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

研究者关注的是涉及多个共存尺度的材料模型，对于这些模型，临时计算方法经常产生不满意的结果。通过严格的数学研究，她发现了有效的参数和近似值，否则不明显。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，该研究涉及光子带隙材料，地球物理反演和非均质电化学表面产生的结构。研究者回答了关于小尺度参数的相互作用的开放性问题，在奇点表面的解展开，以及连续和离散反演之间的联系。通常，许多物质的行为，无论是地质沉积物还是设计的电磁材料，都依赖于不同长度尺度上材料成分之间的相互作用。研究者研究这类问题，其中的进展在应用领域和数学领域都有影响。新的光子模拟技术有助于下一代纳米级材料和光学器件的发展。量身定制的计算方法允许更精确的地球物理勘探。研究者和合作者还分析了进一步了解某些电化学过程的基本模型。研究生和本科生都可以通过适当层次的参与项目而受益。

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.