We propose a matrix-free parallel two-level-deflation preconditioner combined with the Complex Shifted Laplacian preconditioner(CSLP) for the two-dimensional Helmholtz problems. The Helmholtz equation is widely studied in seismic exploration, antennas, and medical imaging. It is one of the hardest problems to solve both in terms of accuracy and convergence, due to scalability issues of the numerical solvers. Motivated by the observation that for large wavenumbers, the eigenvalues of the CSLP-preconditioned system shift towards zero, deflation with multigrid vectors, and further high-order vectors were incorporated to obtain wave-number-independent convergence. For large-scale applications, high-performance parallel scalable methods are also indispensable. In our method, we consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The CSLP preconditioner is approximated by one parallel geometric multigrid V-cycle. For the two-level deflation, the matrix-free Galerkin coarsening as well as high-order re-discretization approaches on the coarse grid are studied. The results of matrix-vector multiplications in Krylov subspace methods and the interpolation/restriction operators are implemented based on the finite-difference grids without constructing any coefficient matrix. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.

翻译：我们针对二维亥姆霍兹问题提出了一种结合复位移拉普拉斯预处理子（CSLP）的无矩阵并行双层压缩预处理方法。亥姆霍兹方程在地震勘探、天线设计及医学成像等领域得到广泛研究。由于数值求解器在可扩展性方面存在局限，该方程在精度与收敛性两方面均属于最具挑战性的问题之一。基于大波数条件下CSLP预处理系统的特征值趋近于零这一观察结果，我们引入多重网格向量压缩技术，并结合高阶向量以实现波数无关的收敛性。针对大规模应用场景，高性能并行可扩展方法不可或缺。本文方法采用预条件Krylov子空间法求解有限差分离散线性系统：通过并行几何多重网格V循环近似CSLP预处理子；在双层压缩策略中，分别研究了基于无矩阵Galerkin粗化与粗网格高阶重离散方法。Krylov子空间法中的矩阵向量乘法及插值/限制算子均基于有限差分网格实现，无需构建任何系数矩阵。这些改进直接降低了内存消耗。模型问题的数值实验表明，该方法在中波数范围内实现了波数无关收敛性。所提出的无矩阵并行框架展现出良好的弱可扩展性与强可扩展性。