Determining the twist of an optical fiber

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University of Delaware
This research focuses on recovering the coefficient of a two speed hyperbolic system of partial differential equations from the reflection boundary data, where the source and the receiver are at the same location. We study the associated initial value problem (the forward problem) and then the coefficient determination inverse problem using a fixed point argument. We then implement the inversion scheme numerically. We also study the inverse problem of recovering the coefficient of this system from the transmission boundary data, where the source and receiver are at different locations. We obtain an upper bound of the coefficient in terms of the transmission data, and we also obtain a relation between transmission and reflection data. For multi-dimensional problems, we study the regularity at the origin of spherical harmonic expansions because solutions of some PDEs are constructed using spherical harmonic expansions.