Asymptotics at Resonant Scales: Application to Inhomogeneous Material Simulation, Discretization and Inversion

根据莫斯科DMS-0605021标准，研究人员将重点放在涉及多个共存尺度的材料模型上，对于这些模型，特别计算方法经常产生不令人满意的结果。通过有趣的数学研究，她找到了一些其他方面并不明显的有效参数和近似值。她将计算和分析工具结合起来，创造出可以模拟和成像复合材料的新技术。特别是，这项研究涉及来自光子带隙材料、地球物理反转和非均质选择化学表面的结构。研究人员回答了关于小尺度参数的相互作用、奇异面上的解展开以及连续和离散反演之间的联系等公开问题。通常情况下，许多物质的行为，无论是地质沉积物还是设计的电磁材料，都取决于不同长度尺度上材料成分之间的相互作用。研究人员研究这类问题，在这类问题中，进步对应用领域和数学都有影响。新的光子模拟技术有助于下一代纳米材料和光学器件的发展。量身定做的计算方法允许进行更准确的地球物理勘探。研究人员和合作者还分析了模型，这些模型进一步加深了对某些电化学过程的基本理解。研究生和本科生都可以通过在适当的级别参与该项目而受益。

MoskowDMS-0605021 The investigator focuses on material models involvingmultiple coexisting scales, for which ad-hoc computationalapproaches frequently yield unsatisfactory results. Through arigorous mathematical study, she finds effective parameters andapproximations that are not otherwise obvious. She combinescomputational and analytic tools to produce novel techniques thatcan simulate and image composite materials. In particular, thestudy involves structures arising from photonic bandgapmaterials, geophysical inversion, and heterogeneouselectrochemical surfaces. The investigator answers openquestions about the interaction of small scale parameters,solution expansions at singularity surfaces, and the linksbetween continuous and discrete inversion. Often the behaviors of many material substances, whethergeological deposits or designed electromagnetic materials, dependon interactions between components of the materials at differentscales of length. The investigator studies problems of this sort,in which advances have an impact in the applications area as wellas in mathematics. The new photonic simulation techniques aid inthe development of the next generation of nano-scale materialsand optical devices. The tailored computational approaches allowfor more accurate geophysical prospecting. The investigator andcollaborators also analyze models that further the fundamentalunderstanding of certain electrochemical processes. Bothgraduate and undergraduate students benefit by participating inthe project at appropriate levels.