Recursive Sweeping Preconditioner for the 3D Helmholtz Equation
For computational scientists solving large-scale 3D Helmholtz problems, this method significantly reduces computational costs while maintaining efficiency.
The paper introduces a recursive sweeping preconditioner for the 3D Helmholtz equation, reducing setup cost from O(N^{4/3}) to O(N) and application cost per iteration from O(N log N) to O(N), with only a mild increase in iteration count.
This paper introduces the recursive sweeping preconditioner for the numerical solution of the Helmholtz equation in 3D. This is based on the earlier work of the sweeping preconditioner with the moving perfectly matched layers (PMLs). The key idea is to apply the sweeping preconditioner recursively to the quasi-2D auxiliary problems introduced in the 3D sweeping preconditioner. Compared to the non-recursive 3D sweeping preconditioner, the setup cost of this new approach drops from $O(N^{4/3})$ to $O(N)$, the application cost per iteration drops from $O(N\log N)$ to $O(N)$, and the iteration count only increases mildly when combined with the standard GMRES solver. Several numerical examples are tested and the results are compared with the non-recursive sweeping preconditioner to demonstrate the efficiency of the new approach.