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The College of Literature, Science, and the Arts pres.

Professor Liliana Borcea, the Peter Field Collegiate Professorship in Mathematics, Inaugural Lecture

Mathematical and Computational Aspects of Imaging with Waves

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Wave based imaging is an inverse problem for a wave equation or a system of equations with a wide range of applications in nondestructive testing of structures such as airplane wings, ultrasound for medical diagnosis, radar, sonar, geophysical exploration, etc. It seeks to determine scattering structures in a medium, modeled mathematically by a reflectivity function, from data collected by sensors that probe the medium with signals and measure the resulting waves. Most imaging methods formulate the inverse problem as a least squares data fit optimization, and assume a linear mapping between the unknown reflectivity and the data. The linearization, known as the Born (single scattering) approximation is not accurate in strongly scattering media, so the reconstruction of the reflectivity may be poor. I will describe a new inversion methodology that is based on a reduced order model approach. This borrows ideas from dynamical systems, where the reduced order model is a projection of an operator, called wave propagator, which describes the propagation of the waves in the unknown medium. I will explain how such a reduced order model can be constructed from measurements at the sensors and then I will show how it can be used for improving the existing inversion methodology.
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