Analysis and Computation for Inverse Problems in Differential Equations

许多具有物理意义的物体不能被直接研究。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。当这些问题转化为数学术语时，它们就会以偏微分方程式的形式出现，偏微分方程式是数学科学的通用语言。然而，由于在模型中可能有额外的未知数，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解决。该项目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。在这个项目中，PI从数学和计算的角度处理这种“反问题”的实际方面。主要的挑战是何时可以根据给定的数据量做出唯一的确定，但由于这些逆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可能产生无限接近的数据集。这种缺乏稳定性的方面使得设计和分析算法来有效地恢复未知数具有极大的挑战性。PI将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。PI还研究逆光谱问题，一个经典的例子是给出物体的振动频率，并寻求确定其内部结构。这里的天体可以是金属光束，也可以是恒星，比如太阳。这项提议的一个中心主题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，与爱因斯坦1905年的随机漫步模型一起，植根于19世纪的物理学。在这里，一个非常局部化的扰动以高斯的特征形状传播，而且，该过程是马尔可夫的；在给定的时间步长，状态仅取决于在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，这是有代价的；由此产生的分析要复杂得多，也更具挑战性。该项目在培养研究生和本科生方面也有重要的教育内容。

Many objects of physical interest cannot be studied directly. Examples include the following: 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 one may have additional unknowns in the model, these introduce unknown parameters in the equations that have to be resolved by means of further measurements. Specific problems addressed in this project 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 this project the PI deals with the practical aspects of such "inverse problems" from a mathematical and computational perspective. The main challenge is when a unique determination can be made from a given amount of data, but as 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 lack of stability aspect makes designing and analyzing algorithms for the efficient numerical recovery of the unknowns extremely challenging. The PI will concentrate on developing extremely fast algorithms designed to detect significant features utilizing only minimal data. The PI also looks at inverse spectral problems, and a classic example of which is to be given the vibrational frequencies of a body and seek to determine its internal construction. Here the body can be a metal beam or a star such as the sun. A central theme 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 together with Einstein's 1905 random walk model. Here a very localised disturbance spreads with the characteristic shape of a Gaussian and, further, the process is Markovian; at a given time step the state depends only on that 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. The project also has a significant educational component in the training of graduate and undergraduate students.