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We present a solver for the 2D high-frequency Helmholtz equation inheterogeneous acoustic media, with online parallel complexity that scalesoptimally as $\mathcal{O}(\frac{N}{L})$, where $N$ is the number of volumeunknowns, and $L$ is the number of processors, as long as $L$ grows at mostlike a small fractional power of $N$. The solver decomposes the domain intolayers, and uses transmission conditions in boundary integral form toexplicitly define "polarized traces", i.e., up- and down-going waves sampled atinterfaces. Local direct solvers are used in each layer to precompute traces oflocal Green's functions in an embarrassingly parallel way (the offline part),and incomplete Green's formulas are used to propagate interface data in asweeping fashion, as a preconditioner inside a GMRES loop (the online part).Adaptive low-rank partitioning of the integral kernels is used to speed uptheir application to interface data. The method uses second-order finitedifferences. The complexity scalings are empirical but motivated by an analysisof ranks of off-diagonal blocks of oscillatory integrals. They continue to holdin the context of standard geophysical community models such as BP and Marmousi2, where convergence occurs in 5 to 10 GMRES iterations.

The method of polarized traces for the 2D Helmholtz equation