WORKSHOP: Multi-scale and high-contrast PDE: from modelling, to mathematical analysis, to inversion

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The goal of this proposal is to ask for support from the Council towards the cost of a workshop on PDE which will be held in Oxford between June 28th and July 1st 2011.The mathematical analysis of PDE modelling materials presenting multiple scales have been an active area of research for more than 40 years. The study of the corresponding imaging, or reconstruction, problem is a more recent one. If the material parameters of the PDE present high contrast ratio, the solutions of the PDE become particularly challenging to analyse and to compute. Similar problems occur in time dependent equations in the frequency domain for high frequency. On the other hand, very high frequency regimes, or very contrasted materials, were considered first in imaging, as well-differentiated areas are, at first sight, simpler to locate by ad-hoc methods. Over the last decade the analysis of the inversion problem at moderate frequencies, the rigorous derivation of asymptotics at very high frequencies, and the regularity properties of solutions of elliptic PDE in very heterogeneous media have received a lot of attention.Part of the attention is due to the fact that these problems are particularly challenging. For another part, it is because of the numerous applications of these results in material sciences and in bio-medical imaging. Recently, emerging bio-medical imaging methods based on the observation of non-linear interactions of coupled physical phenomena (such as for example vibro-acoustography) have also become the subject of active research. Progresses on the mathematical understanding of the direct and inverse problems associated to these hybrid imaging methods are crucial to obtain enhanced imaging possibilities, beyond what is obtained by the integration of different imaging modalities taken separately. The focus of this workshop will be to stimulate collaborations between the participants, in the hope of achieving significant progress in (a) complete understanding of the direct problem with high contrast or high frequencies, (b) unified approaches to the inverse problem for both small and large contrast or frequencies, and (c) mathematical modelling of emerging experimental measurement methods. With this goal in mind, we wish to bring together senior experts and young researchers interested in the mathematical problems associated with imaging of multi-scale, or high contrast materials. All the mathematicians participating in the workshop are actively working on different aspects on these problems. Their expertise comprises heterogeneous random media, regularity theory for linear and non-linear PDE with very contrasted coefficients, mathematical invisibility (or cloaking), imaging and numerical reconstruction, numerical methods for high frequency elliptic problems, and emerging biomedical imaging methods. We have also invited an experimental physicist, whose recent work is devoted to new imaging methods for liquid crystals. The mathematical challenges associated with the mathematical formulation and understanding of these experiments and other hybrid measurement methods could be one of the applications of theoretical developments we hope this workshop will produce.