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

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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.