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In this paper, we discuss the usage of overlapping techniques for improving the convergence of preconditioners based on incomplete factorizations. To enable parallelism, these preconditioners are usually applied after the input matrix is permuted into a nested arrow form using k-way nested dissection. This graph partitioning technique uses k-way partitionning by vertex separator to recursively partition the graph of the input matrix into k subgraphs using a subset of its vertices called a separator. The overlapping technique is then based on algebraically extending the associated subdomains of these subgraphs and their corresponding separators obtained from k-way nested dissection by their direct neighbours. A similar approach is known to accelerate the convergence of domain decomposition methods, where the input matrix is partitioned into a number of independent subdomains using k-way vertex partitioning of a graph by edge separators, a different graph decomposition technique. We discuss the effect of the overlapping technique on the convergence of two classes of preconditioners, on the basis of nested factorization and block incomplete LDU factorization.

This paper introduces a new preconditioning technique that is suitable for matrices arising from the discretization of a system of PDEs on unstructured grids. The preconditioner satisfies a so-called filtering property, which ensures that the input matrix is identical with the preconditioner on a given filtering vector. This vector is chosen to alleviate the effect of low-frequency modes on convergence and so decrease or eliminate the plateau that is often observed in the convergence of iterative methods. In particular, the paper presents a general approach that allows to ensure that the filtering condition is satisfied in a matrix decomposition. The input matrix can have an arbitrary sparse structure. Hence, it can be reordered using nested dissection, to allow a parallel computation of the preconditioner and of the iterative process. We show the efficiency of our preconditioner through a set of numerical experiments on symmetric and nonsymmetric matrices.

Spherical harmonic transforms SHT are at the heart of many scientific and practical applications ranging from climate modelling to cosmological observations. In many of these areas, new cutting-edge science goals have been recently proposed requiring simulations and analyses of experimental or observational data at very high resolutions and of unprecedented volumes. Both these aspects pose formidable challenge for the currently existing implementations of the transforms. This paper describes parallel algorithms for computing SHT with two variants of intra-node parallelism appropriate for novel supercomputer architectures, multi-core processors and Graphic Processing Units GPU. It also discusses their performance, alone and embedded within a top-level, Message Passing Interface-based parallelisation layer ported from the S2HAT library, in terms of their accuracy, overall efficiency and scalability. We show that our inverse SHT run on GeForce 400 Series GPUs equipped with latest Compute Unified Device Architecture architecture Fermi outperforms the state of the art implementation for a multi-core processor executed on a current Intel Core i7-2600K. Furthermore, we show that an Message Passing Interface/Compute Unified Device Architecture version of the inverse transform run on a cluster of 128 Nvidia Tesla S1070 is as much as 3times faster than the hybrid Message Passing Interface/OpenMP version executed on the same number of quad-core processors Intel Nehalem for problem sizes motivated by our target applications. Performance of the direct transforms is however found to be at the best comparable in these cases. We discuss in detail the algorithmic solutions devised for the major steps involved in the transforms calculation, emphasising those with a major impact on their overall performance and elucidates the sources of the dichotomy between the direct and the inverse operations.Copyright © 2013 John Wiley & Sons, Ltd.

We consider the iterative solution of the linear systems arising from four convection-diffusion model problems: the scalar convection-diffusion problem, Stokes problem, Oseen problem, and Navier-Stokes problem. We give the explicit Kronecker product structure of the coefficient matrices, especially the Kronecker product structure for the convection term. For the latter three model cases, the coefficient matrices have a $2 \times 2$ blocks, and each block is a Kronecker product or a summation of several Kronecker products. We use the Kronecker products and block structures to design the diagonal block preconditioner, the tridiagonal block preconditioner and the constraint preconditioner. We can find that the constraint preconditioner can be regarded as the modification of the tridiagonal block preconditioner and the diagonal block preconditioner based on the cell Reynolds number. That's the reason why the constraint preconditioner is usually better. We also give numerical examples to show the efficiency of this kind of Kronecker product approximation preconditioners.