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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.

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All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

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.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

symPACK: A GPU-Capable Fan-Out Sparse Cholesky Solver

Sparse symmetric positive definite systems of equations are ubiquitous in scientific workloads and applications. Parallel sparse Cholesky factorization is the method of choice for solving such linear systems. Therefore, the development of parallel sparse Cholesky codes that can efficiently run on today’s large-scale heterogeneous distributed-memory platforms is of vital importance. Modern supercomputers offer nodes that contain a mix of CPUs and GPUs. To fully utilize the computing power of these nodes, scientific codes must be adapted to offload expensive computations to GPUs. We present symPACK, a GPU-capable parallel sparse Cholesky solver that uses one-sided communication primitives and remote procedure calls provided by the UPC++ library. We also utilize the UPC++ "memory kinds" feature to enable efficient communication of GPU-resident data. We show that on a number of large problems, symPACK outperforms comparable state-of-the-art GPU-capable Cholesky factorization codes by up to 14x on the NERSC Perlmutter supercomputer.

Bellavita, Julian↗
NASA NTRS1990

Using a multifrontal sparse solver in a high performance, finite element code

We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP.

King, Scott D.↗
NASA NTRS2000

User's Manual for PCSMS (Parallel Complex Sparse Matrix Solver)

PCSMS (Parallel Complex Sparse Matrix Solver) is a computer code written to make use of the existing real sparse direct solvers to solve complex, sparse matrix linear equations. PCSMS converts complex matrices into real matrices and use real, sparse direct matrix solvers to factor and solve the real matrices. The solution vector is reconverted to complex numbers. Though, this utility is written for Silicon Graphics (SGI) real sparse matrix solution routines, it is general in nature and can be easily modified to work with any real sparse matrix solver. The User's Manual is written to make the user acquainted with the installation and operation of the code. Driver routines are given to aid the users to integrate PCSMS routines in their own codes.

Reddy, C. J.↗
DOE OSTI2024

A graphics processing unit accelerated sparse direct solver and preconditioner with block low rank compression

We present the GPU implementation efforts and challenges of the sparse solver package STRUMPACK. The code is made publicly available on github with a permissive BSD license. STRUMPACK implements an approximate multifrontal solver, a sparse LU factorization which makes use of compression methods to accelerate time to solution and reduce memory usage. Multiple compression schemes based on rank-structured and hierarchical matrix approximations are supported, including hierarchically semi-separable, hierarchically off-diagonal butterfly, and block low rank. Here, in this paper, we present the GPU implementation of the block low rank (BLR) compression method within a multifrontal solver. Our GPU implementation relies on highly optimized vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs, rocBLAS and rocSOLVER for AMD GPUs and the Intel oneAPI Math Kernel Library (oneMKL) for Intel GPUs. Additionally, we rely on external open source libraries such as SLATE (Software for Linear Algebra Targeting Exascale), MAGMA (Matrix Algebra on GPU and Multi-core Architectures), and KBLAS (KAUST BLAS). SLATE is used as a GPU-capable ScaLAPACK replacement. From MAGMA we use variable sized batched dense linear algebra operations such as GEMM, TRSM and LU with partial pivoting. KBLAS provides efficient (batched) low rank matrix compression for NVIDIA GPUs using an adaptive randomized sampling scheme. The resulting sparse solver and preconditioner runs on NVIDIA, AMD and Intel GPUs. Interfaces are available from PETSc, Trilinos and MFEM, or the solver can be used directly in user code. We report results for a range of benchmark applications, using the Perlmutter system from NERSC, Frontier from ORNL, and Aurora from ALCF. For a high frequency wave equation on a regular mesh, using 32 Perlmutter compute nodes, the factorization phase of the exact GPU solver is about 6.5× faster compared to the CPU-only solver. The BLR-enabled GPU solver is about 13.8× faster than the CPU exact solver. For a collection of SuiteSparse matrices, the STRUMPACK exact factorization on a single GPU is on average 1.9× faster than NVIDIA’s cuDSS solver.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1987

Solution of the two-dimensional Navier-Stokes equations using sparse matrix solvers

The use of direct sparse matrix solvers in the solution of the Navier-Stokes equations is investigated. The Yale Sparse Matrix Package and its implementation in the solution algorithm is described. The streamfunction-vorticity form of the Navier-Stokes equations are discretized and linearized and the resulting system of equations are solved using this package. Several viscous flow problems are investigated, including flow in a cavity and flow around a NACA0012 airfoil. Massively separated flow around a sine wave airfoil is investigated and high Reynolds number solutions are obtained. A solution of the unsteady flow around a Joukowski airfoil at high angle of attack is presented.

