arXiv subjects

Jed Brown

Publications and source records attributed to Jed Brown.

26 records · Page 2Linked to original sources

arXiv2018

A Parallel Solver for Graph Laplacians

Problems from graph drawing, spectral clustering, network flow and graph partitioning can all be expressed in terms of graph Laplacian matrices. There are a variety of practical approaches to solving these problems in serial. However, as problem sizes increase and single core speeds stagnate, parallelism is essential to solve such problems quickly. We present an unsmoothed aggregation multigrid method for solving graph Laplacians in a distributed memory setting. We introduce new parallel aggregation and low degree elimination algorithms targeted specifically at irregular degree graphs. These algorithms are expressed in terms of sparse matrix-vector products using generalized sum and product operations. This formulation is amenable to linear algebra using arbitrary distributions and allows us to operate on a 2D sparse matrix distribution, which is necessary for parallel scalability. Our solver outperforms the natural parallel extension of the current state of the art in an algorithmic comparison. We demonstrate scalability to 576 processes and graphs with up to 1.7 billion edges.

cs.DC↗

arXiv2018

PETSc/TS: A Modern Scalable ODE/DAE Solver Library

High-quality ordinary differential equation (ODE) solver libraries have a long history, going back to the 1970s. Over the past several years we have implemented, on top of the PETSc linear and nonlinear solver package, a new general-purpose, extensive, extensible library for solving ODEs and differential algebraic equations (DAEs). Package includes support for both forward and adjoint sensitivities that can be easily utilized by the TAO optimization package, which is also part of PETSc. The ODE/DAE integrator library strives to be highly scalable but also to deliver high efficiency for modest-sized problems. The library includes explicit solvers, implicit solvers, and a collection of implicit-explicit solvers, all with a common user interface and runtime selection of solver types, adaptive error control, and monitoring of solution progress. The library also offers enormous flexibility in selection of nonlinear and linear solvers, including the entire suite of PETSc iterative solvers, as well as several parallel direct solvers.

math.NA↗

arXiv2014

Run-time extensibility and librarization of simulation software

Build-time configuration and environment assumptions are hampering progress and usability in scientific software. That which would be utterly unacceptable in non-scientific software somehow passes for the norm in scientific packages. The community needs reusable software packages that are easy use and flexible enough to accommodate next-generation simulation and analysis demands.

cs.SE↗

arXiv2013

Achieving High Performance with Unified Residual Evaluation

We examine residual evaluation, perhaps the most basic operation in numerical simulation. By raising the level of abstraction in this operation, we can eliminate specialized code, enable optimization, and greatly increase the extensibility of existing code.

cs.MS↗

arXiv2012

A low memory, highly concurrent multigrid algorithm

We examine what is an efficient and scalable nonlinear solver, with low work and memory complexity, for many classes of discretized partial differential equations (PDEs) - matrix-free Full multigrid (FMG) with a Full Approximation Storage (FAS) - in the context of current trends in computer architectures. Brandt proposed an extremely low memory FMG-FAS algorithm over 25 years ago that has several attractive properties for reducing costs on modern - memory centric -- machines and has not been developed to our knowledge. This method, segmental refinement (SR), has very low memory requirements because the finest grids need not be held in memory at any one time but can be "swept" through, computing coarse grid correction and any quantities of interest, allowing for orders of magnitude reduction in memory usage. This algorithm has two useful ideas for effectively exploiting future architectures: improved data locality and reuse via "vertical" processing of the multigrid algorithms and the method of $τ$-corrections, which allows for not storing the entire fine grids at any one time. This report develops this algorithm for a model problem and a parallel generalization of the original sweeping technique. We show that FMG-FAS-SR can work as originally predicted, solving systems accurately enough to maintain the convergence rate of the discretization with one FMG iteration, and that the parallel algorithm provides a natural approach to fully exploiting the available parallelism of FMG.

math.NA↗

arXiv2012

Optimizing the Performance of Streaming Numerical Kernels on the IBM Blue Gene/P PowerPC 450 Processor

Several emerging petascale architectures use energy-efficient processors with vectorized computational units and in-order thread processing. On these architectures the sustained performance of streaming numerical kernels, ubiquitous in the solution of partial differential equations, represents a challenge despite the regularity of memory access. Sophisticated optimization techniques are required to fully utilize the Central Processing Unit (CPU). We propose a new method for constructing streaming numerical kernels using a high-level assembly synthesis and optimization framework. We describe an implementation of this method in Python targeting the IBM Blue Gene/P supercomputer's PowerPC 450 core. This paper details the high-level design, construction, simulation, verification, and analysis of these kernels utilizing a subset of the CPU's instruction set. We demonstrate the effectiveness of our approach by implementing several three-dimensional stencil kernels over a variety of cached memory scenarios and analyzing the mechanically scheduled variants, including a 27-point stencil achieving a 1.7x speedup over the best previously published results.

cs.PF↗

arXiv2008

The shallow shelf approximation as a "sliding law" in a thermomechanically coupled ice sheet model

The shallow shelf approximation is a better ``sliding law'' for ice sheet modeling than those sliding laws in which basal velocity is a function of driving stress. The shallow shelf approximation as formulated by \emph{Schoof} [2006a] is well-suited to this use. Our new thermomechanically coupled sliding scheme is based on a plasticity assumption about the strength of the saturated till underlying the ice sheet in which the till yield stress is given by a Mohr-Coulomb formula using a modeled pore water pressure. Using this scheme, our prognostic whole ice sheet model has convincing ice streams. Driving stress is balanced in part by membrane stresses, the model is computable at high spatial resolution in parallel, it is stable with respect to parameter changes, and it produces surface velocities seen in actual ice streams.

physics.geo-ph↗

arXiv2006

On exact solutions and numerics for cold, shallow, and thermocoupled ice sheets

This three section report can be regarded as an extended appendix to (Bueler, Brown, and Lingle 2006). First we give the detailed construction of an exact solution to a standard continuum model of a cold, shallow, and thermocoupled ice sheet. The construction is by calculation of compensatory accumulation and heat source functions which make a chosen pair of functions for thickness and temperature into exact solutions of the coupled system. The solution we construct here is ``TestG'' in (Bueler and others, 2006) and the steady state solution ``Test F'' is a special case. In the second section we give a reference C implementation of these exact solutions. In the last section we give an error analysis of a finite difference scheme for the temperature equation in the thermocoupled model. The error analysis gives three results, first the correct form of the Courant-Friedrichs-Lewy (CFL) condition for stability of the advection scheme, second an equation for error growth which contributes to understanding the famous ``spokes'' of (Payne and others, 2000), and third a convergence theorem under stringent fixed geometry and smoothness assumptions.

physics.geo-ph↗