Numerical and Analytical Studies of Boundary Value Problems for PDE's. Direct and Inverse Problems

小行星9704575 这项资助的研究将集中在分析和数值 偏微分方程的研究将重点放在直接 以及逆问题。就直接问题而言， 与偏微分方程有关的有效边界层特性 将研究具有快速振荡系数的情况。这种形式的PDE 用于模拟复合材料的行为。其他 复合材料的“均匀化”问题，例如，某些 与微观和微观之间的关系有关的问题 宏观失效（包括紧密间隔纤维的脱粘， 脱层）进行研究。关于逆问题， 主要研究内容包括：1）线性椭圆型方程的反“系数”问题 具有空间变化系数的方程和2）逆“源” 半线性椭圆型方程的问题。用于识别的数据 在这两种情况下，由超定（狄利克雷或诺依曼）边界 数据这项工作的分析部分涉及以下问题： 可识别性和连续依赖性。对于反“系数” 问题，裂纹的识别调查，不均匀性 和腐蚀损害将继续进行， 将开始薄膜的“成像”;对于逆“源问题”， 与著名的希弗（或庞培）猜想的联系将是 特别强调的是， 域的“平滑性”和识别能力。数字工作 将致力于设计有效的重建方法， 最大程度地依赖于解决方案的结构信息 潜在的PDE。 这项资助的研究将集中在分析和数值 与连续介质力学有关的问题。正问题研究 应用于几个重要的实际问题，例如， 复合材料强度和潜在失效的评估 （包括断裂、纤维脱粘和分层）。特别 重点放在理解之间的关系 微观和宏观行为。反问题研究 直接应用于1）医学阻抗成像，2） 机械部件（和传感器）的无损检测，以及3） 托卡马克（聚变）装置磁诊断的解释。的一部分 本研究关注的是确定所提出的 用于各种标识的边界数据（证明唯一性和 连续依赖性结果）。这项研究的另一部分涉及 设计有效的算法，例如用于检测和定位 金属部件中的裂纹和不均匀性以及 用真实的测定薄膜（气敏元件）的氧化程度 实验数据将有博士后的积极参与 研究人员和研究生，甚至希望是一些高级研究人员 本科生在各个方面的研究。

9704575 Vogelius The research funded by this grant will focus on analytical and numerical studies of partial differential equations. There will be an emphasis on direct as well as on inverse problems. As far as direct problems are concerned, the effective boundary layer behavior encountered in connection with PDE's with rapidly oscillating coefficients will be investigated. PDE's of this form are used to model the behavior of composite materials. Other ``homogenization" problems for composite materials, for example, certain problems associated with the relationship between microscopic and macroscopic failure (including debonding of closely spaced fibers and delamination) will be studied. Concerning inverse problems, those to be investigated include 1) inverse "coefficient" problems for linear elliptic equations with spatially varying coefficients and 2) inverse "source" problems for semilinear elliptic equations. The data used for identification in both cases consists of overdetermined (Dirichlet or Neumann) boundary data. The analytical component of this work concerns such questions as identifiability and continuous dependence. For the inverse "coefficient" problem, the investigation of the identification of cracks, inhomogeneities and corrosion damage will be continued and a study related to the ``imaging" of thin films will be initiated; for the inverse "source problem," connections with the well known Schiffer (or Pompeiu) conjecture will be investigated with particular emphasis on the relation between the ``smoothness" of the domain and the ability to identify. The numerical work will be devoted to the design of effective reconstruction methods, which to the largest extent possible rely on structural information about the solutions of the underlying PDE. The research funded by this grant will focus on analytical and numerical problems related to continuum mechanics. The research on direct problems has applications to several impor tant practical problems, for instance the assessment of the strength and potential failure of composite materials (including fracture, debonding of fibers and delamination). Special emphasis is put on the understanding of the relationship between microscopic and macroscopic behavior. The research on inverse problems has immediate applications to 1) medical impedance imaging, 2) nondestructive testing of mechanical parts (and sensors) as well as 3) the interpretation of magnetic diagnostics for Tokamak (fusion) devices. Part of this research is concerned with determining the sufficiency of the proposed boundary data for the various identifications (proving uniqueness and continuous dependence results). Another part of this research involves the design of effective algorithms, for instance for the detection and location of cracks and inhomogeneities in metal components as well as for the determination of the level of oxidation of thin films (gas sensors) using real experimental data. There will be an active involvement of post-doctoral researchers as well as graduate students and hopefully even some advanced undergraduate students in various aspects of the research.