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

研讨会：多尺度和高对比度偏微分方程：从建模，到数学分析，到反演

• 批准号: EP/I028668/1
• 负责人: Yves Capdeboscq
• 金额: $ 2.01万
• 依托单位: University of Oxford
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2011
• 资助国家: 英国
• 起止时间: 2011 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FI028668%2F1
• 关键词: WORKSHOP; Multi; scale; contrast; PDE

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.

该提案的目的是向理事会提供支持，以在2011年6月28日至7月1日在牛津举行的PDE研讨会上。对PDE建模材料的数学分析呈现了多个规模的研究材料已经是研究的活跃领域已有40多年了。对相应成像或重建问题的研究是一个较新的问题。如果PDE的材料参数具有较高的对比度，则PDE的解决方案在分析和计算方面变得特别具有挑战性。高频的频域中的时间依赖方程发生了类似的问题。另一方面，首先考虑了非常高的频率状态或非常形成鲜明的材料，而差异化的区域首先是通过临时方法更简单的。在过去的十年中，对中等频率的反转问题，在非常高的频率下的渐近衍射的严格推导以及在非常异构媒体中椭圆形PDE解决方案的规律性特性引起了很多关注。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众多应用。最近，基于观察到耦合物理现象的非线性相互作用的新兴生物医学成像方法（例如振动声学）也已成为主动研究的主题。对与这些混合成像方法相关的直接和反问题的数学理解的进展对于获得增强的成像可能性至关重要，超越了通过分别采用的不同成像方式的整合而获得的。该研讨会的重点将是刺激参与者之间的合作，以期在（a）完全了解高对比度或高频的直接问题上，（b）小型和大对比度或频率的统一方法，以及（c）出现的实验测量方法的数学模型。考虑到这一目标，我们希望将对与多尺度或高对比度材料成像相关的数学问题感兴趣的高级专家和年轻研究人员汇集在一起​​。参加研讨会的所有数学家都在积极研究这些问题的不同方面。它们的专业知识包括异质的随机介质，具有非常对比系数的线性和非线性PDE的规则性理论，数学隐形性（或披肩），成像和数值重建，高频椭圆问题的数值方法以及出现生物医学成像方法。我们还邀请了一位实验物理学家，他最近的工作致力于用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表述和理解相关的数学挑战可能是我们希望该研讨会能够产生的理论发展的应用之一。