Search NASA⌕ Search

SEARCH · Search NASA

Results for “sparse direct solvers”

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.

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

At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
NASA NTRS1994

Performance issues for iterative solvers in device simulation

Due to memory limitations, iterative methods have become the method of choice for large scale semiconductor device simulation. However, it is well known that these methods still suffer from reliability problems. The linear systems which appear in numerical simulation of semiconductor devices are notoriously ill-conditioned. In order to produce robust algorithms for practical problems, careful attention must be given to many implementation issues. This paper concentrates on strategies for developing robust preconditioners. In addition, effective data structures and convergence check issues are also discussed. These algorithms are compared with a standard direct sparse matrix solver on a variety of problems.

Fan, Qing↗
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↗
DOE OSTI2023

Combining Sparse Approximate Factorizations with Mixed-precision Iterative Refinement

The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed-precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed-precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so, we first develop a new error analysis for LU- and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that mixed-precision iterative refinement combined with approximate sparse factorization can lead to considerable reductions of both the time and memory consumption.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Sparse Approximate Multifrontal Factorization with Composite Compression Methods

This article presents a fast and approximate multifrontal solver for large sparse linear systems. In a recent work by Liu et al., we showed the efficiency of a multifrontal solver leveraging the butterfly algorithm and its hierarchical matrix extension, HODBF (hierarchical off-diagonal butterfly) compression to compress large frontal matrices. The resulting multifrontal solver can attain quasi-linear computation and memory complexity when applied to sparse linear systems arising from spatial discretization of high-frequency wave equations. To further reduce the overall number of operations and especially the factorization memory usage to scale to larger problem sizes, in this article we develop a composite multifrontal solver that employs the HODBF format for large-sized fronts, a reduced-memory version of the nonhierarchical block low-rank format for medium-sized fronts, and a lossy compression format for small-sized fronts. This allows us to solve sparse linear systems of dimension up to 2.7 × larger than before and leads to a memory consumption that is reduced by 70% while ensuring the same execution time. The code is made publicly available in GitHub.

97 MATHEMATICS AND COMPUTING↗
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↗
NASA NTRS1990

A flux-split solution procedure for unsteady inlet flows

The unstart and restart of an axisymmetric inlet is investigated using a flux-split procedure applied to the Euler and Reduced Navier Stokes (RNS) equations. A time consistent direct sparse matrix solver is applied to compute the transient flow field both internal and external to the inlet. Time varying oblique and normal shocks are captured. The code is quite general and is applicable for subsonic, transonic and supersonic free streams. The current analysis is concerned with supersonic flight conditions.

Pordal, H. S.↗
NASA NTRS1990

Transient behavior of supersonic flow through inlets

A solution technique to compute inlet flow behavior is presented. The phenomena of inlet unstart and restart are investigated using a flux-split procedure applied to the Euler and Reduced Navier Stokes (RNS) equations. A time consistent direct sparse matrix solver in conjunction with a domain decomposition strategy is applied to compute the transient flow behavior both internal and external to the inlet. Time varying shocks and time varying recirculation regions are efficiently analyzed. The code is quite general and is suitable for the computation of flow for a wide variety of geometries and over a wide range of Mach and Reynolds numbers.

Pordal, H. S.↗
NASA NTRS1991

A pressure flux-split technique for computation of inlet flow behavior

A method for calculating the flow field in aircraft engine inlets is presented. The phenomena of inlet unstart and restart are investigated. Solutions of the reduced Navier-Stokes (RNS) equations are obtained with a time consistent direct sparse matrix solver that computes the transient flow field both internal and external to the inlet. Time varying shocks and time varying recirculation regions can be efficiently analyzed. The code is quite general and is suitable for the computation of flow for a wide variety of geometries and over a wide range of Mach and Reynolds numbers.

