Elliptic Inverse Problems

椭圆反问题

• 批准号: 9805629
• 负责人: Ian Knowles
• 金额: $ 6.57万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-15 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9805629&HistoricalAwards=false
• 关键词: Elliptic; Inverse; Problems

9805629 KnowlesThis project centers on parameter identification inverse problems. Typically, in such a problem one is given certain information about the solution of a partial differential equation and one is hopeful of using this information to compute one or more of the coefficients in the equation. Such problems are typically mathematically ill-posed and, to make matters worse, in processing the data in many practical applications, one must contend not only with measurement error but sometimes also with the fact that the readings may only be taken at a rather sparse collection of measurement points. In this proposal we outline a continuation of work on a new and very promising approach to a broad class of parameter identification problems. At the core of the method is the Dirichlet principle for elliptic boundary value problems, that the solution may be obtained by the minimization of a certain energy functional; that the same energy functional can also be used to compute coefficients in the elliptic equation, if an appropriate constrained minimization is employed, is the significant observation that drives much of the work in this proposal. Extensive numerical experiments have shown that the same numerical stability that is associated with the Dirichlet principle approach to finding solutions is also present (assuming the appropriate stabilization is employed) in the constrained minimization; in particular the method appears to be quite robust in the presence of noise in the data. The parameter identification problems amenable to this approach are, roughly speaking, those that involve elliptic equations that may besolved by means of a Dirichlet principle; as is well known this is a large class, having considerable practical significance. In addition, many parameter identification inverse problems involving parabolic (diffusion) and hyperbolic (wave propagation) equations may be restated to a form covered by this theory, and thus the eventual impact is expected to be even greater.The first part of the project involves producing working algorithms to identify, from easily obtainable measurements, the various parameters needed to quantify flow in porous media. This is a crucial step in the modeling of underground water systems and a necessary first step when one wishes for example to manage water resources efficiently, or to predict the effects on an aquifer caused by environmental factors such as flooding or industrial pollution. The second part of the project is concerned with the problem of imaging inside the human body with electrical impedance tomography. Here, one attempts, by means of low voltage electrical measurements taken at the surface of a subject, to form an image of the interior. Such imaging has the advantage of being both non-invasive and non-destructive, and inexpensive; current disadvantages include poor image quality, and it is hoped that improvements can be accomplished by methods related to those used above. Related practical applications of these ideas include non-destructive evaluations involving the determination of gas pores, impurities, and cracks in cast metals; such problems are of interest in a variety of manufacturing situations, including the inspection of airplane parts, and also in the manufacture and testing of nuclear reactor containment vessels.

9805629 Knowles 该项目以参数辨识逆问题为中心。 通常，在这样的问题中，给出了有关偏微分方程的解的某些信息，并且人们希望使用该信息来计算方程中的一个或多个系数。 此类问题通常在数学上是不适定的，更糟糕​​的是，在许多实际应用中处理数据时，人们不仅必须应对测量误差，有时还必须应对这样的事实：读数只能在相当稀疏的测量点集合中获取。在本提案中，我们概述了针对广泛的参数识别问题的一种新的且非常有前途的方法的继续工作。 该方法的核心是椭圆边值问题的狄利克雷原理，通过最小化某个能量泛函来求得解；如果采用适当的约束最小化，相同的能量泛函也可以用于计算椭圆方程中的系数，这是推动本提案中大部分工作的重要观察。 大量的数值实验表明，与狄利克雷原理方法寻找解决方案相关的相同数值稳定性也存在于约束最小化中（假设采用了适当的稳定性）；特别是在数据中存在噪声的情况下，该方法似乎非常稳健。 粗略地说，适用于这种方法的参数识别问题是那些涉及可以通过狄利克雷原理解决的椭圆方程的问题；众所周知，这是一个大类，具有相当大的现实意义。 此外，许多涉及抛物线（扩散）和双曲（波传播）方程的参数识别反问题可以重述为该理论所涵盖的形式，因此最终的影响预计会更大。该项目的第一部分涉及生成工作算法，以从容易获得的测量中识别量化多孔介质中的流动所需的各种参数。 这是地下水系统建模中的关键步骤，也是当人们希望有效管理水资源或预测洪水或工业污染等环境因素对含水层造成的影响时必要的第一步。 该项目的第二部分涉及电阻抗断层扫描人体内部成像的问题。 在这里，人们试图通过在对象表面进行的低压电气测量来形成内部图像。 这种成像的优点是既非侵入性、非破坏性，又便宜；目前的缺点是图像质量较差，希望可以通过与上述相关的方法来改进。 这些想法的相关实际应用包括无损评估，涉及铸造金属中气孔、杂质和裂纹的测定；这些问题在各种制造情况下都令人感兴趣，包括飞机零件的检查，以及核反应堆安全壳的制造和测试。