SEARCH · Search NASA

Results for “sparse linear solver”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

SourceResearch connectionFrom yearThrough year

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 55 records · Page 3

All sources NASA NTRS DOE OSTI

DOE OSTI2020

Scalable line and plane relaxation in a parallel structured multigrid solver

The efficient solution of sparse, linear systems that arise through the discretization of partial differential equations remains a key challenge for a range of high performance scientific simulations. One approach for reducing data movement and improving performance is by exposing and exploiting structure in a problem through the use of robust structured multilevel solvers. By choosing coarsening that preserves the structure of the problem, these methods maintain efficient structured computation and communication throughout the multigrid hierarchy. However, when coarsening is not permitted to be dependent on the operator, anisotropy must be addressed by the smoother — producing error compatible for coarse-grid correction with structured coarsening. Here, the components required in a scalable parallel structured solver are described with a focus on memory and communication efficiency of robust smoothers. While the implementation of communication and memory reduction techniques in smoothers integrated in a complete 3D solver present a significant engineering challenge, a novel approach is proposed that addresses these challenges systematically through a change to the solver’s execution model. Enabled by user-level threading paired with a set of data and communication abstractions, this approach permits seamless aggregation of communication in plane smoothers — directly reusing code for a 2D distributed multilevel cycle. Results show an effective reduction in communication costs for coarse-grid problems, and result in a speedup of 8.7x in smoothing routines shown in Fig. 12 using this approach. This produces a significant improvement to strong scalability while maintaining favorable weak scaling behavior. Finally, a parallel scaling study using a series of refined meshes is included that demonstrates the effectiveness of this approach in an application of interest.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Xyce(™) Parallel Electronic Simulator v.7.5

The Xyce Parallel Electronic Simulator simulates electronic circuit behavior in DC, AC, HB, MPDE and transient mode using standard analog (DAE) and/or device (PDE) device models including several age and radiation aware devices. It supports a variety of computing platforms (both serial and parallel) computers. Lastly, it uses a variety of modern solution algorithms dynamic parallel load-balancing and iterative solvers.! ! Xyce is primarily used to simulate the voltage and current behavior of a circuit network (a network of electronic devices connected via a conductive network). As a tool, it is mainly used for the design and analysis of electronic circuits.! ! Kirchoff's conservation laws are enforced over a network using modified nodal analysis. This results in a set of differential algebraic equations (DAEs). The resulting nonlinear problem is solved iteratively using a fully coupled Newton method, which in turn results in a linear system that is solved by either a standard sparse-direct solver or iteratively using Trilinos linear solver packages, also developed at Sandia National Laboratories.

Source record↗

DOE OSTI2018

Exploring Deep Learning and Sparse Matrix Format Selection

We proposed to explore the use of Deep Neural Networks (DNN) for addressing the longstanding barriers. The recent rapid progress of DNN technology has created a large impact in many fields, which has significantly improved the prediction accuracy over traditional machine learning techniques in image classifications, speech recognitions, machine translations, and so on. To some degree, these tasks resemble the decision makings in many HPC tasks, including the aforementioned format selection for SpMV and linear solver selection. For instance, sparse matrix format selection is akin to image classification—such as, to tell whether an image contains a dog or a cat; in both problems, the right decisions are primarily determined by the spatial patterns of the elements in an input. For image classification, the patterns are of pixels, and for sparse matrix format selection, they are of non-zero elements. DNN could be naturally applied if we regard a sparse matrix as an image and the format selection or solver selection as classification problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

PyAMG: Algebraic Multigrid Solvers in Python

PyAMG is a Python package of algebraic multigrid (AMG) solvers and supporting tools for approximating the solution to large, sparse linear systems of algebraic equations, Ax = b, where A is an n × n sparse matrix. Sparse linear systems arise in a range of problems in science, from fluid flows to solid mechanics to data analysis. While the direct solvers available in SciPy’s sparse linear algebra package (scipy.sparse.linalg) are highly efficient, in many cases iterative methods are preferred due to overall complexity. However, the iterative methods in SciPy, such as CG and GMRES, often require an efficient preconditioner in order to achieve a lower complexity. Preconditioning is a powerful tool whereby the conditioning of the linear system and convergence rate of the iterative method are both dramatically improved. PyAMG constructs multigrid solvers for use as a preconditioner in this setting. A summary of multigrid and algebraic multigrid solvers can be found in Olson (2015a), in Olson (2015b), and in Falgout (2006); a detailed description can be found in Briggs et al. (2000) and Trottenberg et al. (2001).

