Elliptic Inverse Problems

椭圆反问题

• 批准号: 0107492
• 负责人: Ian Knowles
• 金额: $ 7.8万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-09-01 至 2004-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0107492&HistoricalAwards=false
• 关键词: Elliptic; Inverse; Problems

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.

这是一个新的方法参数估计逆问题的工作的延续。 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，后者的性质至关重要。 这个想法的核心是观察到自伴椭圆方程的狄利克雷原理可以重新公式化，以产生方程中的系数，而不是解。 最近的工作表明，该技术很容易扩展到抛物型和双曲型系统以及，这大大扩展了适用范围。 由于这些方法直接面对非线性反问题，它们在计算上比其中问题初始线性化的算法更昂贵。 另一方面，当成功时，直接方法倾向于给出更好的图像，没有围绕线性化方法的各种伪影问题。 该方法也表现出显着的稳定性和准确性，在面对显着的不适定性。 该建议主要集中在两个通用的情况下，（尚未解决）的问题，重建的所有系数函数的地下水流动和运输的方程和电阻抗层析成像问题。 选择这些例子的部分原因是为了表明这一思路的广泛适用性。 第一个被选为一类问题的代表，其中测量的解决方案是从内部的区域，而第二个是一个例子代表的情况下，只有边界信息的解决方案是可用的。 还说明了扩大利用反射地震数据对海底区域成像和利用微波脉冲雷达探测地雷的情况。 这些方法也可能有深刻的理论含义，以及在证明相关的反问题的唯一性定理的方向。复杂的物理过程往往是由线性常微分方程或偏微分方程组的数学表示。 该建模过程的一个关键部分涉及确定建模某些过程的方程中的所有系数函数。 在许多实际情况下，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，人们不能容易地测量大多数地下参数，而在医学成像中，人们总是试图“非侵入性地”推断内部属性。“另一方面，人们通常可以对特定物理过程的影响进行有用的测量。 例如，在地下水流中，人们可以在威尔斯的网格中测量水随时间的高度（水头），并且在电阻抗层析成像中，人们可以在身体的表面处施加电流并测量所得的表面电压。 在数学上，在这些示例中的每一个中，给出关于基本方程的解的数据，目的是使用该数据来估计参数函数中的一些或全部。 这就是反问题的本质。 这个项目继续研究逆问题的计算算法，包括医学成像、地雷探测、海底地震勘探和地下水建模。 预期的好处将包括大大提高低能量电子层析成像的图像质量，并有可能制作完整的流动和污染物模型，用于地下含水层的管理和补救。