# Parallel Controllability Methods For the Helmholtz Equation

## TL;DR

This paper introduces parallel controllability methods for solving high-frequency Helmholtz equations by transforming the problem into the time domain, resulting in scalable algorithms that handle large-scale problems efficiently.

## Contribution

The paper develops robust, parallel controllability algorithms for the Helmholtz equation using first and second-order wave formulations, applicable to general boundary-value problems.

## Key findings

- Achieves high accuracy and convergence in Helmholtz solutions
- Demonstrates strong scalability on massively parallel architectures
- Handles problems with up to a billion unknowns

## Abstract

The Helmholtz equation is notoriously difficult to solve with standard numerical methods, increasingly so, in fact, at higher frequencies. Controllability methods instead transform the problem back to the time-domain, where they seek the time-harmonic solution of the corresponding time-dependent wave equation. Two different approaches are considered here 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, convergence and strong scalability of controllability methods for the solution of high frequency Helmholtz equations with up to a billion unknowns on massively parallel architectures.