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On Numerics and Inverse Problems

出版	University of Washington Libraries, 2015
URL	http://books.google.com.hk/books?id=56WGjwEACAAJ&hl=&source=gbs_api

註釋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[infinity]0(X --> D'(Y) expressed in the oscillatory integral form Af(y) = [indefinite integral] eiø (y, [xi]) a(y, [xi]) f̂([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.