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Sparse Approximate Multifrontal Factorization with Butterfly Compression for High-Frequency Wave Equations

In this work, we present a fast and approximate multifrontal solver for large-scale sparse linear systems arising from finite-difference, finite-volume or finite-element discretization of high-frequency wave equations. The proposed solver leverages the butterfly algorithm and its hierarchical matrix extension for compressing and factorizing large frontal matrices via graph-distance guided entry evaluation or randomized matrix-vector multiplication-based schemes. Complexity analysis and numerical experiments demonstrate $\mathcal{O}(N\log^2 N)$ computation and $\mathcal{O}(N)$ memory complexity when applied to an $N\times N$ sparse system arising from 3D high-frequency Helmholtz and Maxwell problems.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2016

An Optimized Multicolor Point-Implicit Solver for Unstructured Grid Applications on Graphics Processing Units

In the field of computational fluid dynamics, the Navier-Stokes equations are often solved using an unstructuredgrid approach to accommodate geometric complexity. Implicit solution methodologies for such spatial discretizations generally require frequent solution of large tightly-coupled systems of block-sparse linear equations. The multicolor point-implicit solver used in the current work typically requires a significant fraction of the overall application run time. In this work, an efficient implementation of the solver for graphics processing units is proposed. Several factors present unique challenges to achieving an efficient implementation in this environment. These include the variable amount of parallelism available in different kernel calls, indirect memory access patterns, low arithmetic intensity, and the requirement to support variable block sizes. In this work, the solver is reformulated to use standard sparse and dense Basic Linear Algebra Subprograms (BLAS) functions. However, numerical experiments show that the performance of the BLAS functions available in existing CUDA libraries is suboptimal for matrices representative of those encountered in actual simulations. Instead, optimized versions of these functions are developed. Depending on block size, the new implementations show performance gains of up to 7x over the existing CUDA library functions.

Zubair, Mohammad↗

DOE OSTI2019

symPACK v2.0

SymPACK is a parallel direct solver for sparse symmetric systems of linear equations. SymPACK is implemented for large- scale distributed-memory platforms. It employs several innovative ideas, including the use of an asynchronous task paradigm, one-sided communication, and dynamic scheduling.

Ng, EsmondG↗

DOE OSTI2019

UPC++: A High-Performance Communication Framework for Asynchronous Computation

© 2019 IEEE UPC++ is a C++ library that supports high-performance computation via an asynchronous communication framework. This paper describes a new incarnation that differs substantially from its predecessor, and we discuss the reasons for our design decisions. We present new design features, including future-based asynchrony management, distributed objects, and generalized Remote Procedure Call (RPC). We show microbenchmark performance results demonstrating that one-sided Remote Memory Access (RMA) in UPC++ is competitive with MPI-3 RMA; on a Cray XC40 UPC++ delivers up to a 25% improvement in the latency of blocking RMA put, and up to a 33% bandwidth improvement in an RMA throughput test. We showcase the benefits of UPC++ with irregular applications through a pair of application motifs, a distributed hash table and a sparse solver component. Our distributed hash table in UPC++ delivers near-linear weak scaling up to 34816 cores of a Cray XC40. Our UPC++ implementation of the sparse solver component shows robust strong scaling up to 2048 cores, where it outperforms variants communicating using MPI by up to 3.1x. UPC++ encourages the use of aggressive asynchrony in low-overhead RMA and RPC, improving programmer productivity and delivering high performance in irregular applications.

