Simple parallel and distributed algorithms for spectral graph sparsification

We describe a simple algorithm for spectral graph sparsification, based on iterative computations of weighted spanners and uniform sampling. Leveraging the algorithms of Baswana and Sen for computing spanners, we obtain the first distributed spectral sparsification algorithm. We also obtain a parallel algorithm with improved work and time guarantees. Combining this algorithm with the parallel framework of Peng and Spielman for solving symmetric diagonally dominant linear systems, we get a parallel solver which is much closer to being practical and significantly more efficient in terms of the total work.Comment: replaces "A simple parallel and distributed algorithm for spectral sparsification". Minor change

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'

Reducing communication in sparse solvers

Sparse matrix operations dominate the cost of many scientific applications. In parallel, the performance and scalability of these operations is limited by irregular point-to-point communication. Multiple methods are investigated throughout this dissertation for reducing the cost associated with communication throughout sparse matrix operations. Algorithmic changes reduce communication requirements, but also affect accuracy of the operation, leading to reduced convergence of scientific codes. We investigate a method of systematically removing relatively small non-zeros throughout an algebraic multigrid hierarchy, yielding significant reductions to the cost of sparse matrix-vector multiplication that outweigh affects of reduced accuracy of the multiplication. Therefore, the reduction in per-iteration communication costs outweigh the cost of extra solver iterations. As a result, sparsification yields improvement of both the performance and scalability of algebraic multigrid. Alterations to the parallel implementation of MPI communication also yield reduced costs with no effect on accuracy. We investigate methods of agglomerating messages on-node before injecting into the network, reducing the amount of costly inter-node communication. This node-aware communication yields improvements to both performance and scalability of matrix operations, particularly in strong scaling studies. Furthermore, we show an improvement in the cost of algebraic multigrid as a result of reduced communication costs in sparse matrix operations. Finally, performance models can be used to analyze the costs of matrix operations, indicating the source of dominant communication costs, such as initializing messages or transporting bytes of data. We investigate methods of improving traditional performance models of irregular point-to-point communication through the addition of node-awareness, queue search costs, and network contention penalties

HIGH-PERFORMANCE SPECTRAL METHODS FOR COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS

Recent research shows that by leveraging the key spectral properties of eigenvalues and eigenvectors of graph Laplacians, more efficient algorithms can be developed for tackling many graph-related computing tasks. In this dissertation, spectral methods are utilized for achieving faster algorithms in the applications of very-large-scale integration (VLSI) computer-aided design (CAD) First, a scalable algorithmic framework is proposed for effective-resistance preserving spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving post-scaling and iterative solution refinement schemes. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms. Based on the framework of the spectral graph reduction, a Sparsified graph-theoretic Algebraic Multigrid (SAMG) is proposed for solving large Symmetric Diagonally Dominant (SDD) matrices. The proposed SAMG framework allows efficient construction of nearly-linear sized graph Laplacians for coarse-level problems while maintaining good spectral approximation during the AMG setup phase by leveraging a scalable spectral graph sparsification engine. Our experimental results show that the proposed method can offer more scalable performance than existing graph-theoretic AMG solvers for solving large SDD matrices in integrated circuit (IC) simulations, 3D-IC thermal analysis, image processing, finite element analysis as well as data mining and machine learning applications. Finally, the spectral methods are applied to power grid and thermal integrity verification applications. This dissertation introduces a vectorless power grid and thermal integrity verification framework that allows computing worst-case voltage drop or thermal profiles across the entire chip under a set of local and global workload (power density) constraints. To address the computational challenges introduced by the large 3D mesh-structured thermal grids, we apply the spectral graph reduction approach for highly-scalable vectorless thermal (or power grids) verification of large chip designs. The effectiveness and efficiency of our approach have been demonstrated through extensive experiments