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

Multilevel convergence analysis of multigrid-reduction-in-time

This paper presents a multilevel convergence framework for multigrid-reduction-in-time (MGRIT) as a generalization of previous two-grid estimates. The framework provides a priori upper bounds on the convergence of MGRIT V- and F-cycles, with different relaxation schemes, by deriving the respective residual and error propagation operators. The residual and error operators are functions of the time stepping operator, analyzed directly and bounded in norm, both numerically and analytically. We present various upper bounds of different computational cost and varying sharpness. These upper bounds are complemented by proposing analytic formulae for the approximate convergence factor of V-cycle algorithms that take the number of fine grid time points, the temporal coarsening factors, and the eigenvalues of the time stepping operator as parameters. The paper concludes with supporting numerical investigations of parabolic (anisotropic diffusion) and hyperbolic (wave equation) model problems. We assess the sharpness of the bounds and the quality of the approximate convergence factors. Observations from these numerical investigations demonstrate the value of the proposed multilevel convergence framework for estimating MGRIT convergence a priori and for the design of a convergent algorithm. We further highlight that observations in the literature are captured by the theory, including that two-level Parareal and multilevel MGRIT with F-relaxation do not yield scalable algorithms and the benefit of a stronger relaxation scheme. An important observation is that with increasing numbers of levels MGRIT convergence deteriorates for the hyperbolic model problem, while constant convergence factors can be achieved for the diffusion equation. The theory also indicates that L-stable Runge-Kutta schemes are more amendable to multilevel parallel-in-time integration with MGRIT than A-stable Runge-Kutta schemes.Comment: 26 pages; 17 pages Supplementary Material

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth

Optimized Schwarz Waveform Relaxation for Advection Reaction Diffusion Equations in Two Dimensions

Optimized Schwarz Waveform Relaxation methods have been developed over the last decade for the parallel solution of evolution problems. They are based on a decomposition in space and an iteration, where only subproblems in space-time need to be solved. Each subproblem can be simulated using an adapted numerical method, for example with local time stepping, or one can even use a different model in different subdomains, which makes these methods very suitable also from a modeling point of view. For rapid convergence however, it is important to use effective transmission conditions between the space-time subdomains, and for best performance, these transmission conditions need to take the physics of the underlying evolution problem into account. The optimization of these transmission conditions leads to a mathematically hard best approximation problem of homographic type. We study in this paper in detail this problem for the case of linear advection reaction diffusion equations in two spatial dimensions. We prove comprehensively best approximation results for transmission conditions of Robin and Ventcel type. We give for each case closed form asymptotic values for the parameters, which guarantee asymptotically best performance of the iterative methods. We finally show extensive numerical experiments, and we measure performance corresponding to our analysisComment: 42 page

[Research activities in applied mathematics, fluid mechanics, and computer science]

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, fluid mechanics, and computer science during the period April 1, 1995 through September 30, 1995