Abstract
This note reviews certain classical applications of microlocal analysis in inverse problems. The text is based on lecture notes for a postgraduate level minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019.
Highlights
Microlocal analysis originated in the 1950s, and it is a substantial mathematical theory with many different facets and applications
ΨDOs were introduced by Kohn and Nirenberg [1], and Fourier integral operators (FIOs) and wave front sets were studied systematically by Hörmander [2]
In this note we will give a very brief idea of the different points of view to microlocal analysis mentioned in the introduction, as
Summary
Microlocal analysis originated in the 1950s, and it is a substantial mathematical theory with many different facets and applications. A kind of “variable coefficient Fourier analysis” for solving variable coefficient PDEs; or as a theory of pseudodifferential operators (ΨDOs) and Fourier integral operators (FIOs); or as a phase space (or time-frequency) approach to studying functions, operators and their singularities (wave front sets). Microlocal analysis is a powerful tool in the study of geodesic X-ray transforms related to seismic imaging applications. This note is organized as follows—in Section 2, we will motivate the theory of ΨDOs and discuss some of its properties without giving proofs. That studies inverse problems in rather general settings by using constructions like the ones in Sections 5 and 6. Αn ) where the α j are nonnegative integers Such an n-tuple α α is called a multi-index. All coefficients, boundaries and so forth are assumed to be C ∞ for ease of presentation
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