Browsing by Author "Cocola, Jorio"

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    Microlocal Analysis of Hyperbolic Equations with Memory and Applications
    (2019-04-24) Cocola, Jorio; de Hoop, Maarten
    This thesis introduces a framework for extending microlocal analysis to hyperbolic equations with memory. These equations describe wave motion in attenuating and dispersive media and have been widely employed in various domain of science, spanning from global seismology to medical imaging. Although mathematical problems in these domains have been successfully studied with microlocal analysis, its application to hyperbolic equation with memory has somewhat lagged behind. This thesis proposes to bridge that gap. In the first part, I construct parametrices for first order hyperbolic initial value problems with a zero order memory term. These parametrices are then used to prove a propagation of singularities theorem and to explain the arising of stationary singularities. I then discuss an open problem on the diagonalization of second order wave equations with memory. Finally, in the last part, I leverage the analysis developed in the previous chapters to extend time reversal methods (TR) for these types of wave equations. The presence of the memory term, precludes the time reversibility of wave propagation creating an obstruction in the naive employment of TR methods. I derive and analyze a microlocal “boundary compensation method” that allows overcoming this issue and that, coupled with a reverse-time continuation from the boundary, is applied to solve an inverse source problem.