We consider the use of multipreconditioning, which allows for multiple preconditioners to be applied in parallel, on high-frequency Helmholtz problems. Typical applications present challenging sparse linear systems which are complex non-Hermitian and, due to the pollution effect, either very large or else still large but under-resolved in terms of the physics. These factors make finding general purpose, efficient and scalable solvers difficult and no one approach has become the clear method of choice. In this work we take inspiration from domain decomposition strategies known as sweeping methods, which have gained notable interest for their ability to yield nearly-linear asymptotic complexity and which can also be favourable for high-frequency problems. While successful approaches exist, such as those based on higher-order interface conditions, perfectly matched layers (PMLs), or complex tracking of wave fronts, they can often be quite involved or tedious to implement. We investigate here the use of simple sweeping techniques applied in different directions which can then be incorporated in parallel into a multipreconditioned GMRES strategy. Preliminary numerical results on a two-dimensional benchmark problem will demonstrate the potential of this approach.

翻译：本文探讨将多重预处理技术应用于高频亥姆霍兹问题，该技术允许并行施加多个预处理算子。此类应用通常面临具有挑战性的稀疏线性系统，这些系统是复杂的非厄米特矩阵，并且由于数值污染效应，其规模要么非常庞大，要么虽规模稍小但物理分辨率不足。这些因素使得寻找通用、高效且可扩展的求解器变得困难，目前尚未形成明确的主流方法。本研究从被称为扫描方法的区域分解策略中汲取灵感，该类方法因能实现近似线性的渐近复杂度而备受关注，并且对高频问题也表现出良好适应性。尽管已存在成功的实现方案（例如基于高阶界面条件、完全匹配层或波前复杂追踪的方法），但这些方案往往实现过程较为复杂或繁琐。本文研究将应用于不同方向的简单扫描技术并行整合到多重预处理的GMRES策略中的可行性。针对二维基准问题的初步数值结果将展示该方法的潜在优势。