Abstract
Overview In this talk we describe the composition calculus of Fourier Integral Operators (FIOs) with fold and cusp singularities. Such operators appear in many inverse scattering problems, where the composition calculus can be used as a tool for recovering images. In these problems, caustics occur and create artifacts which make the reconstruction more complicated and challenging. The goal is to understand these artifacts, find their strength and try to remove them.
Keywords
- Single Source
- Pseudodifferential Operator
- Fourier Integral Operator
- Inverse Scattering Problem
- Generate Acoustics Wave
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Beylkin, G. Imaging of discontinuities in the inverse problem by inversion of a generalized Radon transform, Jour. Math. Phys. 28 (1985), 99–108.
Duistermaat, J.J., Guillemin, V., The spectrum of positive elliptic operators and periodic bicharacteristics, Inv. math., 29 (1975), 39–79.
Felea, R. Composition calculus of Fourier integral operators with fold and blowdown singularities, Comm. P.D.E, 30 (13) (2005), 1717–1740.
Felea, R., Greanleaf, A., An FIO calculus for marine seismic imaging: folds and cross caps, Communications in PDEs, 33 (1), (2008), 45–77.
Felea, R., Greanleaf, A., Fourier integral operators with open umbrellas and seismic inversion for cusp caustics, Math Ress Lett, 17 (5) (2010), 867–886.
Felea, R., Greenleaf, A., Pramanik, M., An FIO calculus for marine seismic imaging, II: Sobolev estimates, Math. Annalen, (2011).
Givental, A. Lagrangian imbeddings of surfaces and unfolded Whitney umbrella. (English) Func. Anal. Appl. 20 (3) (1986), 197–203.
Greenleaf, A., Uhlmann, G., Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Jour. Func. Anal., 89 (1990), 220–232.
Greenleaf, A., Uhlmann,G., Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier, Grenoble, 40 (1990), 443–466.
Guillemin, V., Uhlmann, G., Oscillatory integrals with singular symbols. Duke Math. J. 48 (1) (1981), 251–267.
Hörmander, L., Fourier integral operators, I. Acta mathematica, 127 (1971), 79–183.
Melrose, R.B., Uhlmann, G.A., Lagrangian intersection and the Cauchy problem. Comm. Pure Appl. Math., 32 (4) (1979), 483–519.
Nolan, C.J., Scattering in the presence of fold caustics. SIAM J. Appl. Math. 61 (2) (2000), 659–672.
Nolan, C.J., Symes, W.W., Global solutions of a linearized inverse problem for the acoustic wave equation, Comm. in PDE 22, (1997), 919–952.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Basel
About this paper
Cite this paper
Felea, R. (2013). Microlocal Analysis of FIOs with Singularities. In: Grieser, D., Teufel, S., Vasy, A. (eds) Microlocal Methods in Mathematical Physics and Global Analysis. Trends in Mathematics(). Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0466-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-0348-0466-0_4
Published:
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-0465-3
Online ISBN: 978-3-0348-0466-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)