Discrete Symbol Calculus
Name
Demanet-2011-Discrete Symbol Calculus.pdf
Size
1.45 MB
Format
Adobe PDF
Checksum (MD5)
5dad5a611607077a02fd94fe9cc8cb5b
Author(s)
Demanet, Laurent
Ying, Lexing
Alternative Title
SIAM Review
Date Issued
February 2011
Journal
SIAM Review
Publisher
Society for Industrial and Applied Mathematics
Citation
Demanet, Laurent and Lexing Ying. "Discrete Symbol Calculus." SIAM Review 53.1 (2011): 71-104. (c) 2011 Society for Industrial and Applied Mathematics
Version
Final published version
Abstract
This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space $x$ and frequency $\xi$. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow fast-converging, nonasymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines to be written. The classical results of closedness of such symbol classes under multiplication, inversion, and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution $N$ very weakly, typically only through $\log N$ factors. We present three applications to computational problems related to wave propagation: (1) preconditioning the Helmholtz equation, (2) decomposing wave fields into one-way components, and (3) depth extrapolation in reflection seismology. The software is made available in the software sections of math.mit.edu/$\sim$laurent and www.math.utexas.edu/users/lexing.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1137/080731311