Bachan, John↗

DOE OSTI2019

A communication-avoiding 3D sparse triangular solver

We present a novel distributed memory algorithm to improve the strong scalability of the solution of a sparse triangular system. This operation appears in the solve phase of direct methods for solving general sparse linear systems, Ax = b. Our 3D sparse triangular solver employs several techniques, including a 3D MPI process grid, elimination tree parallelism, and data replication, all of which reduce the per-process communication when combined. We present analytical models to understand the communication cost of our algorithm and show that our 3D sparse triangular solver can reduce the per-process communication volume asymptotically by a factor of O(n1/4) and O(n1/6) for problems arising from the finite element discretizations of 2D "planar" and 3D "non-planar" PDEs, respectively. We implement our algorithm for use in SuperLU_DIST3D, using a hybrid MPI+OpenMP programming model. Our 3D triangular solve algorithm, when run on 12k cores of Cray XC30, outperforms the current state-of-the-art 2D algorithm by 7.2x for planar and 2.7x for the non-planar sparse matrices, respectively.

Sao, Piyush↗

NASA NTRS1990

An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, part 2

It is shown how the look-ahead Lanczos process (combined with a quasi-minimal residual QMR) approach) can be used to develop a robust black box solver for large sparse non-Hermitian linear systems. Details of an implementation of the resulting QMR algorithm are presented. It is demonstrated that the QMR method is closely related to the biconjugate gradient (BCG) algorithm; however, unlike BCG, the QMR algorithm has smooth convergence curves and good numerical properties. We report numerical experiments with our implementation of the look-ahead Lanczos algorithm, both for eigenvalue problem and linear systems. Also, program listings of FORTRAN implementations of the look-ahead algorithm and the QMR method are included.

Freund, Roland W.↗

NASA NTRS1993

Three-dimensional unstructured grid Euler computations using a fully-implicit, upwind method

A method has been developed to solve the Euler equations on a three-dimensional unstructured grid composed of tetrahedra. The method uses an upwind flow solver with a linearized, backward-Euler time integration scheme. Each time step results in a sparse linear system of equations which is solved by an iterative, sparse matrix solver. Local-time stepping, switched evolution relaxation (SER), preconditioning and reuse of the Jacobian are employed to accelerate the convergence rate. Implicit boundary conditions were found to be extremely important for fast convergence. Numerical experiments have shown that convergence rates comparable to that of a multigrid, central-difference scheme are achievable on the same mesh. Results are presented for several grids about an ONERA M6 wing.

Whitaker, David L.↗

NASA NTRS1995

A General Purpose Sparse Matrix Parallel Solvers Package

A general purpose solver package for constructing and solving a range of sparse linear systems arising from discretization of PDEs on unstructured meshes is developed. Once the sparse symmetric complex matrix is constructed, it can be solved by either a preconditioned bi-conjugate gradient solver, a two-stage Cholesky LDLT factorization solver, or a hybrid solver combining the above two methods.

solver sparse matrix solvers package PDE↗

NASA NTRS1995

A General Purpose Sparse Matrix Parallel Solvers Package

solver sparse matrix solvers package PDE↗

DOE OSTI2019

Fast and Robust Linear Solvers based on Hierarchical Matrices (LDRD Final Report)

This report is the final report for the LDRD project "Fast and Robust Linear Solvers using Hierarchical Matrices". The project was a success. We developed two novel algorithms for solving sparse linear systems. We demonstrated their effectiveness on ill-conditioned linear systems from ice sheet simulations. We showed that in many cases, we can obtain near-linear scaling. We believe this approach has strong potential for difficult linear systems and should be considered for other Sandia and DOE applications. We also report on some related research activities in dense solvers and randomized linear algebra.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Fast and Scalable Sparse Triangular Solver for Multi-GPU Based HPC Architectures

