Discrete Symbol Calculus

Laurent Demanet, Lexing Ying
2011 SIAM Review  
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 ξ. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow to write fast-converging, non-asymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines. The classical results of closedness of such symbol classes
more » ... ch 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 wavefields into one-way components and 3) depth extrapolation in reflection seismology. The software can be downloaded from the software section of either author's website.
doi:10.1137/080731311 fatcat:7x4mehzpzbgnvjtvau4ckjpd3m