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.Comment: 14 page

Sparsifying preconditioner for the time-harmonic Maxwell's equations

This paper presents the sparsifying preconditioner for the time-harmonic Maxwell's equations in the integral formulation. Following the work on sparsifying preconditioner for the Lippmann-Schwinger equation, this paper generalizes that approach from the scalar wave case to the vector case. The key idea is to construct a sparse approximation to the dense system by minimizing the non-local interactions in the integral equation, which allows for applying sparse linear solvers to reduce the computational cost. When combined with the standard GMRES solver, the number of preconditioned iterations remains small and essentially independent of the frequency. This suggests that, when the sparsifying preconditioner is adopted, solving the dense integral system can be done as efficiently as solving the sparse system from PDE discretization

Localized Sparsifying Preconditioner for Periodic Indefinite Systems

This paper introduces the localized sparsifying preconditioner for the pseudospectral approximations of indefinite systems on periodic structures. The work is built on top of the recently proposed sparsifying preconditioner with two major modifications. First, the local potential information is utilized to improve the accuracy of the preconditioner. Second, an FFT based method to compute the local stencil is proposed to reduce the setup time of the algorithm. Numerical results show that the iteration number of this improved method grows only mildly as the problem size grows, which implies that solving pseudospectral approximation systems is computationally as efficient as solving sparse systems, up to a mildly growing factor

Preconditioning orbital minimization method for planewave discretization

We present an efficient preconditioner for the orbital minimization method when the Hamiltonian is discretized using planewaves (i.e., pseudospectral method). This novel preconditioner is based on an approximate Fermi operator projection by pole expansion, combined with the sparsifying preconditioner to efficiently evaluate the pole expansion for a wide range of Hamiltonian operators. Numerical results validate the performance of the new preconditioner for the orbital minimization method, in particular, the iteration number is reduced to $O(1)$ and often only a few iterations are enough for convergence

Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation

This paper presents an efficient preconditioner for the Lippmann-Schwinger equation that combines the ideas of the sparsifying and the sweeping preconditioners. Following first the idea of the sparsifying preconditioner, this new preconditioner starts by transforming the dense linear system of the Lippmann-Schwinger equation into a nearly sparse system. The key novelty is a newly designed perfectly matched layer (PML) stencil for the boundary degrees of freedoms. The resulting sparse system gives rise to fairly accurate solutions and hence can be viewed as an accurate discretization of the Helmholtz equation. This new PML stencil also paves the way for applying the moving PML sweeping preconditioner to invert the resulting sparse system approximately. When combined with the standard GMRES solver, this new preconditioner for the Lippmann-Schwinger equation takes only a few iterations to converge for both 2D and 3D problems, where the iteration numbers are almost independent of the frequency. To the best of our knowledge, this is the first method that achieves near-linear cost to solve the 3D Lippmann-Schwinger equation in high frequency cases