Scalable Parallel Methods for the Helmholtz Equation via Exact Controllability

Large-scale Helmholtz problems are notoriously difficult to solve with standard iterative methods, in fact increasingly so, the higher the frequency ω > 0. Controllability methods (CM) offer an alternative approach for the numerical solution of the Helmholtz equation. Instead of solving the problem directly in the frequency domain, we first transform it back to the time domain where we seek the time-periodic solution y(.,t) of the corresponding time-dependent wave equation with known period T = (2π)/ω. By minimizing a cost functional, which penalizes the mismatch after one period, CM iteratively steer y towards the desired periodic state. Here, we consider two different approaches based either on the first or second-order formulation of the wave equation. Both are extended to general boundary-value problems governed by the Helmholtz equation and lead to robust and inherently parallel algorithms. Numerical results illustrate the accuracy and strong scalability of CM with up to a billion unknowns on massively parallel architectures.