Time-Optimal Tree Computations on Sparse Meshes

The main goal of this work is to fathom the suitability of the mesh with multiple broadcasting architecture (MMB) for some tree-related computations. We view our contribution at two levels: on the one hand, we exhibit time lower bounds for a number of tree-related problems on the MMB. On the other hand, we show that these lower bounds are tight by exhibiting time-optimal tree algorithms on the MMB. Specifically, we show that the task of encoding and/or decoding n-node binary and ordered trees cannot be solved faster than Ω(log n) time even if the MMB has an infinite number of processors. We then go on to show that this lower bound is tight. We also show that the task of reconstructing n-node binary trees and ordered trees from their traversais can be performed in O(1) time on the same architecture. Our algorithms rely on novel time-optimal algorithms on sequences of parentheses that we also develop

Computing the eigenvalues of symmetric H2-matrices by slicing the spectrum

The computation of eigenvalues of large-scale matrices arising from finite element discretizations has gained significant interest in the last decade. Here we present a new algorithm based on slicing the spectrum that takes advantage of the rank structure of resolvent matrices in order to compute m eigenvalues of the generalized symmetric eigenvalue problem in $\mathcal{O}(n m \log^\alpha n)$ operations, where $\alpha>0$ is a small constant

Optimal, scalable forward models for computing gravity anomalies

We describe three approaches for computing a gravity signal from a density anomaly. The first approach consists of the classical "summation" technique, whilst the remaining two methods solve the Poisson problem for the gravitational potential using either a Finite Element (FE) discretization employing a multilevel preconditioner, or a Green's function evaluated with the Fast Multipole Method (FMM). The methods utilizing the PDE formulation described here differ from previously published approaches used in gravity modeling in that they are optimal, implying that both the memory and computational time required scale linearly with respect to the number of unknowns in the potential field. Additionally, all of the implementations presented here are developed such that the computations can be performed in a massively parallel, distributed memory computing environment. Through numerical experiments, we compare the methods on the basis of their discretization error, CPU time and parallel scalability. We demonstrate the parallel scalability of all these techniques by running forward models with up to $10^8$ voxels on 1000's of cores.Comment: 38 pages, 13 figures; accepted by Geophysical Journal Internationa

Preparing sparse solvers for exascale computing.

Sparse solvers provide essential functionality for a wide variety of scientific applications. Highly parallel sparse solvers are essential for continuing advances in high-fidelity, multi-physics and multi-scale simulations, especially as we target exascale platforms. This paper describes the challenges, strategies and progress of the US Department of Energy Exascale Computing project towards providing sparse solvers for exascale computing platforms. We address the demands of systems with thousands of high-performance node devices where exposing concurrency, hiding latency and creating alternative algorithms become essential. The efforts described here are works in progress, highlighting current success and upcoming challenges. This article is part of a discussion meeting issue 'Numerical algorithms for high-performance computational science'

Adaptive multiresolution computations applied to detonations

A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge--Kutta--Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allow to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.Comment: Zeitschrift f\"ur Physicalische Chemie, accepte

PT-Scotch: A tool for efficient parallel graph ordering

The parallel ordering of large graphs is a difficult problem, because on the one hand minimum degree algorithms do not parallelize well, and on the other hand the obtainment of high quality orderings with the nested dissection algorithm requires efficient graph bipartitioning heuristics, the best sequential implementations of which are also hard to parallelize. This paper presents a set of algorithms, implemented in the PT-Scotch software package, which allows one to order large graphs in parallel, yielding orderings the quality of which is only slightly worse than the one of state-of-the-art sequential algorithms. Our implementation uses the classical nested dissection approach but relies on several novel features to solve the parallel graph bipartitioning problem. Thanks to these improvements, PT-Scotch produces consistently better orderings than ParMeTiS on large numbers of processors