Mathematical Sciences: Elliptic Inverse Problems

研究人员诺尔斯研究参数识别反问题。通常，在这样的问题中，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(有时是不适定的)，更糟糕的是，在处理大多数实际应用中的数据时，人们不仅必须与测量误差作斗争，有时还必须与读数可能仅在相当稀疏的测量点集合中获得的事实作斗争。因此，在许多领域，有充分的证据表明，还没有足够可靠的参数识别算法来产生广泛的实际应用。在这个项目中，研究人员开发了一种新的方法来解决广泛的参数识别问题。这种方法的核心是椭圆边值问题的Dirichlet原理，即解可以通过某个能量泛函的最小化来获得；如果采用适当的约束最小化，同样的能量泛函也可以用来计算椭圆型方程中的系数，这是关键。最近的实验表明，与Dirichlet原理求解方法相同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的情况下表现出相当的鲁棒性。这种方法适用的参数辨识问题，粗略地说，是那些涉及可用狄利克雷原理求解的椭圆型方程的问题；众所周知，这是一大类具有相当实际意义的问题。该项目的大部分工作涉及将这些想法应用于产生工作算法，以解决含水层透过率问题(地下水系统建模的关键步骤)和利用电阻抗断层成像技术在人体内成像的问题。在存在数据误差的情况下处理参数识别的反问题在许多领域具有基本的实际重要性，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。这样的问题在数学上是不适定的，因为数据中的微小变化(例如由测量误差引起)可能会导致计算参数中出现无法控制的大误差。在这一建议中，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，它已经显示出处理这类适度不适定问题的巨大希望，而且有迹象表明，这种方法可能有助于解决最严重的不适定例子。该项目集中在两个测试案例上，含水层透过率问题(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技术在人体内(非侵入性和非破坏性)成像的问题。如果可以在测试用例中利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在上面概述的领域和其他地方的广泛相关应用中使用。

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.