Discrete Symbol Calculus

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

Fast wave computation via Fourier integral operators

“…Discrete symbol calculus (DSC) is a natural answer to this question [13]. With the help of DSC, constructing P and P −1 takes only a number of steps that is polylogarithmic in N .…”

“…Concerning fast computation of general pseudodifferential and Fourier integral operators (FIO), Candès and the authors have reported on different ideas in [13,7,8]. Many of the algorithmic tools used in those papers are present in different forms or different contexts; let us for instance mention work on angular decompositions of the symbol of pseudodifferential operators [1], work on the butterfly algorithm and its applications in [21,23,38,37,35], work on fast "beamforming" methods for filtered backprojection for Radon and generalized Radon transforms (a problem similar to wave propagation) in [36,22,2], work on the plane-wave time-domain fast multipole method summarized in Chapters 18 and 19 of [9], and work on "phase-screen" methods in geophysics [11].…”

“…This section is a summary of [13]. We use the discrete symbol calculus (DSC) framework to represent the operators P = L 1/2 and P −1 = L −1/2 .…”

Fast algorithm for extracting the diagonal of the inverse matrix with application to the electronic structure analysis of metallic systems

“…This type of algorithm can be quite efficient if fast algorithms for calculating Gv are available (without the explicit knowledge of G). Examples of such fast algorithms include multigrid methods [6], the fast multipole method [13] (combined with iterative solvers), and the discrete symbol calculus [9], to name a few. One shortcoming of this approach is that the accuracy is relatively low for general G due to its Monte Carlo nature.…”

Approximate inversion of the wave-equation Hessian via randomized matrix probing

“…Because it s a function of both space and wave-vector, we can refer to H as a dip-dependent and scale-dependent scaling. Under this pseudodifferential model, it is possible to define B i from elementary symbols, by adapting the "discrete symbol calculus" approach of (Demanet and Ying (2011)). In other words, each B i is a pseudodifferential operator with a symbol that uses a single basis function such as Fourier sine or cosine.…”