# KAUST-IAMCS Workshop on Modeling and Simulation of Wave Propagation and Applications

## Jack Poulson, University of Texas at Austin A Parallel Sweeping Preconditioner for High-Frequency Heterogeneous 3D Helmholtz Equations

### Abstract

A parallelization of a recently introduced sweeping preconditioner for high frequency heterogeneous Helmholtz equations is presented along with experimental results for the full SEG/EAGE Overthrust seismic model at 30 Hz, using eight grid points per characteristic wavelength; to the best of our knowledge, this is the largest 3D Helmholtz calculation to date, and our algorithm only required fifteen minutes to complete on 8,192 cores. While the setup and application costs of the sweeping preconditioner are trivially $$\Theta \left( N^{\frac{4}{3}} \right)$$ and $$\Theta \left( N \log N \right)$$, this paper provides strong empirical evidence that the number of iterations required for the convergence of GMRES equipped with the sweeping preconditioner is essentially independent of the frequency of the problem. Generalizations to time-harmonic Maxwell and linear-elastic wave equations are also briefly discussed since the techniques behind our parallelization are not specific to the Helmholtz equation.