Wave Holtz: Theory and High Performance Applications

Abstract

The solution to the Helmholtz equation describes the spatial waveform of a time-harmonic wave and arises in acoustics, electromagnetism, and seismology. Its efficient numerical treatment makes engineering problems tractable but presents two central difficulties. First, the waves must be sufficiently resolved. Engineering problems often span hundreds to thousands of wavelengths, over which phase error accumulates, demanding many points per wavelength and yielding systems with millions to billions of degrees of freedom. Second, standard discretizations produce a complex-valued, highly indefinite linear system on which general-purpose iterative methods perform poorly or fail outright, making specialized methods for the equation a challenging, active research topic. This dissertation advances one such method: WaveHoltz, which relates the Helmholtz equation to its time-dependent counterpart, the wave equation. WaveHoltz evolves the wave equation and filters its solution over a short time interval to estimate the Helmholtz solution; this estimate is fed back as an initial condition, iteratively improving the estimate until convergence; this iteration can be accelerated by standard Krylov methods. On the theoretical side, convergence of the discrete iteration is established, the temporal error of the wave solve is eliminated, and WaveHoltz-HMM, a homogenization method with error estimates, is introduced. On the computational side, a high-performance GPU implementation combining multigrid-accelerated time stepping with massively parallel algorithms and a GPU-tailored domain decomposition approach made efficient by using WaveHoltz as a subdomain solver are presented. Altogether, WaveHoltz proves robust and efficient for solving the Helmholtz equation.