Mathematical inverse problems arising in acoustic imaging

这个研究项目涉及由声学中的一个自然问题引发的数学逆问题。我们能在多大程度上用声音“看”？换句话说，通过向一种介质发射声脉冲并测量产生的回声，可以推断出介质的物理性质到什么程度？我们能做的不仅仅是创建一个粗略的图像（例如，目前用超声医学扫描所做的），还能确定诸如声阻抗密度等物理参数的实际值吗？最近的数学研究表明，后者虽然目前还不可行，但实际上是可能的。声波的传播是由一类特殊的偏微分方程（PDE）来模拟的。表征声音传播的物体或物理介质的物理参数在这些偏微分方程中作为称为系数的特定术语出现。例如，在医学成像中，当一个人用超声波传感器记录声学回声时，他实际上是在记录PDE的部分溶液，并且他想要从记录的部分溶液中确定未知的方程系数。这在数学中被称为逆问题（与计算给定PDE的解的经典正向问题相反）。逆问题理论正在迅速发展，但我们仍然不理解这一学科的许多基本问题，特别是，目前还不明白如何最好地从方程的部分解计算控制声音传播的PDE系数-换句话说，如何从记录的声学回声中获得最准确的图像！提出的研究计划的总体目标是发展：(1)求解逆问题的新数学方法；(2)与包括物理学家和工程师在内的更广泛的科学界感兴趣的波现象相关的新计算技术；(3)将数学见解转化为实际的成像技术。这项研究既涉及纯数学，包括对PDE和相关领域的分析，也涉及计算方法，包括算法的设计、实现和分析，并将涉及一个由才华横溢的研究生、博士后研究人员和国际合作者组成的多元化团队。实现该计划的目标不仅将产生新的数学和提高我们对波浪现象的科学理解，而且还提供了增强诊断成像能力的可能性，以及对我们所依赖的建筑物和结构进行无损检测。

This research program concerns mathematical inverse problems motivated by a natural question from acoustics. To what extent can we "see" with sound? In other words, to what extent can physical properties of a medium be inferred by transmitting a sound pulse toward it and measuring the resulting echoes? Can we do more than create a rough picture (as is currently done with ultrasound medical scans, for example) and determine the actual values of physical parameters such as density of acoustic impedance? Recent mathematical research suggests that the latter, while not currently feasible, may in fact be possible. The propagation of acoustic waves is modelled by a special class of what are known as partial differential equations (PDE). The physical parameters that characterize a body or physical medium through which sound is propagating occur as specific terms in these PDE called coefficients. In medical imaging, for example, when one records acoustic echoes with an ultrasonic sensor, one is in effect recording part of the solution to a PDE, and one wants to determine the coefficients of the equation, which are unknown, from the recorded partial solution. This is known in mathematics as an inverse problem (as opposed to the classical forward problem of computing a solution to a given PDE). The theory of inverse problems is under rapid development, but we still do not understand many basic questions in the subject, and in particular, it is not currently understood how best to compute the coefficient of the PDE governing sound propagation from partial solutions to the equation---in other words, how to make the most accurate possible picture from recorded acoustic echoes! The overall goals of the proposed research program are to develop: (1) novel mathematical methods for solving inverse problems; (2) new computational techniques related to wave phenomena of interest to the broader scientific community, including physicists and engineers; (3) to translate mathematical insights into practical real-world imaging technologies. The research involves both pure mathematics, including the analysis of PDE and related fields, as well as computational methods, including the design, implementation and analysis of algorithms, and will involve a diverse team of talented graduate students, postdoctoral researchers and international collaborators. Achieving the program's objectives will not only yield new mathematics and improve our scientific understanding of wave phenomena, but it offers the possibility of an enhanced capacity for diagnostic imaging, and non-destructive testing of the buildings and structures we rely on.