Uniqueness and Reconstructions Methods for Inverse Problems

反问题的唯一性和重构方法

• 批准号: 1319052
• 负责人: William Rundell
• 金额: $ 28万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-15 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1319052&HistoricalAwards=false
• 关键词: Uniqueness; Reconstructions; Methods; Inverse; Problems

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.

该提案中解决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，我们专注于开发极快的算法，旨在仅利用最少的数据来检测重要特征。该提案的一个核心特征是研究所谓的反常扩散模型的反问题。经典扩散基于布朗运动，根源于 19 世纪的物理学。这里，非常局部的扰动以钟形曲线的特征形状传播，此外，给定时间步长的过程状态仅取决于前一时间步长的状态。虽然这适用于多种模型，但对于那些表现出“历史”或“记忆”效应的模型来说却失败了。 这包括过去二十年开发的许多材料以及经济预测，例如股票和商品市场建模。事实证明，反常扩散反问题中的病态程度可能与经典案例的病态程度有很大不同，这表明确实涉及基础的新物理学。从数学和计算的角度来看，这是有代价的；由此产生的分析要复杂得多且更具挑战性。许多具有物理意义的物体无法直接研究。示例包括对身体内部进行成像、确定固体物体内的裂缝以及难以接近物体的电导率等材料参数。当这些问题转化为数学术语时，它们采用偏微分方程的形式，即数学科学的通用语言。然而，由于模型中有额外的未知数，这些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。在这个提案中，我们从数学和计算的角度处理此类“逆问题”的实际问题。我们感兴趣的是何时可以从给定数量的数据中做出唯一的确定，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种解决方案，两个截然不同的对象可能会产生无限接近的数据集。这使得设计和分析用于未知数的有效数值恢复的算法极具挑战性。逆问题可以具有多种复杂程度。有些，例如地震建模，需要大规模的计算资源并积累大量的数据。其他人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或便携式机器中的缺陷，从而以非侵入性方式检测肿瘤。该提案在本科生培训方面也具有重要的教育意义。反问题的许多独特特征可以通过考虑振动、热传导和声散射中的应用来看出，并且可以具有重要的实践部分。实验设备容易获得，而且价格便宜。金属板形成导电的二维介质，锯子切割绝缘夹杂物，廉价的热敏电阻可用于测量数据。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。 “隐藏”物体的神秘感可以通过黑色、浅色不透明、透声扬声器布来增添。我们已经积累了大部分此类设备，其中一些在以前的本科生研究经验中得到了很好的使用。