Interconnected Hierarchical Rank-structured Methods for Directly Solving and Preconditioning the Helmholtz Equation

In this paper, we design interconnected hierarchical rank-structured methods for the fast direct solution and effective preconditioning of the Helmholtz equation. First, for essentially elliptic PDEs, a direct method is proposed to exploit interconnected structures within two hierarchical layers: the hierarchical partitioning of a large problem into sub-problems for smaller subdomains, and the hierarchically semiseparable (HSS) approximation for the sub-problems. Interconnected low-rank structures enable extensive reuse of information among computations associated with different subdomains, which greatly reduces the cost and memory usage for the construction of structured representations as needed in fast structured direct solutions. In contract, previous structured direct solvers usually require expensive structured constructions which contribute to the dominant cost. Our direct method can quickly solve either the low-frequency or the damped Helmholtz equation. The factorization cost can be as low as O(n) in 2D and O(n logn) in 3D, where n is the matrix size. For the high-frequency case, we build a preconditioner by creating artificial damping near the boundary of subdomains. We give the first result on how the magnitude of the damping parameter affects the compressibility of HSS representation of boundary maps. For O(ω2−γ) damping where ω is the angular frequency and γ ∈ [0, 1], the factorization of the preconditioner costs O(ω2+γ) in 2D and O(ω3+3γ) in 3D. For the interior impedance problem, we show that the condition number of the preconditioned system is O(ω1−γ), and prove GMRES convergence for general cases based on the Elman estimates. Comparing with solving the preconditioner problem iteratively, the rank-structured approach gives a robust solution for various choices of damping, and has a predictable tradeoff between cost the and effectiveness.