Mathematical inverse problems arising in acoustic imaging

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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.