Pordal, H. S.↗
NASA NTRS1991

Inviscid steady/unsteady flow calculations

The solution of the Euler equations using a flux splitting procedure is considered for low subsonic to high supersonic flows. Steady and unsteady, internal and external flow fields, are computed. For transient flows, a direct sparse matrix solver is applied to compute the flow field at each instant of time. Oscillation free normal and oblique shocks are captured. Unstart and restart of a simplified two-dimensional inlet is investigated.

Pordal, H. S.↗
NASA NTRS1990

A flux-split solution procedure for unsteady flow calculations

The solution of reduced Navier Stokes (RNS) equations is considered using a flux-split procedure. Unsteady flow in a two dimensional engine inlet is computed. The problems of unstart and restart are investigated. A sparse matrix direct solver combined with domain decomposition strategy is used to compute the unsteady flow field at each instant of time. Strong shock-boundary layer interaction, time varying shocks and time varying recirculation regions are efficiently captured.

Pordal, H. S.↗
NASA NTRS1990

A review of reduced Navier-Stokes computations for compressible viscous flows

A reduced form of the Navier-Stokes equations, defined by a single composite of the Euler, boundary layer, and triple deck approximations, is considered for the computation of viscous interacting flows. Global pressure or pseudopotential relaxation methods are integrated with coupled sparse matrix direct solvers or coupled strongly implicit ILU algorithms to efficiently capture sharp shocks and regions of recirculation. Solutions are obtained for a variety of 2D and 3D geometries and for Mach numbers (M) spanning the range from incompressible (M = 0) to supersonic (M = 6) flow.

Rubin, S. G.↗
NASA NTRS1993

A three-dimensional pressure flux-split RNS application to sub/supersonic flow in inlets and ducts

The reduced Navier-Stokes (RNS) formulation is combined with a pressure based flux split procedure for the computational analysis of three dimensional flow. Only lowest order diffusion, that is required to satisfy the no-slip boundary condition is retained in this approximation. The governing equations are appropriately differenced such that the physical boundary conditions are closely coupled and combined to form a closed discrete system. No numerical or characteristic type boundary conditions are required. A global pressure relaxation procedure, i.e. multi-sweep PNS, is considered. A sparse matrix direct solver is applied to the crossplane solution. At the outflow boundary only the pressure or pressure gradient is prescribed. The resulting formulation has been applied to the solution of three-dimensional subsonic/supersonic internal flows in inlets and ducts.

Khosla, P. K.↗
DOE OSTI2026

Scientific Core Library Stack (SCLS) v2026

SCLS (Scientific Core Library Stack) is an opinionated build and packaging system for scientific computing libraries developed at Lawrence Berkeley National Laboratory. It produces a coherent, reproducible stack of numerical libraries — including BLAS/LAPACK, MPI, sparse direct and iterative solvers, graph partitioners, and parallel I/O libraries (e.g., PETSc, SLEPc, HDF5, NetCDF, MUMPS, OpenBLAS) — that work together without manual repair by downstream scientific software. From a single recipe-and-flavor model, SCLS produces native RPM packages for RHEL-family Linux, DEB packages for Debian/Ubuntu, direct Unix-style prefix installs for HPC and locked-down environments, and native macOS builds. Multiple build "flavors" (e.g., GCC+OpenBLAS, GCC+MKL, Intel+MKL, debug) coexist in distinct prefixes on the same host. Compared to general-purpose meta-build frameworks, SCLS is deliberately curated rather than infinitely configurable. It enforces deterministic, audit-friendly behavior: explicit build dependencies, no silent feature autodetection, a clear open-source license policy, and rpath-based runtime linkage so installs integrate cleanly with standard package-manager workflows.