Designing efficient and scalable sparse linear algebra kernels on modern multi-GPU based HPC systems is a daunting task due to significant irregular memory references and workload imbalance across the GPUs. This is particularly the case for \textit{Sparse Triangular Solver (SpTRSV)} which introduces additional two-dimensional computation dependencies among subsequent computation steps. Dependency information is exchanged and shared among GPUs, thus warrant for efficient memory allocation, data partitioning, and workload distribution as well as fine-grained communication and synchronization support. In this work, we demonstrate that directly adopting unified memory can adversely affect the performance of SpTRSV on multi-GPU architectures, despite linking via fast interconnect like NVLinks and NVSwitches. Alternatively, we employ the latest NVSHMEM technology based on Partitioned Global Address Space programming model to enable efficient fine-grained communication and drastic synchronization overhead reduction. Furthermore, to handle workload imbalance, we propose a malleable task-pool execution model which can further enhance the utilization of GPUs. By applying these techniques, our experiments on the NVIDIA multi-GPU supernode V100-DGX-1 and DGX-2 systems demonstrate that our design can achieve on average 3.53x (up to 9.86x) speedup on a DGX-1 system and 3.66x (up to 9.64x) speedup on a DGX-2 system with 4-GPUs over the Unified-Memory design. The comprehensive sensitivity and scalability studies also show that the proposed zero-copy SpTRSV is able to fully utilize the computing and communication resources of the multi-GPU system.

Xie, Chenhao↗

DOE OSTI2024

Batched sparse direct solver design and evaluation in SuperLU_DIST

Over the course of interactions with various application teams, the need for batched sparse linear algebra functions has emerged in order to make more efficient use of the GPUs for many small and sparse linear algebra problems. In this paper, we present our recent work on a batched sparse direct solver for GPUs. The sparse LU factorization is computed by the levels of the elimination tree, leveraging the batched dense operations at each level and a new batched Scatter GPU kernel. The sparse triangular solve is computed by the level sets of the directed acyclic graph (DAG) of the triangular matrix. Batched operations overcome the large overhead associated with launching many small kernels. For medium sized matrix batches with not-so-small bandwidth, using an NVIDIA A100 GPU, our new batched sparse direct solver is orders of magnitude faster than a batched banded solver and uses less than one-tenth of the memory.

Boukaram, Wajih↗

DOE OSTI2021

Simulations of plasmas and fluids using anti-symmetric models

ALMA (Anti-symmetric, Large-Moment, Accelerated) is a fast, flexible, and scalable toolkit designed to solve hyperbolic conservation law systems in hybrid supercomputers. Here this manuscript describes the theoretical background and implementation of ALMA, which uses the anti-symmetric formulation of fluids to obtain simple, robust, and easily paralellizable code. Practical GPU acceleration is realized on entire applications with an overall gain factor of 2 to 4. ALMA also provides a parallel, GPU accelerated sparse solver based on geometric multigrid, capable of diagonalizing linear systems with 239 unknowns. Here we demonstrate ALMA's scaling and performance in petascale supercomputers and use standard fluid models to verify the overall approach with canonical benchmark problems.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2019

Modeling the asynchronous Jacobi method without communication delays

Asynchronous iterative methods for solving linear systems are gaining renewed interest due to the high cost of synchronization points in massively parallel codes. Historically, theory on asynchronous iterative methods has focused on asymptotic behavior, while the transient behavior remains poorly understood. In this paper, we study a model of the asynchronous Jacobi method without communication delays, which we call simplified asynchronous Jacobi. Simplified asynchronous Jacobi can be used to model asynchronous Jacobi implemented in shared memory or distributed memory with fast communication networks. Our analysis uses the idea of a propagation matrix, which is similar in concept to an iteration matrix. We show that simplified asynchronous Jacobi can continue to reduce the residual when some processes are slower than other processes. We also show that simplified asynchronous Jacobi can converge when synchronous Jacobi does not. We verify our analysis of simplified asynchronous Jacobi using results from asynchronous Jacobi implemented in shared and distributed memory.

97 MATHEMATICS AND COMPUTING↗