将研究波传播包括电磁波、声波、弹性波等的散射和反散射问题。我们将发展一类新的基础算法及相关的可计算建模来求解波传播中的反问题,主要研究内容包括（1）大规模多尺度散射问题可计算建模,（2）多频数据下的介质、源项反演算法,（3）测不准原理下的收敛性和稳定性分析，（4）多频算法在地球物理反演中的应用。这类新的数值求解反散射问题的方法从理论上将克服反问题不适定性等著名共性难题。本项目的主要特色就是将基础算法与可计算建模相结合, 发展新的计算方法, 解决相关共性难题, 力争在理论、算法、建模及地球物理反演应用上取得突破。

项目组围绕复杂介质波传播中的反问题的数学可计算建模、数学理论、基础算法和实验验证的相关内容，开展深入研究。项目组在波动方程正反散射问题(包括声波、电磁波和弹性波)，偏微分方程基础算法，以及反问题应用，特别是微地震监测等方面取得了一系列的成果。