A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems

In this paper we construct and analyse a level-dependent coarsegrid correction scheme for indefinite Helmholtz problems. This adapted multigrid method is capable of solving the Helmholtz equation on the finest grid using a series of multigrid cycles with a grid-dependent complex shift, leading to a stable correction scheme on all levels. It is rigourously shown that the adaptation of the complex shift throughout the multigrid cycle maintains the functionality of the two-grid correction scheme, as no smooth modes are amplified in or added to the error. In addition, a sufficiently smoothing relaxation scheme should be applied to ensure damping of the oscillatory error components. Numerical experiments on various benchmark problems show the method to be competitive with or even outperform the current state-of-the-art multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or BiCGStab.Comment: 21 page

Preconditioning harmonic unsteady potential flow calculations

This paper considers finite element discretisations of the Helmholtz equation and its generalisation arising from harmonic acoustics perturbations to a non-uniform steady potential flow. A novel elliptic, positive definite preconditioner, with a multigrid implementation, is used to accelerate the iterative convergence of Krylov subspace solvers. Both theory and numerical results show that for a model 1D Helmholtz test problem the preconditioner clusters the discrete system's eigenvalues and lowers its condition number to a level independent of grid resolution. For the 2D Helmholtz equation, grid independent convergence is achieved using a QMR Krylov solver, significantly outperforming the popular SSOR preconditioner. Impressive results are also presented on more complex domains, including an axisymmetric aircraft engine inlet with non-stagnant mean flow and modal boundary conditions

Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio

Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the pressure correction has to be solved at every time step in a thin spherical shell representing the global atmosphere. This elliptic solve can be one of the computationally most demanding components in semi-implicit semi-Lagrangian time stepping methods which are very popular as they allow for larger model time steps and better overall performance. With increasing model resolution, algorithmically efficient and scalable algorithms are essential to run the code under tight operational time constraints. We discuss the theory and practical application of bespoke geometric multigrid preconditioners for equations of this type. The algorithms deal with the strong anisotropy in the vertical direction by using the tensor-product approach originally analysed by B\"{o}rm and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the analysis to three dimensions under slightly weakened assumptions, and numerically demonstrate its efficiency for the solution of the elliptic PDE for the global pressure correction in atmospheric forecast models. For this we compare the performance of different multigrid preconditioners on a tensor-product grid with a semi-structured and quasi-uniform horizontal mesh and a one dimensional vertical grid. The code is implemented in the Distributed and Unified Numerics Environment (DUNE), which provides an easy-to-use and scalable environment for algorithms operating on tensor-product grids. Parallel scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR supercomputer.Comment: 22 pages, 6 Figures, 2 Table

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

Absolute value preconditioning for symmetric indefinite linear systems

We introduce a novel strategy for constructing symmetric positive definite (SPD) preconditioners for linear systems with symmetric indefinite matrices. The strategy, called absolute value preconditioning, is motivated by the observation that the preconditioned minimal residual method with the inverse of the absolute value of the matrix as a preconditioner converges to the exact solution of the system in at most two steps. Neither the exact absolute value of the matrix nor its exact inverse are computationally feasible to construct in general. However, we provide a practical example of an SPD preconditioner that is based on the suggested approach. In this example we consider a model problem with a shifted discrete negative Laplacian, and suggest a geometric multigrid (MG) preconditioner, where the inverse of the matrix absolute value appears only on the coarse grid, while operations on finer grids are based on the Laplacian. Our numerical tests demonstrate practical effectiveness of the new MG preconditioner, which leads to a robust iterative scheme with minimalist memory requirements

Local Fourier Analysis of the Complex Shifted Laplacian preconditioner for Helmholtz problems

In this paper we solve the Helmholtz equation with multigrid preconditioned Krylov subspace methods. The class of Shifted Laplacian preconditioners are known to significantly speed-up Krylov convergence. However, these preconditioners have a parameter beta, a measure of the complex shift. Due to contradictory requirements for the multigrid and Krylov convergence, the choice of this shift parameter can be a bottleneck in applying the method. In this paper, we propose a wavenumber-dependent minimal complex shift parameter which is predicted by a rigorous k-grid Local Fourier Analysis (LFA) of the multigrid scheme. We claim that, given any (regionally constant) wavenumber, this minimal complex shift parameter provides the reader with a parameter choice that leads to efficient Krylov convergence. Numerical experiments in one and two spatial dimensions validate the theoretical results. It appears that the proposed complex shift is both the minimal requirement for a multigrid V-cycle to converge, as well as being near-optimal in terms of Krylov iteration count.Comment: 20 page