Uniqueness and Reconstructions Methods for Inverse Problems

这项建议涉及的具体问题包括从表面测量恢复内部物体的位置和形状，或根据声学或电磁散射数据确定障碍物。特别是，我们专注于开发极快的算法，旨在仅使用最少的数据来检测重要特征。这一建议的一个主要特点是研究所谓的反常扩散模型的反问题。经典扩散以布朗运动为基础，起源于19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，但对于那些表现出“历史”或“记忆”效应的模型来说，它就失败了。这包括在过去二十年中开发的许多材料以及经济预测，如股票和商品市场建模。事实证明，异常扩散反问题的病态程度可能与经典情况下的非常不同，这表明确实涉及到了基本的新物理。从数学和计算的角度来看，这是要付出代价的；由此产生的分析要复杂得多，也更具挑战性。许多具有物理意义的物体不能直接研究。例如，成像身体内部，确定固体物体内的裂缝，以及材料参数，如无法接触到的物体的导电性。当这些问题被转化为数学术语时，它们就会以偏微分方程式的形式出现，这是数学科学中的语言。然而，由于我们在模型中有额外的未知数，这些未知数在方程中引入了未知参数，必须通过进一步的测量来额外求解。在这项建议中，我们从数学和计算的角度处理这种“反问题”的实际方面。我们感兴趣的是何时可以从给定的数据量做出唯一的确定，但这些逆问题的特征通常是严重的“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可能产生无限接近的数据集。这一方面使得设计和分析用于有效地恢复未知数的算法非常具有挑战性。反问题可以有多个复杂程度。有些方法，如地震模拟，需要大规模的计算资源和大量的数据积累。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，使手持扫描仪能够定位结构材料中的缺陷，或者使便携式机器能够以非侵入性的方式检测肿瘤。该提案在培养本科生方面也有重要的教育内容。从振动、热传导和声学散射的应用来看，反问题的许多明显特征都可以看出，并且可以有重要的实际操作部分。实验设备很容易买到，而且很便宜。金属板可以制造导电的2D介质，锯片可以作为绝缘夹杂，廉价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转换的软件。黑色、浅不透明、声音透明的音箱布可以增加隐藏物体的神秘感。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.