A Class of Iterative Solvers for the Helmholtz Equation: Factorizations, Sweeping Preconditioners, Source Transfer, Single Layer Potentials, Polarized Traces, and Optimized Schwarz Methods

Solving time-harmonic wave propagation problems by iterative methods is a difficult task, and over the last two decades an important research effort has gone into developing preconditioners for the simplest representative of such wave propagation problems, the Helmholtz equation. A specific class of these new preconditioners is considered here. They were developed by researchers with various backgrounds using formulations and notations that are very different, and all are among the most promising preconditioners for the Helmholtz equation. The goal of the present article is to show that this class of preconditioners is based on a common mathematical principle, and that they can all be formulated in the context of domain decomposition methods known as optimized Schwarz methods. This common formulation allows us to explain in detail how and why all these methods work. The domain decomposition formulation also allows us to avoid technicalities in the implementation description we give of these recent methods. The equivalence of these methods with optimized Schwarz methods translates at the discrete level into equivalence with approximate block LU decomposition preconditioners, and in each case we give the algebraic version, including a detailed description of the approximations used. While we choose to use the Helmholtz equation for which these methods were developed, our notation is completely general and the algorithms we give are written for an arbitrary second-order elliptic operator. The algebraic versions are even more general, assuming only a connectivity pattern in the discretization matrix. All the new methods studied here are based on sequential decomposition of the problem in space into a sequence of subproblems, and they have in their optimal form the property to lead to nilpotent iterations, like an exact block LU factorization. Using our domain decomposition formulation, we finally present an algorithm for two-dimensional decompositions, i.e., decompositions that contain cross points, which is still nilpotent in its optimal form. Its approximation is currently an active area of research, and it would have been difficult to discover such an algorithm without the domain decomposition framework.