Parallel scalability study of three dimensional additive Schwarz preconditioners in non-overlapping domain decomposition

In this paper we study the parallel scalability of variants of additive Schwarz preconditioners for three dimensional non-overlapping domain decomposition methods. To alleviate the computational cost, both in terms of memory and floating-point complexity, we investigate variants based on a sparse approximation or on mixed 32- and 64-bit calculation. The robustness of the preconditioners is illustrated on a set of linear systems arising from the finite element discretization of elliptic PDEs through extensive parallel experiments on up to 1000 processors. Their efficiency from a numerical and parallel performance view point are studied

Parallel Selected Inversion for Space-Time Gaussian Markov Random Fields

Performing a Bayesian inference on large spatio-temporal models requires extracting inverse elements of large sparse precision matrices for marginal variances. Although direct matrix factorizations can be used for the inversion, such methods fail to scale well for distributed problems when run on large computing clusters. On the contrary, Krylov subspace methods for the selected inversion have been gaining traction. We propose a parallel hybrid approach based on domain decomposition, which extends the Rao-Blackwellized Monte Carlo estimator for distributed precision matrices. Our approach exploits the strength of Krylov subspace methods as global solvers and efficiency of direct factorizations as base case solvers to compute the marginal variances using a divide-and-conquer strategy. By introducing subdomain overlaps, one can achieve a greater accuracy at an increased computational effort with little to no additional communication. We demonstrate the speed improvements on both simulated models and a massive US daily temperature data.Comment: 17 pages, 7 figure

Sixth NASTRAN (R) Users' Colloquium

Papers are presented on NASTRAN programming, and substructuring methods, as well as on fluids and thermal applications. Specific applications and capabilities of NASTRAN were also delineated along with general auxiliary programs

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.Peer ReviewedPostprint (author's final draft

The principles and practice of the Xylophone Bar Magnetometer

PhD ThesisThis thesis reports on work undertaken to analyse, design, optimise, and fabricate a high-Quality factor mechanical resonant magnetometer, based on a Xylophone Bar Resonator (XBR). The principle of operation is based on the use of nodal supports to mechanically isolate a transverse beam vibrating in its fundamental mode. A control model is developed for the device, incorporating the effect of electromechanical parametric amplification. The device response and performance is shown to be strongly dependent on the Q factor of the sense element. The need for a quantitative model of XBR dynamics in order to design an optimal XBR is thus established. Using a Rayleigh-Ritz based approach, a model of the modal dynamics of an XBR is developed for the first time. In order to examine the efficacy of the nodal supports, a new model for support loss for resonators with two supports is developed and presented. Analytical models for other sources of dissipation are adapted for the first time to the XBR case. Combining these developments with a system level model allows for the development of a quantitative predictor of the fundamental and electronic noise limits on performance for an XBR. The model is solved over the operational range of geometric parameters, yielding optimisation criteria for the geometry. Corresponding predictions for the force and magnetic field sensitivity are presented. Based on the results, an optimised XBR design is exhibited for a macroscopic metal flexural XBM to be fabricated via Wire EDM. The fabricated devices are characterised, constituting the first demonstration of a macroscopic flexural XBR. The resulting Q factors and sensitivities are shown to be in agreement with the predictions. Fruitful directions for further work are suggested throughout the thesis and summarised in the conclusions. The original contribution to knowledge made by the thesis can be summarised as the development of an original and detailed theory of the principles of XBR optimisation for high Q, and demonstration of an operational macroscopic flexural XBM for the first time

On the parallel scalability of hybrid linear solvers for large 3D problems

Large-scale scientific applications and industrial simulations are nowadays fully integrated in many engineering areas. They involve the solution of large sparse linear systems. The use of large high performance computers is mandatory to solve these problems. The main topic of this research work was the study of a numerical technique that had attractive features for an efficient solution of large scale linear systems on large massively parallel platforms. The goal is to develop a high performance hybrid direct/iterative approach for solving large 3D problems. We focus specifically on the associated domain decomposition techniques for the parallel solution of large linear systems. We have investigated several algebraic preconditioning techniques, discussed their numerical behaviours, their parallel implementations and scalabilities. We have compared their performances on a set of 3D grand challenge problems

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 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 shows its rapid trend to further diversity and depth