Discrete Symbol Calculus

12 years 7 months ago

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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 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 wav...

Laurent Demanet, Lexing Ying

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