Mathematical Sciences: Elliptic Inverse Problems

研究者研究参数辨识反问题。通常，在这样一个问题中，人们使用关于偏微分方程解的某些给定信息来计算方程中的一个或多个系数。这样的问题总是不适定的（有时很严重），更糟糕的是，在大多数实际应用中处理数据时，人们不仅要与测量误差作比较，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比较。因此，在许多领域，充分可靠的参数识别算法尚未得到广泛的实际应用，这是有据可查的。在这个项目中，研究者开发了一种新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，这是关键的观察结果。最近的实验表明，与狄利克雷原理方法相关联的数值稳定性也存在于约束最小化中，而且该方法在数据中存在噪声时似乎相当稳健。这种方法所适用的参数辨识问题，粗略地说，是那些涉及到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。该项目的大部分工作涉及将这些想法应用于解决含水层透射率问题（地下水系统建模的关键步骤）和使用电阻抗断层成像的人体内部成像问题的工作算法。在存在数据误差的情况下处理参数识别的逆问题在许多领域具有根本的实际重要性，包括但绝不限于医学和工业成像、高能物理、地球物理和水文学。这些问题在数学上是病态的，因为数据中的微小变化（例如由测量误差引起的）可能导致计算参数中出现无法控制的大误差。在本提案中，提出了一种新的优化方法来解决一类广泛的参数识别问题，该方法已经显示出处理该类中中度不适定问题的重大希望，并且有迹象表明这种方法可能有助于解决最严重的不适定示例。该项目集中于两个测试案例，含水层透射率问题（例如，监测地下水系统中污染物流动的必要和关键步骤）和使用电阻抗断层成像（非侵入性和非破坏性）在人体内部成像的问题。如果在测试用例中可以使用这些想法开发出适当有效的算法，那么类似的算法应该可以用于上述领域和其他地方的广泛相关应用程序。

Knowles The investigator studies parameter identification inverse problems. Typically, in such a problem one uses certain given information about the solution of a partial differential equation to compute one or more of the coefficients in the equation. Such problems are invariably ill-posed (sometimes badly so) and, to make matters worse, in processing the data in most 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 consequence, in many areas, it is well documented that algorithms for parameter identification that are sufficiently reliable to engender widespread practical use are not yet available. In this project the investigator develops a new approach to a broad class of parameter identification problems. At the core of the approach 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 key observation. Recent experiments have shown that the same numerical stability that is associated with the Dirichlet principle approach to finding solutions is also present in the constrained minimization, and furthermore the method appears to be quite robust in the presence of noise in the data. The class of parameter identification problems to which the approach is applicable is, roughly speaking, those that involve elliptic equations that may be solved by means of a Dirichlet principle; as is well known, this is a large class, having considerable practical significance. The bulk of the work in the project involves applying these ideas to producing working algorithms for the solution of the aquifer transmissivity problem (a crucial step in the modeling of under ground water systems) and the problem of imaging inside the human body with electrical impedance tomography. Inverse problems dealing with parameter identification in the presence of data error are of fundamental practical importance in a number of areas, including, but by no means limited to, medical and industrial imaging, high energy physics, geophysics, and hydrology. Such problems are mathematically ill-posed in the sense that small variations in the data (caused by measurement error for example) can cause uncontrollably large errors to appear in the calculated parameters. In this proposal a new optimization approach to a broad class of parameter identification problems is presented that already shows significant promise for handling the moderately ill-posed problems in this class, and there are indications that such an approach may help solve the most egregiously ill-posed examples. The project concentrates on two test cases, the aquifer transmissivity problem (a necessary and crucial step in monitoring the flow of contaminants in underground water systems, for example) and the problem of (noninvasive and nondestructive) imaging inside the human body using electrical impedance tomography. If suitably effective algorithms can be developed with these ideas in the test cases, similar algorithms should be possible for a wide range of related applications in the areas outlined above, and elsewhere.