Bender, Erich E.↗
DOE OSTI2022

High performance sparse multifrontal solvers on modern GPUs

Here, we have ported the numerical factorization and triangular solve phases of the sparse direct solver STRUMPACK to GPU. STRUMPACK implements sparse LU factorization using the multifrontal algorithm, which performs most of its operations in dense linear algebra operations on so-called frontal matrices of various sizes. Our GPU implementation off-loads these dense linear algebra operations, as well as the sparse scatter–gather operations between frontal matrices. For the larger frontal matrices, our GPU implementation relies on vendor libraries such as cuBLAS and cuSOLVER for NVIDIA GPUs and rocBLAS and rocSOLVER for AMD GPUs. For the smaller frontal matrices we developed custom CUDA and HIP kernels to reduce kernel launch overhead. Overall, high performance is achieved by identifying submatrix factorizations corresponding to sub-trees of the multifrontal assembly tree which fit entirely in GPU memory. The multi-GPU setting uses SLATE (Software for Linear Algebra Targeting Exascale) as a modern GPU-aware replacement for ScaLAPACK. On 4 nodes of SUMMIT the code runs ~10X faster when using all 24 V100 GPUs compared to when it only uses the 168 POWER9 cores. On 8 SUMMIT nodes, using 48 V100 GPUs, the sparse solver reaches over 50TFlop/s. Compared to SuperLU, on a single V100, for a set of 17 matrices our implementation is faster for all but one matrix, and is on average 5X (median 4X) faster

97 MATHEMATICS AND COMPUTING↗
NASA NTRS2002

A Method for Optimizing Non-Axisymmetric Liners for Multimodal Sound Sources

Central processor unit times and memory requirements for a commonly used solver are compared to that of a state-of-the-art, parallel, sparse solver. The sparse solver is then used in conjunction with three constrained optimization methodologies to assess the relative merits of non-axisymmetric versus axisymmetric liner concepts for improving liner acoustic suppression. This assessment is performed with a multimodal noise source (with equal mode amplitudes and phases) in a finite-length rectangular duct without flow. The sparse solver is found to reduce memory requirements by a factor of five and central processing time by a factor of eleven when compared with the commonly used solver. Results show that the optimum impedance of the uniform liner is dominated by the least attenuated mode, whose attenuation is maximized by the Cremer optimum impedance. An optimized, four-segmented liner with impedance segments in a checkerboard arrangement is found to be inferior to an optimized spanwise segmented liner. This optimized spanwise segmented liner is shown to attenuate substantially more sound than the optimized uniform liner and tends to be more effective at the higher frequencies. The most important result of this study is the discovery that when optimized, a spanwise segmented liner with two segments gives attenuations equal to or substantially greater than an optimized axially segmented liner with the same number of segments.

Watson, W. R.↗
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 OSTI2024

Sparse Linear Solvers for Large-scale Electromagnetic Transient Simulations

Linear solvers form the basis for electromagnetic transient (EMT) simulations. There is a need to speed up EMT simulations as larger regions are analyzed using EMT simulations. For the same, the performance of linear solvers plays an important role. Exploiting the sparsity of the matrices generated in EMT simulations could assist with speed-up. Scalability is also crucial as power grids expand, demanding solutions capable of accommodating the increasing system size. Recent studies from the North American Electric Reliability Corporation (NERC) increasingly emphasize that EMT simulation models of the power grid will grow larger with the inclusion of power electronics components. Parallelisms in sparsity patterns exploit modern central processing units (CPUs), multi-core CPUs, and graphics processing units (GPUs) architectures in sparse solver designs. Therefore, this paper explores publicly available existing linear solvers and investigates their efficiency in large-scale power grid simulations. A large-scale power grid is developed by increasing the size of the IEEE 39 bus test system to up to 39000 bus systems.

Hsu, Kuan-Chieh↗
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 OSTI2023

Newly Released Capabilities in the Distributed-Memory SuperLU Sparse Direct Solver

We present the new features available in the recent release of SuperLU_DIST, Version 8.1.1. SuperLU_DIST is a distributed-memory parallel sparse direct solver. The new features include (1) a 3D communication-avoiding algorithm framework that trades off inter-process communication for selective memory duplication, (2) multi-GPU support for both NVIDIA GPUs and AMD GPUs, and (3) mixed-precision routines that perform single-precision LU factorization and double-precision iterative refinement. Apart from the algorithm improvements, we also modernized the software build system to use CMake and Spack package installation tools to simplify the installation procedure. Throughout the article, we describe in detail the pertinent performance-sensitive parameters associated with each new algorithmic feature, show how they are exposed to the users, and give general guidance of how to set these parameters. We illustrate that the solver’s performance both in time and memory can be greatly improved after systematic tuning of the parameters, depending on the input sparse matrix and underlying hardware.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Optimal Power Flow Derived Sparse Linear Solver Benchmarks

