Mathematical Sciences: Elliptic Inverse Problems

诺尔斯 研究了参数辨识反问题。 通常，在这样的问题中，人们使用关于偏微分方程的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（有时很糟糕），更糟糕的是，在大多数实际应用中，在处理数据时，人们不仅必须与测量误差作斗争，而且有时还要与这样一个事实作斗争，即读数只能在相当稀疏的测量点集合上进行。 因此，在许多领域，有充分的证据表明，用于参数识别的算法是足够可靠的，以产生广泛的实际用途还没有。 在这个项目中，研究人员开发了一种新的方法来广泛的一类参数识别问题。 该方法的核心是椭圆边值问题的狄利克雷原理，即可以通过最小化某个能量泛函来获得解;如果采用适当的约束最小化，则相同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，同样的数值稳定性，这是与狄利克雷原理的方法来寻找解决方案也存在于约束最小化，此外，该方法似乎是相当强大的存在噪声的数据。 类的参数识别问题的方法是适用的，粗略地说，那些涉及椭圆方程，可以解决的手段狄利克雷原则;众所周知，这是一个大类，具有相当大的实际意义。 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，以解决含水层传输问题（地下水系统建模的关键步骤）和电阻抗断层成像人体内的问题。 逆问题处理参数识别的数据错误的存在下，在一些领域，包括但不限于，医学和工业成像，高能物理，地球物理学和水文学的基本实际重要性。 这样的问题在数学上是不适定的，因为数据中的小变化（例如由测量误差引起的）可能导致计算参数中出现不可控制的大误差。 在这个建议中，提出了一种新的优化方法，广泛的一类参数识别问题，已经显示出显着的承诺，为处理适度不适定的问题，在这一类，有迹象表明，这种方法可能有助于解决最异常不适定的例子。 该项目集中在两个测试案例，含水层transmittance问题（一个必要的和关键的步骤，在监测地下水系统中的污染物的流动，例如）和（非侵入性和非破坏性）成像的问题，在人体内使用电阻抗断层扫描。 如果在测试用例中可以用这些想法开发出适当有效的算法，那么在上面概述的领域和其他地方的广泛相关应用中，类似的算法应该是可能的。

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.