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

9704575 Vogelius本基金资助的研究将集中于偏微分方程的分析和数值研究。我们将着重于正问题和逆问题。就直接问题而言，将研究与系数快速振荡的偏微分方程有关的有效边界层行为。这种形式的PDE用于模拟复合材料的行为。复合材料的其他“均质化”问题，例如，与微观和宏观破坏（包括紧密间隔纤维的脱粘和分层）之间关系相关的某些问题将被研究。在反问题方面，研究的问题包括：1)空间变系数线性椭圆方程的“系数”反问题；2)半线性椭圆方程的“源”反问题。在这两种情况下用于识别的数据由超定（狄利克雷或诺伊曼）边界数据组成。这项工作的分析部分涉及诸如可识别性和持续依赖性等问题。对于逆“系数”问题，将继续对裂纹、不均匀性和腐蚀损伤的识别进行研究，并开展薄膜“成像”的相关研究；对于逆“源问题”，将研究与著名的Schiffer（或Pompeiu）猜想的联系，特别强调域的“平滑”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底层PDE解的结构信息。该基金资助的研究将集中于与连续介质力学有关的分析和数值问题。对直接问题的研究可以应用于几个重要的实际问题，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。特别强调的是对微观和宏观行为之间关系的理解。逆问题的研究在以下领域有直接的应用：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）设备的磁诊断解释。本研究的一部分是关于确定所提出的各种识别的边界数据的充分性（证明唯一性和连续依赖性结果）。这项研究的另一部分涉及有效算法的设计，例如用于检测和定位金属部件中的裂纹和不均匀性，以及使用真实实验数据确定薄膜的氧化水平（气体传感器）。博士后研究人员，研究生，甚至一些高级本科生都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.