Abstract
Pseudodifferential operators were introduced in the mid 1900s as a powerful new tool in the development of the theory of partial differential equations. More recently, it has been observed that these operators may form the basis for novel numerical techniques used in the analysis and simulation of physical systems including wave propagation and medical imaging, as well as for advances in signal processing. This course will focus on the numerical implementations of pseudodifferential operators and practical applications. Of particular interest are: the variety of ways to implement these operators, including via fast transforms, decomposition into product-convolution operators, Gabor multipliers, and wavelet transform; speed of implementations; relation to asymptotic expansions; real experience with numerical implementations including in geophysical applications.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional conference series in applied mathematics, Academic Press, 1992.
J. Du and M. W. Wong, A product formula for localization operators, Bulletin of the Korean Mathematical Society, 37 (2000), pp. 77–84.
H. G. Feichtinger and K. Nowak, A first survey of Gabor multipliers, Advances in Gabor analysis, Applied Numerical Harmonic Analysis, Birkhauser, Boston, 2003, pp. 99–128.
G. B. Folland, Harmonic Analysis in Phase Space, vol. 122 of Annals of Mathematical Studies, Princeton University Press, Princeton, New Jersey, 1989.
P. Gibson and M. Lamoureux, Maximally symmetric, minimally redundant partitions of unity in the plane, Comptes rendus mathematiques, 26 (2004), pp. 65–72.
P. C. Gibson, J. Grossman, M. P. Lamoureux, and G. F. Margrave, A fast, discrete Gabor transform by partition of unity. Preprint, 2002.
P. C. Gibson, M. P. Lamoureux, and G. F. Margrave, Representation of linear operators by Gabor multipliers. Preprint, 2003.
K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor Analysis and Algorithms, H. Feichtinger and T. Strohmer, editors, Birkhauser, Boston, 1998, pp. 211–232.
K. Gröchenig, Foundations of time–frequency analysis, Birkhauser, Boston, 2001.
J. P. Grossman, G. F. Margrave, and M. P. Lamoureux, Constructing adaptive nonuniform Gabor frames from partitions of unity, tech. rep., CREWES, University of Calgary, 2002.
grossman, Fast wavefield extrapolation by phase-shift in the nonuniform Gabor domain, tech. rep., CREWES, University of Calgary, 2002.
J. P. Grossman, G. F. Margrave, M. P. Lamoureux, and R. Aggarwala, Constant-Q wavelet estimation via a nonstationary Gabor spectral model, tech. rep., CREWES, University of Calgary, 2001.
G. F. Margrave, L. Dong, P. C. Gibson, J. P. Grossman, D. C. Henley, V. Iliescu, and M. P. Lamoureux, Gabor deconvolution: extending Wiener’s method to nonstationarity, The CSEG Recorder, 28 (2003).
G. F. Margrave, P. C. Gibson, J. P. Grossman, D. C. Henley, V. Iliescu, and M. P. Lamoureux, The Gabor transform, pseudodifferential operators, and seismic deconvolution, Integrated Computer-Aided Engineering, 9 (2004), pp. 1–13.
G. F. Margrave, D. C. Henley, M. P. Lamoureux, V. Iliescu, and J. P. Grossman, An update on Gabor deconvolution, tech. rep., CREWES, University of Calgary, 2002.
G. F. Margrave and M. P. Lamoureux, Gabor deconvolution, tech. rep., CREWES, University of Calgary, 2001.
margrave, Gabor deconvolution, The CSEG Recorder, 2006 Special Issue (2006), pp. 30–37.
Y. Meyer and R. Coifman, Wavelets: Calderon–Zygmund and Multilinear Operators, vol. 48 of Cambridge studies in advanced mathematics, Cambridge University Press, Cambridge, 1997.
A. V. Oppeheim and R. W. Schafer, Discrete-time Signal Processing, Prentice Hall, New Jersey, 1998.
X. S. Raymond, Elementary Introduction to the Theory of Pseudodifferential Operators, CRC Press, Florida, 1991.
R. Rochberg and K. Tachizawa, Pseudodifferential operators, Gabor frames, and local trigonometric bases, Gabor Analysis and Algorithms, H. Feichtinger and T. Strohmer, editors, Birkhauser, Boston, 1998, pp. 171–192.
G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1997.
M. W. Wong, Weyl transforms, Springer-Verlag, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lamoureux, M.P., Margrave, G.F. (2008). An Introduction to Numerical Methods of Pseudodifferential Operators. In: Rodino, L., Wong, M.W. (eds) Pseudo-Differential Operators. Lecture Notes in Mathematics, vol 1949. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68268-4_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-68268-4_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-68266-0
Online ISBN: 978-3-540-68268-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)