Fast Alternating BiDirectional Preconditioner for the 2D High-Frequency Lippmann-Schwinger Equation

This paper presents a fast iterative solver for Lippmann-Schwinger equation for highfrequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solver is based on the sparsifying preconditioner [63] and a domain decomposition approach similar to the method of polarized traces [64]. The iterative solver has two levels, the outer level in which a sparsifying preconditioner for the Lippmann-Schwinger equation is constructed, and the inner level, in which the resulting sparsified system is solved fast using an iterative solver preconditioned with a bidirectional matrix-free variant of the method of polarized traces. The complexity of the construction and application of the preconditioner is O(N) and O(N logN) respectively, where N is the number of degrees of freedom. Numerical experiments in 2D indicate that the number of iterations in both levels depends weakly on the frequency resulting in a method with an overall O(N logN) complexity.