Discrete Symbol Calculus

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.