Mathematical inverse problems arising in acoustic imaging

声学成像中出现的数学反问题

• 批准号: RGPIN-2022-04547
• 负责人: Gibson, Peter
• 金额: $ 1.31万
• 依托单位: York University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758975
• 关键词: Mathematical; inverse; problems; arising; acoustic

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.

该研究计划涉及由声学中的自然问题激发的数学反问题。我们在多大程度上可以用声音“看到”？换句话说，通过向其传输声脉冲并测量所得回声来推断介质的物理特性在多大程度上可以推断出来？我们可以做的不仅仅是创建粗糙的图片（例如，就像当前使用超声医学扫描一样），并确定物理参数的实际值，例如声学阻抗的密度？最近的数学研究表明，后者虽然目前不可行，但实际上可能是可能的。 声波的传播由特殊类别的部分微分方程（PDE）建模。表征声音传播的身体或物理介质的物理参数以这些PDE的特定术语发生，称为系数。例如，在医学成像中，当一个人记录具有超声传感器的声学回声时，实际上是将部分记录到PDE的溶液中，并且希望从记录的部分溶液中确定方程的系数，这是未知的。这在数学中被称为一个反问题（与计算给定PDE的解决方案的经典前进问题相反）。反问题的理论正在迅速发展，但我们仍然不了解该主题中的许多基本问题，尤其是，目前尚不理解如何最好地计算PDE的系数，从部分解决方案到方程式到方程式 - 换句话说，如何从录制的声音回声中制作出最准确的图片！拟议的研究计划的总体目标是开发：（1）解决反问题的新型数学方法； （2）与更广泛的科学界有关的波浪现象有关的新计算技术，包括物理学家和工程师； （3）将数学见解转化为实用的现实成像技术。这项研究既涉及纯数学，包括对PDE和相关领域的分析，以及计算方法，包括算法的设计，实施和分析，并且将涉及一组有才华的研究生，博士后研究人员和国际合作者组成的多样化团队。 实现该计划的目标不仅会产生新的数学并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，以及对我们依赖的建筑物和结构的非破坏性测试。