A fast algorithm for particle simulations
Journal of Computational Physics
A fast direct algorithm for the solution of the Laplace equation on regions with fractal boundaries
Journal of Computational Physics
Computation of Pseudo-Differential Operators
SIAM Journal on Scientific Computing
Algorithms for Numerical Analysis in High Dimensions
SIAM Journal on Scientific Computing
Fast Computation of Fourier Integral Operators
SIAM Journal on Scientific Computing
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Multilevel Projection-Based Nested Krylov Iteration for Boundary Value Problems
SIAM Journal on Scientific Computing
Compact Fourier Analysis for Designing Multigrid Methods
SIAM Journal on Scientific Computing
Wave atoms and time upscaling of wave equations
Numerische Mathematik
A Fast Method for Linear Waves Based on Geometrical Optics
SIAM Journal on Numerical Analysis
A perfectly matched layer for the absorption of electromagnetic waves
Journal of Computational Physics
Matrix Probing and its Conditioning
SIAM Journal on Numerical Analysis
Computers & Mathematics with Applications
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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 $\xi$. 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.