Sparsifying preconditioner for pseudospectral approximations of indefinite systems on periodic structures

This paper introduces the sparsifying preconditioner for the pseudospectral approximation of highly indefinite systems on periodic structures, which include the frequency-domain response problems of the Helmholtz equation and the Schr\"odinger equation as examples. This approach transforms the dense system of the pseudospectral discretization approximately into an sparse system via an equivalent integral reformulation and a specially-designed sparsifying operator. The resulting sparse system is then solved efficiently with sparse linear algebra algorithms and serves as a reasonably accurate preconditioner. When combined with standard iterative methods, this new preconditioner results in small iteration counts. Numerical results are provided for the Helmholtz equation and the Schr\"odinger in both 2D and 3D to demonstrate the effectiveness of this new preconditioner.