Due to the changing nature of the power grid, it is increasingly important to be able to solve a high-fidelity optimal power-flow models on large power networks. This high-fidelity problem, called AC Optimal Power Flow (ACOPF), is a nonlinear, nonconvex optimization problem. One of the few reliable ways of solving such a problem is interior point methods. These methods result in sparse linear systems where the coefficient matrix is symmetric, indefinite and nearly always ill-conditioned. As such, they are particularly challenging for sparse linear solvers and represent a considerable computational bottleneck in solving the ACOPF problem. In this paper, we introduce a repository of linear systems captured from ACOPF problems when solved by the open-source optimizer IPOPT. These matrices are meant to be used as a test suite for sparse linear solver development.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1999

Object-Oriented Design for Sparse Direct Solvers

We discuss the object-oriented design of a software package for solving sparse, symmetric systems of equations (positive definite and indefinite) by direct methods. At the highest layers, we decouple data structure classes from algorithmic classes for flexibility. We describe the important structural and algorithmic classes in our design, and discuss the trade-offs we made for high performance. The kernels at the lower layers were optimized by hand. Our results show no performance loss from our object-oriented design, while providing flexibility, case of use, and extensibility over solvers using procedural design.

Dobrian, Florin↗
DOE OSTI2023

Unified Communication Optimization Strategies for Sparse Triangular Solver on CPU and GPU Clusters

This paper presents a unified communication optimization framework for sparse triangular solve (SpTRSV) algorithms on CPU and GPU clusters. The framework builds upon a 3D communication-avoiding (CA) layout of Px × Py × Pz processes that divides a sparse matrix into Pz submatrices, each handled by a Px × Py 2D grid with block-cyclic distribution. We propose three communication optimization strategies: First, a new 3D SpTRSV algorithm is developed, which trades the inter-grid communication and synchronization with replicated computation. This design requires only one inter-grid synchronization, and the inter-grid communication is efficiently implemented with sparse allreduce operations. Second, broadcast and reduction communication trees are used to reduce message latency of the intra-grid 2D communication on CPU clusters. Finally, we leverage GPU-initiated one-sided communication to implement the communication trees on GPU clusters. With these nested inter- and intra-grid communication optimization strategies, the proposed 3D SpTRSV algorithm can attain up to 3.45x speedups compared to the baseline 3D SpTRSV algorithm using up to 2048 Cori Haswell CPU cores. In addition, the proposed GPU 3D SpTRSV algorithm can achieve up to 6.5x speedups compared to the proposed CPU 3D SpTRSV algorithm with Pz up to 64. Finally it is remarkable that the proposed GPU 3D SpTRSV can scale to 256 GPUs using the Perlmutter system while the existing 2D SpTRSV algorithm can only scale up to 4 GPUs.

Sao, Piyush↗
DOE OSTI2020

Where did the tumor start? An inverse solver with sparse localization for tumor growth models

In this work, we present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction–diffusion partial differential equation (PDE) for the tumor concentration. We use a PDE-constrained optimization formulation for the inverse problem. The unknown parameters are the reaction coefficient (proliferation), the diffusion coefficient (infiltration), and the initial condition field for the tumor PDE. Segmentation of magnetic resonance imaging (MRI) scans drive the inverse problem where segmented tumor regions serve as partial observations of the tumor concentration. Like most cases in clinical practice, we use data from a single time snapshot. Moreover, the precise time relative to the initiation of the tumor is unknown, which poses an additional difficulty for inversion. We perform a frozen-coefficient spectral analysis and show that the inverse problem is severely ill-posed. We introduce a biophysically motivated regularization on the structure and magnitude of the tumor initial condition. In particular, we assume that the tumor starts at a few locations (enforced with a sparsity constraint on the initial condition of the tumor) and that the initial condition magnitude in the maximum norm is equal to one. We solve the resulting optimization problem using an inexact quasi-Newton method combined with a compressive sampling algorithm for the sparsity constraint. Our implementation uses PETSc and AccFFT libraries. We conduct numerical experiments on synthetic and clinical images to highlight the improved performance of our solver over a previously existing solver that uses standard two-norm regularization for the calibration parameters. The existing solver is unable to localize the initial condition. Our new solver can localize the initial condition and recover infiltration and proliferation. In clinical datasets (for which the ground truth is unknown), our solver results in qualitatively different solutions compared to the two-norm regularized solver.

97 MATHEMATICS AND COMPUTING↗