An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling

We present a sparse linear system solver that is based on a multifrontal variant of Gaussian elimination, and exploits low-rank approximation of the resulting dense frontal matrices. We use hierarchically semiseparable (HSS) matrices, which have low-rank off-diagonal blocks, to approximate the frontal matrices. For HSS matrix construction, a randomized sampling algorithm is used together with interpolative decompositions. The combination of the randomized compression with a fast ULV HSS factorization leads to a solver with lower computational complexity than the standard multifrontal method for many applications, resulting in speedups up to 7 fold for problems in our test suite. The implementation targets many-core systems by using task parallelism with dynamic runtime scheduling. Numerical experiments show performance improvements over state-of-the-art sparse direct solvers. The implementation achieves high performance and good scalability on a range of modern shared memory parallel systems, including the Intel Xeon Phi (MIC). The code is part of a software package called STRUMPACK -- STRUctured Matrices PACKage, which also has a distributed memory component for dense rank-structured matrices

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'

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

Telescopic hybrid fast solver for 3D elliptic problems with point singularities

This paper describes a telescopic solver for two dimensional h adaptive grids with point singularities. The input for the telescopic solver is an h refined two dimensional computational mesh with rectangular finite elements. The candidates for point singularities are first localized over the mesh by using a greedy algorithm. Having the candidates for point singularities, we execute either a direct solver, that performs multiple refinements towards selected point singularities and executes a parallel direct solver algorithm which has logarithmic cost with respect to refinement level. The direct solvers executed over each candidate for point singularity return local Schur complement matrices that can be merged together and submitted to iterative solver. In this paper we utilize a parallel multi-thread GALOIS solver as a direct solver. We use Incomplete LU Preconditioned Conjugated Gradients (ILUPCG) as an iterative solver. We also show that elimination of point singularities from the refined mesh reduces significantly the number of iterations to be performed by the ILUPCG iterative solver

A summary of my twenty years of research according to Google Scholars

I am David Pardo, a researcher from Spain working mainly on numerical analysis applied to geophysics. I am 40 years old, and over a decade ago, I realized that my performance as a researcher was mainly evaluated based on a number called \h-index". This single number contains simultaneously information about the number of publications and received citations. However, dif- ferent h-indices associated to my name appeared in di erent webpages. A quick search allowed me to nd the most convenient (largest) h-index in my case. It corresponded to Google Scholars. In this work, I naively analyze a few curious facts I found about my Google Scholars and, at the same time, this manuscript serves as an experiment to see if it may serve to increase my Google Scholars h-index

Applications of a hyper-graph grammar system in adaptive finite-element computations

This paper describes application of a hyper-graph grammar system for modeling a three-dimensional adaptive finite element method. The hyper-graph grammar approach allows obtaining a linear computational cost of adaptive mesh transformations and computations performed over refined meshes. The computations are done by a hyper-graph grammar driven algorithm applicable to three-dimensional problems. For the case of typical refinements performed towards a point or an edge, the algorithm yields linear computational cost with respect to the mesh nodes for its sequential execution and logarithmic cost for its parallel execution. Such hyper-graph grammar productions are the mathematical formalism used to describe the computational algorithm implementing the finite element method. Each production indicates the smallest atomic task that can be executed concurrently. The mesh transformations and computations by using the hyper-graph grammar-based approach have been tested in the GALOIS environment. We conclude the paper with some numerical results performed on a shared-memory Linux cluster node, for the case of three-dimensional computational meshes refined towards a point, an edge and a face

Graph grammar-based multi-frontal parallel direct solver for two-dimensional isogeometric analysis

This paper introduces the graph grammar based model for developing multi-thread multi-frontal parallel direct solver for two dimensional isogeometric finite element method. Execution of the solver algorithm has been expressed as the sequence of graph grammar productions. At the beginning productions construct the elimination tree with leaves corresponding to finite elements. Following sequence of graph grammar productions generates element frontal matrices at leaf nodes, merges matrices at parent nodes and eliminates rows corresponding to fully assembled degrees of freedom. Finally, there are graph grammar productions responsible for root problem solution and recursive backward substitutions. Expressing the solver algorithm by graph grammar productions allows us to explore the concurrency of the algorithm. The graph grammar productions are grouped into sets of independent tasks that can be executed concurrently. The resulting concurrent multi-frontal solver algorithm is implemented and tested on NVIDIA GPU, providing O(NlogN) execution time complexity where N is the number of degrees of freedom. We have confirmed this complexity by solving up to 1 million of degrees of freedom with 448 cores GPU. © 2012 Published by Elsevier Ltd

Objective multiscale analysis of random heterogeneous materials

The multiscale framework presented in [1, 2] is assessed in this contribution for a study of random heterogeneous materials. Results are compared to direct numerical simulations (DNS) and the sensitivity to user-defined parameters such as the domain decomposition type and initial coarse scale resolution is reported. The parallel performance of the implementation is studied for different domain decompositions