Parallel accelerated cyclic reduction preconditioner for three-dimensional elliptic PDEs with variable coefficients

We present a robust and scalable preconditioner for the solution of large-scale linear systems that arise from the discretization of elliptic PDEs amenable to rank compression. The preconditioner is based on hierarchical low-rank approximations and the cyclic reduction method. The setup and application phases of the preconditioner achieve log-linear complexity in memory footprint and number of operations, and numerical experiments exhibit good weak and strong scalability at large processor counts in a distributed memory environment. Numerical experiments with linear systems that feature symmetry and nonsymmetry, definiteness and indefiniteness, constant and variable coefficients demonstrate the preconditioner applicability and robustness. Furthermore, it is possible to control the number of iterations via the accuracy threshold of the hierarchical matrix approximations and their arithmetic operations, and the tuning of the admissibility condition parameter. Together, these parameters allow for optimization of the memory requirements and performance of the preconditioner.Comment: 24 pages, Elsevier Journal of Computational and Applied Mathematics, Dec 201

Shifted Laplacian multigrid for the elastic Helmholtz equation

The shifted Laplacian multigrid method is a well known approach for preconditioning the indefinite linear system arising from the discretization of the acoustic Helmholtz equation. This equation is used to model wave propagation in the frequency domain. However, in some cases the acoustic equation is not sufficient for modeling the physics of the wave propagation, and one has to consider the elastic Helmholtz equation. Such a case arises in geophysical seismic imaging applications, where the earth's subsurface is the elastic medium. The elastic Helmholtz equation is much harder to solve than its acoustic counterpart, partially because it is three times larger, and partially because it models more complicated physics. Despite this, there are very few solvers available for the elastic equation compared to the array of solvers that are available for the acoustic one. In this work we extend the shifted Laplacian approach to the elastic Helmholtz equation, by combining the complex shift idea with approaches for linear elasticity. We demonstrate the efficiency and properties of our solver using numerical experiments for problems with heterogeneous media in two and three dimensions

High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension

The nonlinear Helmholtz equation (NLH) models the propagation of electromagnetic waves in Kerr media, and describes a range of important phenomena in nonlinear optics and in other areas. In our previous work, we developed a fourth order method for its numerical solution that involved an iterative solver based on freezing the nonlinearity. The method enabled a direct simulation of nonlinear self-focusing in the nonparaxial regime, and a quantitative prediction of backscattering. However, our simulations showed that there is a threshold value for the magnitude of the nonlinearity, above which the iterations diverge. In this study, we numerically solve the one-dimensional NLH using a Newton-type nonlinear solver. Because the Kerr nonlinearity contains absolute values of the field, the NLH has to be recast as a system of two real equations in order to apply Newton's method. Our numerical simulations show that Newton's method converges rapidly and, in contradistinction with the iterations based on freezing the nonlinearity, enables computations for very high levels of nonlinearity. In addition, we introduce a novel compact finite-volume fourth order discretization for the NLH with material discontinuities.The one-dimensional results of the current paper create a foundation for the analysis of multi-dimensional problems in the future.Comment: 47 pages, 8 figure

Sweeping Preconditioner for the Helmholtz Equation: Moving Perfectly Matched Layers

This paper introduces a new sweeping preconditioner for the iterative solution of the variable coefficient Helmholtz equation in two and three dimensions. The algorithms follow the general structure of constructing an approximate $LDL^t$ factorization by eliminating the unknowns layer by layer starting from an absorbing layer or boundary condition. The central idea of this paper is to approximate the Schur complement matrices of the factorization using moving perfectly matched layers (PMLs) introduced in the interior of the domain. Applying each Schur complement matrix is equivalent to solving a quasi-1D problem with a banded LU factorization in the 2D case and to solving a quasi-2D problem with a multifrontal method in the 3D case. The resulting preconditioner has linear application cost and the preconditioned iterative solver converges in a number of iterations that is essentially indefinite of the number of unknowns or the frequency. Numerical results are presented in both two and three dimensions to demonstrate the efficiency of this new preconditioner.Comment: 25 page

Fast, adaptive, high order accurate discretization of the Lippmann-Schwinger equation in two dimension

We present a fast direct solver for two dimensional scattering problems, where an incident wave impinges on a penetrable medium with compact support. We represent the scattered field using a volume potential whose kernel is the outgoing Green's function for the exterior domain. Inserting this representation into the governing partial differential equation, we obtain an integral equation of the Lippmann-Schwinger type. The principal contribution here is the development of an automatically adaptive, high-order accurate discretization based on a quad tree data structure which provides rapid access to arbitrary elements of the discretized system matrix. This permits the straightforward application of state-of-the-art algorithms for constructing compressed versions of the solution operator. These solvers typically require $O(N^{3/2})$ work, where $N$ denotes the number of degrees of freedom. We demonstrate the performance of the method for a variety of problems in both the low and high frequency regimes.Comment: 18 page