Sparsifying preconditioner for pseudospectral approximations of indefinite systems on periodic structures
For computational scientists solving frequency-domain wave problems, this provides a more efficient preconditioner for pseudospectral methods, though it is incremental over existing sparse preconditioning techniques.
The paper introduces a sparsifying preconditioner for pseudospectral approximations of highly indefinite systems (e.g., Helmholtz and Schrödinger equations) on periodic structures, transforming dense systems into sparse ones for efficient iterative solution. Numerical results show small iteration counts in 2D and 3D.
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 Schrödinger 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 Schrödinger in both 2D and 3D to demonstrate the effectiveness of this new preconditioner.