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We present a parallel preconditioning method for the iterative solution of the time-harmonic elastic wave equation which makes use of higher-order spectral elements to reduce pollution error. In particular, the method leverages perfectly matched layer boundary conditions to efficiently approximate the Schur complement matrices of a block LDL T factorization. Both sequential and parallel versions of the algorithm are discussed and results for large-scale problems from exploration geophysics are presented.

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SummaryDiscrete representations of the Helmholtz operator generally give rise to extremely difficult linear systems from an iterative solver perspective. This is due in part to the large oscillatory near null space of the linear system. Typical iterative methods do not effectively reduce error components in the subspace associated with this near null space. Traditional coarse grids used within multilevel solvers also cannot capture these components because of their oscillatory nature. While the shifted Laplacian is a popular method allowing multigrid preconditioners to achieve improved convergence, it is still unsatisfactory for higher‐frequency problems. This paper analyzes the shifted Laplacian through polynomial smoothers to show its effect on the spectrum of the discrete Helmholtz operator. This analysis reveals desirable features of the shifted Laplacian preconditioners as well as limitations. Shifted Laplacian techniques are first extended to smoothed aggregation algebraic multigrid. Motivated by analysis, we propose augmenting the algebraic multigrid shifted Laplacian with a two‐grid error correction step. This additional step consists of the generalized minimal residual iterations applied to a projected version of the unshifted equations on a single auxiliary coarse grid. Results indicate that augmentation can improve the convergence significantly and can reduce the overall solve time on two‐dimensional and three‐dimensional problems. Copyright © 2015 John Wiley & Sons, Ltd.

This paper is concerned with the uncertainty quantification of high-frequency acoustic scattering from objects with random shape in two-dimensional space. Several new methods are introduced to efficiently estimate the mean and variance of the random radar cross section in all directions. In the physical domain, the scattering problem is solved using the boundary integral formulation and Nyström discretization; recently developed fast algorithms are adapted to accelerate the computation of the integral operator and the evaluation of the radar cross section. In the random domain, it is discovered that due to the highly oscillatory nature of the solution, the stochastic collocation method based on sparse grids does not perform well. For this particular problem, satisfactory results are obtained by using quasi-Monte Carlo methods. Numerical results are given for several test cases to illustrate the properties of the proposed approach.