Elliptic Inverse Problems

[805629]本课题主要研究参数辨识逆问题。通常，在这样的问题中，人们会得到关于偏微分方程解的某些信息，人们希望利用这些信息来计算方程中的一个或多个系数。这些问题通常在数学上是病态的，更糟糕的是，在处理许多实际应用中的数据时，人们不仅要与测量误差作斗争，有时还要与读数可能只在一个相当稀疏的测量点集合上进行的事实作斗争。在本建议中，我们概述了继续研究一种新的和非常有前途的方法来解决一类广泛的参数识别问题。该方法的核心是椭圆边值问题的Dirichlet原理，即可以通过最小化某一能量泛函来获得解；如果采用适当的约束最小化，同样的能量泛函也可以用于计算椭圆方程中的系数，这是驱动本建议中大部分工作的重要观察结果。大量的数值实验表明，在约束最小化中也存在与狄利克雷原理求解方法相关的相同的数值稳定性（假设采用适当的稳定化）；特别是在数据中存在噪声的情况下，该方法显得相当稳健。适用于这种方法的参数识别问题，粗略地说，是那些涉及可以用狄利克雷原理求解的椭圆方程的问题；众所周知，这是一个很大的班级，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）方程的参数识别逆问题可能被重述为该理论所涵盖的形式，因此最终的影响预计会更大。该项目的第一部分包括生成工作算法，通过容易获得的测量数据来识别量化多孔介质中流动所需的各种参数。这是建立地下水系统模型的关键一步，也是必要的第一步，例如，当人们希望有效地管理水资源，或预测洪水或工业污染等环境因素对含水层的影响时。该项目的第二部分是关于用电阻抗断层成像在人体内部成像的问题。在这里，人们试图通过在物体表面进行低电压电测量来形成物体内部的图像。这种成像具有非侵入性和非破坏性的优点，而且价格低廉；目前的缺点包括图像质量差，希望通过与上述使用的方法相关的方法可以实现改进。这些想法的相关实际应用包括非破坏性评估，包括测定铸造金属中的气孔、杂质和裂纹；这些问题在各种制造情况下都很重要，包括飞机部件的检查，以及核反应堆安全壳的制造和测试。

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.