The Helmholtz equation is related to seismic exploration, sonar, antennas, and medical imaging applications. It is one of the most challenging problems to solve in terms of accuracy and convergence due to the scalability issues of the numerical solvers. For 3D large-scale applications, high-performance parallel solvers are also needed. In this paper, a matrix-free parallel iterative solver is presented for the three-dimensional (3D) heterogeneous Helmholtz equation. We consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The Complex Shifted Laplace Preconditioner (CSLP) is employed since it results in a linear increase in the number of iterations as a function of the wavenumber. The preconditioner is approximately inverted using one parallel 3D multigrid cycle. For parallel computing, the global domain is partitioned blockwise. The matrix-vector multiplication and preconditioning operator are implemented in a matrix-free way instead of constructing large, memory-consuming coefficient matrices. Numerical experiments of 3D model problems demonstrate the robustness and outstanding strong scaling of our matrix-free parallel solution method. Moreover, the weak parallel scalability indicates our approach is suitable for realistic 3D heterogeneous Helmholtz problems with minimized pollution error.

翻译：亥姆霍兹方程与地震勘探、声呐、天线及医学成像应用密切相关。由于数值求解器的可扩展性问题，它在求解精度和收敛性方面是最具挑战性的问题之一。针对三维大规模应用，还需要高性能并行求解器。本文提出了一种用于三维（3D）非均匀亥姆霍兹方程的无矩阵并行迭代求解器。我们采用预条件Krylov子空间方法来求解通过有限差分离散获得的线性系统。采用复频移拉普拉斯预条件子（CSLP），因为其迭代次数随波数线性增加。该预条件子通过一个并行三维多重网格循环进行近似求逆。在并行计算中，全局域被分块划分。矩阵-向量乘积和预条件算子以无矩阵方式实现，无需构建占用大量内存的系数矩阵。三维模型问题的数值实验证明了我们的无矩阵并行求解方法的鲁棒性和出色的强可扩展性。此外，弱并行可扩展性表明，我们的方法适用于具有最小污染误差的实际三维非均匀亥姆霍兹问题。