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

9704575沃格利乌斯由这笔赠款资助的研究将侧重于偏微分方程式的分析和数值研究。课程将强调正问题和逆问题。对于直接问题，将研究与具有快速振荡系数的偏微分方程组有关的有效边界层行为。这种形式的PDE被用来模拟复合材料的行为。还将研究复合材料的其他“均质化”问题，例如，与微观和宏观破坏之间的关系有关的某些问题(包括紧密排列的纤维脱粘和分层)。关于反问题，需要研究的问题包括：1)具有空间变系数的线性椭圆型方程的反“系数”问题；2)半线性椭圆型方程的反“源”问题。在这两种情况下，用于识别的数据都是超定的(狄利克雷特或诺伊曼)边界数据。这项工作的分析部分涉及可辨识性和连续依赖性等问题。对于“系数”逆问题，将继续研究裂纹、不均匀和腐蚀损伤的识别，并将启动与薄膜“成像”有关的研究；对于“源”逆问题，将研究与著名的Schiffer(或Pompeiu)猜想之间的联系，特别强调区域的“光滑性”与识别能力之间的关系。数值工作将致力于设计有效的重建方法，这些方法最大程度地依赖于关于基本偏微分方程组的解的结构信息。这笔拨款资助的研究将集中在与连续介质力学有关的分析和数值问题上。正问题的研究在几个重要的实际问题上都有应用，例如复合材料的强度和潜在破坏(包括断裂、纤维脱粘和分层)的评估。特别强调对微观行为和宏观行为之间关系的理解。逆问题的研究直接应用于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)装置的磁性诊断解释。这项研究的一部分是关于确定所提出的边界数据对于各种身份识别的充分性(证明唯一性和连续依赖结果)。这项研究的另一部分涉及设计有效的算法，例如，检测和定位金属部件中的裂纹和不均匀，以及使用真实的实验数据确定薄膜(气体传感器)的氧化程度。博士后研究人员以及研究生，甚至一些高级本科生将积极参与研究的各个方面。

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.