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

该提案的目标是请求理事会支持将于2011年6月28日至7月1日在牛津举行的偏微分方程研讨会的费用。呈现多个尺度的偏微分方程建模材料的数学分析40多年来一直是一个活跃的研究领域。相应的成像或重建问题的研究是最近的一个。如果PDE的材料参数呈现高对比度，则PDE的解对于分析和计算变得特别具有挑战性。类似的问题出现在高频频域中的时间相关方程中。另一方面，在成像中首先考虑非常高的频率区域或对比度非常高的材料，因为乍一看，区分良好的区域更容易通过特设方法定位。在过去的十年中，中等频率下的反演问题的分析，在非常高的频率下的渐近性的严格推导，以及在非常不均匀介质中的椭圆偏微分方程的解的正则性得到了很多关注，部分关注是由于这些问题特别具有挑战性。另一方面，这是因为这些结果在材料科学和生物医学成像中的大量应用。最近，基于对耦合物理现象的非线性相互作用的观察的新兴生物医学成像方法（例如振动声成像）也已经成为活跃研究的主题。与这些混合成像方法相关的正问题和逆问题的数学理解的进展对于获得增强的成像可能性是至关重要的，超出了通过单独采取的不同成像方式的集成所获得的。这次讲习班的重点是促进与会者之间的合作，以期在以下方面取得重大进展：（a）完全理解高对比度或高频率的直接问题;（B）统一处理小对比度或高频率和大对比度或高频率的反问题的方法;（c）对新出现的实验测量方法进行数学建模。考虑到这一目标，我们希望将对多尺度或高对比度材料成像相关数学问题感兴趣的高级专家和年轻研究人员聚集在一起。参加研讨会的所有数学家都在积极地研究这些问题的不同方面。他们的专业知识包括异质随机介质，具有非常对比系数的线性和非线性PDE的规律性理论，数学不可见性（或隐身），成像和数值重建，高频椭圆问题的数值方法，以及新兴的生物医学成像方法。我们还邀请了一位实验物理学家，他最近的工作致力于液晶的新成像方法。与这些实验和其他混合测量方法的数学公式和理解相关的数学挑战可能是我们希望本次研讨会产生的理论发展的应用之一。

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.