Uniqueness and Reconstructions Methods for Inverse Problems

本提案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，我们专注于开发极其快速的算法，旨在利用最少的数据检测重要特征。这个建议的一个中心特征是研究所谓的反常扩散模型的逆问题。经典扩散是建立在布朗运动的基础上的，它起源于19世纪的物理学。在这里，一个非常局部的扰动以钟形曲线的特征形状扩散，而且，在给定时间步长的过程状态仅取决于前一个时间步长的状态。虽然这适用于广泛的模型，但它不适用于那些表现出“历史”或“记忆”效应的模型。这包括在过去二十年中开发的许多材料以及经济预测，如股票和商品市场模型。结果表明，异常扩散逆问题的病态程度可能与经典情况非常不同，这表明确实涉及到基本的新物理学。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。许多具有物理意义的物体不能直接研究。例如，人体内部成像，确定固体物体内部的裂缝，以及材料参数，如不可接近物体的导电性。当这些问题被翻译成数学术语时，它们采用偏微分方程的形式，这是数学科学的通用语言。然而，由于我们在模型中有额外的未知数，这些引入了方程中的未知参数，这些参数必须通过进一步的测量来额外解决。在这个建议中，我们从数学和计算的角度来处理这种“逆问题”的实际方面。我们感兴趣的是什么时候可以从给定的数据量中做出唯一的决定，但这些反问题的特点往往是严重的“病态”，这意味着即使问题只有一个解决方案，两个非常不同的对象也可能产生无限接近的数据集。这方面使得设计和分析算法的有效数值恢复的未知极具挑战性。逆问题可以有多个复杂尺度。有些，如地震建模，需要大规模的计算资源和积累相当数量的数据。其他依赖于以最少的数据收集获得极快的计算；开发算法，使手持扫描仪能够定位结构材料中的缺陷，或便携式机器能够以无创方式检测肿瘤。该建议在培养本科生方面也具有重要的教育成分。从振动、热传导和声散射的应用中可以看出逆问题的许多独特特征，并且可以具有重要的动手成分。实验设备很容易买到，而且很便宜。金属板制成导电的二维介质，锯子切割绝缘夹杂物，廉价的热敏电阻可用来测量数据。扬声器产生入射波，麦克风产生接收器，在模拟信号和数字数据之间转换的软件在大多数笔记本电脑上都有。“隐藏”物体的神秘感可以通过黑色、光不透明、声学透明的扬声器布来增加。我们已经积累了很多这样的设备，其中一些在以前的本科生研究经历中得到了很好的应用。

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.