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Discrete Symbol Calculus

Author(s)
Demanet, Laurent; Ying, Lexing
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SIAM Review
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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.
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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.
Date issued
2011-02
URI
http://hdl.handle.net/1721.1/64793
Department
Massachusetts Institute of Technology. Department of Mathematics
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
ISSN
0036-1445
1095-7200

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