Messe, Christian [Lawrence Berkeley National Labor↗
NASA NTRS2005

Parallel Finite Element Domain Decomposition for Structural/Acoustic Analysis

A domain decomposition (DD) formulation for solving sparse linear systems of equations resulting from finite element analysis is presented. The formulation incorporates mixed direct and iterative equation solving strategics and other novel algorithmic ideas that are optimized to take advantage of sparsity and exploit modern computer architecture, such as memory and parallel computing. The most time consuming part of the formulation is identified and the critical roles of direct sparse and iterative solvers within the framework of the formulation are discussed. Experiments on several computer platforms using several complex test matrices are conducted using software based on the formulation. Small-scale structural examples are used to validate thc steps in the formulation and large-scale (l,000,000+ unknowns) duct acoustic examples are used to evaluate the ORIGIN 2000 processors, and a duster of 6 PCs (running under the Windows environment). Statistics show that the formulation is efficient in both sequential and parallel computing environmental and that the formulation is significantly faster and consumes less memory than that based on one of the best available commercialized parallel sparse solvers.

Nguyen, Duc T.↗
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 OSTI2011

Efficient scalable algorithms for hierarchically semiseparable matrices

Hierarchically semiseparable (HSS) matrix algorithms are emerging techniques in constructing the superfast direct solvers for both dense and sparse linear systems. Here, we develope a set of novel parallel algorithms for the key HSS operations that are used for solving large linear systems. These include the parallel rank-revealing QR factorization, the HSS constructions with hierarchical compression, the ULV HSS factorization, and the HSS solutions. The HSS tree based parallelism is fully exploited at the coarse level. The BLACS and ScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the ne-grained level. We have appplied our new parallel HSS-embedded multifrontal solver to the anisotropic Helmholtz equations for seismic imaging, and were able to solve a linear system with 6.4 billion unknowns using 4096 processors, in about 20 minutes. The classical multifrontal solver simply failed due to high demand of memory. To our knowledge, this is the first successful demonstration of employing the HSS algorithms in solving the truly large-scale real-world problems. Our parallel strategies can be easily adapted to the parallelization of the other rank structured methods.

01 COAL, LIGNITE, AND PEAT↗
DOE OSTI2016

A Distributed-Memory Package for Dense Hierarchically Semi-Separable Matrix Computations Using Randomization

In this paper, we present a distributed-memory library for computations with dense structured matrices. A matrix is considered structured if its off-diagonal blocks can be approximated by a rank-deficient matrix with low numerical rank. Here, we use Hierarchically Semi-Separable (HSS) representations. Such matrices appear in many applications, for example, finite-element methods, boundary element methods, and so on. Exploiting this structure allows for fast solution of linear systems and/or fast computation of matrix-vector products, which are the two main building blocks of matrix computations. The compression algorithm that we use, that computes the HSS form of an input dense matrix, relies on randomized sampling with a novel adaptive sampling mechanism. We discuss the parallelization of this algorithm and also present the parallelization of structured matrix-vector product, structured factorization, and solution routines. The efficiency of the approach is demonstrated on large problems from different academic and industrial applications, on up to 8,000 cores. Finally, this work is part of a more global effort, the STRUctured Matrices PACKage (STRUMPACK) software package for computations with sparse and dense structured matrices. Hence, although useful on their own right, the routines also represent a step in the direction of a distributed-memory sparse solver.

98 NUCLEAR DISARMAMENT, SAFEGUARDS, AND PHYSICAL P↗
DOE OSTI2015

Solving a Large Scale Thermal Radiation Problem Using an Interoperable Executive Library Framework on Petascale Supercomputers

We present a novel methodology to compute the transient thermal condi- tion of a set of objects in an open space environment. The governing energy equa- tion and the convective energy transfer are solved by the sparse iterative solvers. The average radiating energy on a set of surfaces is represented by a linear system of the radiosity equations, which is factorized by an out-of-core parallel Cholesky decomposition solver. The coupling and interplay of the direct radiosity solver us- ing GPUs and the CPU-based sparse solver are handled by a light weight software integrator called Interoperable Executive Library (IEL). IEL manages the distribu- tion of data and memory, coordinates communication among parallel processes, and also directs execution of the set of loosely coupled physics tasks as warranted by the thermal condition of the simulated object and its surrounding environment.

Wong, Kwai↗