An Introduction to Numerical Methods of Pseudodifferential Operators

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.

One-way wave propagation with amplitude based on pseudo-differential operators

“…It will be presented in the following subsection. Other methods can be found in [6,7,25]. We solved the ODE by using the classical fourth-order Runge-Kutta scheme [26].…”

Discrete Symbol Calculus

“…For computation of pseudodifferential operators, see also the work by Lamoureux, Margrave, and Gibson [26].…”

“…Expansions of principal symbols a 0 (x, ξ/|ξ|) (homogeneous of degree 0 is ξ) in spherical harmonics in ξ is a useful tool in the theory of pseudodifferential operators [38], and has also been used for fast computations by Bao and Symes in [1]. For computation of pseudodifferential operators, see also the work by Lamoureux, Margrave, and Gibson [26].…”

Imaging seismic reflections

“…It will be presented in the following subsection. Other methods can be found in [47,27,45]. We solved the ODE by using the classical 4 th -order Runge-Kutta scheme [41].…”

“…To show that F a is a FIO we derive an explicit formulation valid for a small time interval around a localized scattering event. Let {ρ i } i∈I be a finite smooth partition on D such that 45) and F b likewise. S a is the solution operator (4.32).…”