Uniqueness and Reconstructions Methods for Inverse Problems

本建议中涉及的具体问题包括从表面测量中恢复内部物体的位置和形状，或从声学或电磁散射数据中确定障碍物。特别是，我们专注于开发非常快速的算法，旨在利用最少的数据来检测重要的功能。这个建议的一个中心特点是所谓的反常扩散模型的反问题的调查。经典扩散是基于布朗运动的，起源于世纪的物理学。在这里，一个非常局部化的扰动以钟形曲线的特征形状传播，而且，在给定的时间步长处的过程的状态仅取决于在前一时间步长处的状态。虽然这适用于广泛的模型，但对于那些表现出“历史”或“记忆”效应的模型来说，这是失败的。 这包括过去二十年开发的许多材料以及经济预测，如股票和商品市场建模。事实证明，病态程度的反常扩散反问题可以是非常不同的经典情况下，这表明确实涉及到基本的新物理。从数学和计算的角度来看，这是有代价的;结果分析相当复杂和具有挑战性，许多物理感兴趣的对象无法直接研究。例如，身体内部成像，固体物体内裂缝的确定，以及材料参数，如不可接近物体的导电性。当这些问题被转化为数学术语时，它们采取偏微分方程的形式，数学科学的语言弗兰卡。然而，由于我们在模型中有额外的未知数，这些在方程中引入了未知参数，这些参数必须通过进一步的测量来解决。在这个建议中，我们处理的实际方面，这样的“逆问题”从数学和计算的角度来看。我们感兴趣的是，当一个唯一的决定可以从一个给定的数据量，但这些逆问题的特点往往是严重的“病态”，这意味着，即使只有一个解决方案的问题，两个非常不同的对象可能会产生数据集是无限接近。这一方面使得设计和分析算法的有效数值恢复的未知数极具挑战性。逆问题可以具有多个复杂性尺度。一些，如地震建模，需要大规模的计算资源和积累大量的数据。其他人则依赖于以最少的数据收集获得极快的计算;开发算法，使手持式扫描仪能够定位结构材料中的缺陷或便携式机器以非侵入性方式检测肿瘤。该提案在本科生培训方面也有重要的教育内容。反问题的许多独特的功能，可以看到从考虑在振动，热传导和声散射的应用，并可以有一个显着的动手组件。实验设备是现成的和便宜的。金属板可以制作导电的2D介质，锯子可以切割绝缘夹杂物，便宜的热敏电阻可以用来测量数据。扬声器产生入射波，麦克风产生接收器，大多数笔记本电脑上都有在模拟信号和数字数据之间进行转换的软件。 神秘的“隐藏”的对象可以添加黑色，光不透明，声学透明的扬声器布。我们已经积累了很多这样的设备，其中一些在以前的本科研究经验中使用得很好。

Specific problems addressed in this proposal include the recovery of the location and shape of interior objects from surface measurements or the determination of obstacles from acoustic or electromagnetic scattering data. In particular, we concentrate on developing extremely fast algorithms designed to detect significant features utilizing only minimal data. One central feature of this proposal is the investigation of inverse problems for so-called anomalous diffusion models. Classical diffusion is based on Brownian motion and has its roots in 19th century physics. Here a very localised disturbance spreads with the characteristic shape of a bell curve and, further, the state of the process at a given time step depends only on the state at the previous time step. While this serves well for a wide range of models, it fails for those that exhibit a "history" or "memory" effect. This includes many materials that been developed over the last twenty years as well as economic forecasting such as stock and commodity market modeling. It turns out that degree of ill-conditioning in anomolous diffusion inverse problems can be very different from those of the classical case suggesting that indeed fundamental new physics is involved. From a mathematical and computational standpoint this comes at a price; the resulting analysis is considerably more complex and challenging.Many objects of physical interest cannot be studied directly. Examples include, imaging the interior of the body, the determination of cracks within solid objects, and material parameters such as the conductivity of inaccessible objects. When these problems are translated into mathematical terms they take the form of partial differential equations, the Lingua Franca of the mathematical sciences. However, since we have additional unknowns in the model, these introduce unknown parameters in the equations that have to be additionally resolved by means of further measurements. In this proposal we deal with the practical aspects of such "inverse problems" from a mathematical and computational perspective. We are interested in when a unique determination can be made from a given amount of data, but these inverse problems are characterized by often severe "ill-conditioning", meaning that even when there is only one solution to the problem, two very different objects may produce data sets that are infinitesimally close. This aspect makes designing and analyzing algorithms for the efficient numerical recovery of the unknowns extremely challenging. Inverse problems can have multiple scales of complexity. Some, such as earthquake modeling require large scale computational resources and amassing considerable amounts of data. Others rely on obtaining extremely fast computations with minimal data collection; developing algorithms that enable a hand-held scanner to locate flaws in structural materials or portable machines to detect tumors in a noninvasive way. The proposal also has a significant educational component in the training of undergraduate students. Many of the distinct features of inverse problems can be seen from considering applications in vibration, heat conduction and acoustic scattering, and can have a significant hands-on component. The experimental equipment is readily available and cheap. Metal plates make conductive 2D media, a saw cuts an insulating inclusion and cheap thermistors can be used to measure data. Loudspeakers make incident waves, microphones make receivers, the software to go between analogue signals and digital data is on most laptops. The mystery of the "hidden" object can be added by black, light opaque, acoustically transparent speaker cloth. We have amassed much of this equipment already, some of it quite well used in previous undergraduate research experiences.