We present the first fast solver for the high-frequency Helmholtz equation that scales optimally in parallel for a single right-hand side. The L-sweeps approach achieves this scalability by departing from the usual propagation pattern, in which information flows in a ${180}^{\circ }$ degree cone from interfaces in a layered decomposition. Instead, with L-sweeps, information propagates in ${90}^{\circ }$ cones induced by a Cartesian domain decomposition (CDD). We extend the notion of accurate transmission conditions to CDDs and introduce a new sweeping strategy to efficiently track the wave fronts as they propagate through the CDD. The new approach decouples the subdomains at each wave front, so that they can be processed in parallel, resulting in better parallel scalability than previously demonstrated in the literature. The method has an overall $\mathcal{O}((N/p)\log \omega )$ empirical run-time for $N=n^d$ total degrees-of-freedom in a d-dimensional problem, frequency ω, and $p=\mathcal{O}(n)$ processors. We introduce the algorithm and provide a complexity analysis for our parallel implementation of the solver. We corroborate all claims in several two- and three-dimensional numerical examples involving constant, smooth, and discontinuous wave speeds.

我们为高频亥姆霍兹方程式提供了第一个快速求解器，该方程式可以在单个右侧并行最佳缩放。L扫描方法通过偏离通常的传播模式来实现这种可扩展性，在常规传播模式中，信息在${180}^{\circ }$锥度从界面上分层分解。相反，通过L扫描，信息传播到${90}^{\circ }$笛卡尔域分解（CDD）诱导的视锥细胞。我们将精确的传输条件的概念扩展到CDD，并引入了一种新的扫描策略，可以在波前传播通过CDD时有效地跟踪波前。新方法在每个波阵面解耦子域，以便可以并行处理它们，从而获得比以前文献中更好的并行可伸缩性。该方法有一个整体$\mathcal{Ø}（（ñ/p）\mathrm{日志}\omega ）$ 的经验运行时间 $ñ={ñ}^d$d维问题的总自由度，频率ω和$p=\mathcal{Ø}（ñ）$处理器。我们介绍了该算法，并为我们的求解器并行实现提供了复杂性分析。我们在涉及恒定，平滑和不连续波速的几个二维和三维数值示例中证实了所有主张。