On numerics and inverse problems

Abstract

In this thesis, two projects in inverse problems are described. The first concerns a simple mathematical model of synthetic aperture radar with undirected beam, modeled as a 2D circular Radon transform with centers restricted to a plane curve \gamma. From the work of Stefanov and Uhlmann (2013), it is known that this operator is microlocally non-injective locally, but microlocally injective globally if \gamma is closed. The problem for non-closed \gamma is examined; it is shown that the Radon transform R_\gamma is microlocally non-injective to leading order if a certain geometric condition is satisfied. Known examples where this condition holds are given. Numerical simulations demonstrate R_\gamma's microlocal non-injectivity for a single curve, and for a four-curve setup satisfying the geometric condition. The second project involves the implementation of an algorithm by de Hoop, Uhlmann, Vasy, and Wendt (2013), with refinements, for computing generic Fourier integral operators (FIOs) associated with canonical graphs, possibly involving caustics. The algorithm can be divided into two parts: a local component that approximately evaluates an FIO A: C^\infty_0(X)\to \mathcal D'(Y) expressed in the oscillatory integral form Af(y)=\int e^{i\phi(y,\xi)} a(y,\xi)\,\hat f(\xi)\,d\xi, modulo an error operator of order 1/2 less than the order of A, and a global component that expresses an arbitrary FIO associated with a canonical graph as a finite sum of these local oscillatory integrals composed with appropriate coordinate changes. A numerical implementation of their algorithm is demonstrated and successfully applied to a variety of FIOs associated with canonical graphs. This algorithm is designed to be easy-to-use for future researchers and the code is freely available from the author.