Elliptic Inverse Problems

这是一种新的参数估计反问题方法的继续。所讨论的系数被计算为某些非负泛函的唯一全局最小值，这些非负泛函也往往具有唯一的临界点，如果人们寻求真正有效的数值算法，后一性质是至关重要的。这个想法的核心是观察到自伴椭圆型方程的狄里克莱特原理可以被重新表述为方程中的系数，而不是解。最近的工作表明，这些技术也很容易扩展到抛物型和双曲型方程组，这大大扩大了适用范围。由于这些方法直接面对非线性逆问题，所以它们的计算代价比问题最初线性化的算法要高。另一方面，当成功时，直接方法倾向于提供更好的图像，而不是线性化方法周围的各种伪影问题。这些方法在面对严重的不适定性时也表现出了显著的稳定性和准确性。该建议主要集中在两种一般情况，即地下水流动和运移方程中所有系数函数的重建问题和阻抗层析成像问题(尚未解决)。选择这些例子在一定程度上表明了这一思想圈的广泛适用性。第一类被选为区域内可获得解决方案测量的一类问题的代表，而第二类被选为代表仅有解决方案边界信息的情况的一个例子。还提出了利用反射地震资料对海底区域进行成像和利用微波脉冲雷达进行地雷探测的扩展建议。这些方法还可能在证明相关反问题唯一性定理方面具有深远的理论意义。复杂的物理过程通常由线性常微分方程组或偏微分方程组来数学表示。该建模过程的关键部分包括确定对某些过程进行建模的方程中的所有系数函数。在许多有实际意义的情况下，由于各种原因，直接测量这些函数是不切实际的。例如，在地下水建模中，人们不能轻易地测量大多数地下参数，而在医学成像中，人们总是试图以非侵入性的方式推断内部特性。另一方面，人们可以对特定物理过程的影响进行有用的测量，这通常是真的。例如，在地下水流动中，人们可以测量一段时间内井网中的水的高度(水头)，而在电阻抗断层成像中，人们可以在物体表面施加电流，并测量产生的表面电压。在数学上，在这些例子中的每一个中，人们被给予关于基本方程的解的数据，目的是使用该数据来估计一些或所有参数函数。这就是逆问题的本质。该项目继续研究反问题的计算算法，包括医学成像、地雷探测、海底地震勘探和地下水建模。预期的好处将包括极大地改善低能量电子层析成像的图像质量，以及制作用于地下含水层管理和修复的完整流动和污染物模型的可能性。

This is a continuation of work on a new approach to parameter estimation inverse problems. The coefficients in question are computed as the unique global minimum of certain non-negative functionals that also tend to have unique critical points, the latter property being of crucial importance if one seeks truly effective numerical algorithms. The core of the idea involves the observation that the Dirichlet principle for self-adjoint elliptic equations can be reformulated to produce the coefficients in the equation, rather than the solution. Recent work indicates that the techniques extend readily to parabolic and hyperbolic systems as well, which extends the range of applicability considerably. As these methods confront the nonlinear inverse problems directly, they are computationally more expensive than algorithms wherein the problem is initially linearized. On the other hand, when successful, the direct methods tend to give better images, free from the various artifact problems that surround the linearization methods. The methods also exhibit remarkable stability and accuracy in the face of significant ill-posedness. The proposal concentrates mainly on two generic cases, the (as yet unsolved) problem of the reconstruction of all the coefficient functions in the equations for groundwater flow and transport and the electrical impedance tomography problem. These examples have been chosen in part to indicate the broad applicability of this circle of ideas. The first is chosen as a representative of the class of problems in which measurements of the solution are available from inside the region, while the second is an example representative of the situation in which only boundary information on the solution is available. An indication is also given on an extension to imaging undersea regions from reflection seismological data, and landmine detection using microwave impulse radar. These methods may also have profound theoretical implications as well, in the direction of proving associated inverse problem uniqueness theorems.Complex physical processes are often represented mathematically by systems of linear ordinary or partial differential equations. A crucial part of this modeling process involves the determination of all of the coefficient functions in the equations modeling certain processes. In many situations of practical interest it is, for various reasons, impractical to measure these functions directly. In groundwater modeling, for example, one cannot easily measure most subsurface parameters, and in medical imaging, one is always trying to infer internal properties ``non-invasively." On the other hand, it is often true that one can make useful measurements of the effects of a particular physical process. For example, in groundwater flow, one can measure the height (head) of water, over time, in a grid of wells, and in electrical impedance tomography, one can apply currents at the surface of a body and measure the resulting surface voltages. Mathematically, in each of these examples one is given data on the solution of a underlying equation with the intent of using this data to estimate some or all of the parameter functions. This is the essence of an inverse problem. This project continues work on computational algorithms for inverse problems involving medical imaging, landmine detection, undersea seismic exploration, and groundwater modeling. Expected benefits would include greatly improved image quality in low energy electrical tomography and the possibility of producing complete flow and contaminant models for use in the management and remediation of underground aquifers.