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

Deterministic continutation of stochastic metastable equilibria via Lyapunov equations and ellipsoids

Numerical continuation methods for deterministic dynamical systems have been one of the most successful tools in applied dynamical systems theory. Continuation techniques have been employed in all branches of the natural sciences as well as in engineering to analyze ordinary, partial and delay differential equations. Here we show that the deterministic continuation algorithm for equilibrium points can be extended to track information about metastable equilibrium points of stochastic differential equations (SDEs). We stress that we do not develop a new technical tool but that we combine results and methods from probability theory, dynamical systems, numerical analysis, optimization and control theory into an algorithm that augments classical equilibrium continuation methods. In particular, we use ellipsoids defining regions of high concentration of sample paths. It is shown that these ellipsoids and the distances between them can be efficiently calculated using iterative methods that take advantage of the numerical continuation framework. We apply our method to a bistable neural competition model and a classical predator-prey system. Furthermore, we show how global assumptions on the flow can be incorporated - if they are available - by relating numerical continuation, Kramers' formula and Rayleigh iteration.Comment: 29 pages, 7 figures [Fig.7 reduced in quality due to arXiv size restrictions]; v2 - added Section 9 on Kramers' formula, additional computations, corrected typos, improved explanation

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

The impact of eigenvalue locality on the convergence behavior of the PSD method for two-cyclic matrices

AbstractIn this paper, we analyse the convergence of the preconditioned simultaneous displacement (PSD) method applied to linear systems of the form Au=b where A is a two-cyclic matrix. Convergence conditions and optimum values of the parameters of the method are determined in the cases where the eigenvalues of the associated Jacobi iteration matrix are either all real or all imaginary. It is shown that the convergence behavior of the PSD method is greatly affected by the locality of the eigenvalues of the associated Jacobi iteration matrix. In particular, it is shown that when these eigenvalues are real the PSD method degenerates into the extrapolated Gauss–Seidel method whereas when they are imaginary its convergence is increased by an order of magnitude and becomes equivalent to the extrapolated SOR method. Finally, a comparison with the SSOR method reveals that the PSD method possesses a better convergence behavior in all cases

Schwarz Iterative Methods: Infinite Space Splittings

We prove the convergence of greedy and randomized versions of Schwarz iterative methods for solving linear elliptic variational problems based on infinite space splittings of a Hilbert space. For the greedy case, we show a squared error decay rate of $O((m+1)^{-1})$ for elements of an approximation space $\mathcal{A}_1$ related to the underlying splitting. For the randomized case, we show an expected squared error decay rate of $O((m+1)^{-1})$ on a class $\mathcal{A}_{\infty}^{\pi}\subset \mathcal{A}_1$ depending on the probability distribution.Comment: Revised version, accepted in Constr. Appro

Asynchronous iterative solution for dominant eigenvectors with applications in performance modelling and PageRank

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A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also