Discrete Symbol Calculus

TL;DR

This paper introduces efficient numerical methods for representing and manipulating differential and integral operators as symbols in phase-space, enabling fast computations in wave propagation problems.

Contribution

It develops non-asymptotic expansions for symbols using rational Chebyshev functions and hierarchical splines, with practical algorithms for operations like multiplication and inversion.

Findings

Efficient algorithms for symbol multiplication, inversion, and square root.

Applications to Helmholtz equation preconditioning, wavefield decomposition, and seismic depth-stepping.

Complexity depends weakly on resolution, mainly through logarithmic factors.

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 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 wavefields into one-way components and 3) depth-stepping in reflection seismology.