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Batched Sparse Linear Algebra (Final Report for Subcontract B648960)
This report finalizes design specifications for developing batched kernels for small tensor operations for unassembled matrix-free iterative solvers, batched solvers for partially assembled operators, and batched solvers with support for various sparse formats. The outcome of the project milestones is a set of interfaces to Batched Sparse LA solvers running on hardware accelerators for use in ECP Libraries and Applications. It is part of the development of sparse batched kernels, solvers/preconditioners as well as creating interoperability in xSDK libraries with sparse and dense batched functions to benefit ECP applications. The participants included representatives from ECP libraries (not limited to the xSDK project), applications, and vendors (AMD, Intel, and NVIDIA). Batched sparse linear algebra solvers form the new frontier for algorithmic development and performance engineering. Many applications (ECP and non-ECP alike) require simultaneous solutions of small linear systems of equations that are structurally sparse. To move towards high hardware utilization, it is important to provide these applications with appropriate interfaces to efficient batched sparse solvers running on modern hardware accelerators. We present interface designs in use by HPC software libraries supporting batched sparse linear algebra and the development of sparse batched kernel codes for solvers and preconditioners. We also address the potential interoperability opportunities to keep the software portable between the major hardware accelerators from AMD, Intel, and NVIDIA. The presented interface specifications includes batched band, sparse iterative, and sparse direct solvers. This report summarizes progress in Kokkos Kernels and the xSDK libraries MAGMA, Ginkgo, hypre, SUNDIALS, and SuperLU_dist.
Batched Sparse Linear Algebra Phase 2
The purpose of the work was to help LLNL develop a broadly applicable set of capabilities that focus on batched sparse linear functionality and its software implementation. The enablement occurred in a broader scope of the Exascale Computing Program (ECP) with an eye on the xSDK collection of applications and libraries. These served as the main dissemination targets and in the end benefitted from this project’s outcomes.
Batched Sparse Linear Algebra Phase 2 (Final Report)
The purpose of the work was to help LLNL develop a broadly applicable set of capabilities that focus on batched sparse linear functionality and its software implementation. The enablement occurred in a broader scope of the Exascale Computing Program (ECP) with an eye on the xSDK collection of applications and libraries. These served as the main dissemination targets and in the end benefitted from this project’s outcomes.
Milestone 49 Report: Batched Sparse LA Phase 5 Implementation
Batched sparse linear algebra operations in general, and solvers in particular, have become the major algorithmic development activity and foremost performance engineering effort in the numerical software libraries work on modern hardware with accelerators such as GPUs. Many applications, ECP and non-ECP alike, require simultaneous solutions of many small linear systems of equations that are structurally sparse in one form or another. In order to move towards high hardware utilization levels, it is important to provide these applications with appropriate interface designs to be both functionally efficient and performance portable and give full access to the appropriate batched sparse solvers running on modern hardware accelerators prevalent across DOE supercomputing sites since the inception of ECP. To this end, we present here a summary of recent advances on the interface designs in use by HPC software libraries supporting batched sparse linear algebra and the development of sparse batched kernel codes for solvers and preconditioners. We also address the potential interoperability opportunities to keep the corresponding software portable between the major hardware accelerators from AMD, Intel, and NVIDIA, while maintaining the appropriate disclosure levels conforming to the active NDA agreements. The presented interface specifications include a mix of batched band, sparse iterative, and sparse direct solvers with their accompanying functionality that is already required by the application codes or we anticipated to be needed in the near future. This report summarizes progress in Kokkos Kernels and the xSDK libraries MAGMA, Ginkgo, hypre, PETSc, and SuperLU.
Algorithmic Patterns for $$\mathcal {H}$$-Matrices on Many-Core Processors
In this work, we consider the reformulation of hierarchical ( \(\mathcal {H}\) ) matrix algorithms for many-core processors with a model implementation on graphics processing units (GPUs). \(\mathcal {H}\) matrices approximate specific dense matrices, e.g., from discretized integral equations or kernel ridge regression, leading to log-linear time complexity in dense matrix–vector products. The parallelization of \(\mathcal {H}\) matrix operations on many-core processors is difficult due to the complex nature of the underlying algorithms. While previous algorithmic advances for many-core hardware focused on accelerating existing \(\mathcal {H}\) matrix CPU implementations by many-core processors, we here aim at totally relying on that processor type. As main contribution, we introduce the necessary parallel algorithmic patterns allowing to map the full \(\mathcal {H}\) matrix construction and the fast matrix–vector product to many-core hardware. In this work, crucial ingredients are space filling curves, parallel tree traversal and batching of linear algebra operations. The resulting model GPU implementation hmglib is the, to the best of the authors knowledge, first entirely GPU-based Open Source \(\mathcal {H}\) matrix library of this kind. We investigate application examples as present in kernel ridge regression, Gaussian Process Regression and kernel-based interpolation. In this context, an in-depth performance analysis and a comparative performance study against a standard multi-core CPU \(\mathcal {H}\) matrix library highlights profound speedups of our many-core parallel approach.
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.
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.
Geopotential Error Analysis from Satellite Gradiometer and Global Positioning System Observables on Parallel Architecture
The recovery of a high resolution geopotential from satellite gradiometer observations motivates the examination of high performance computational techniques. The primary subject matter addresses specifically the use of satellite gradiometer and GPS observations to form and invert the normal matrix associated with a large degree and order geopotential solution. Memory resident and out-of-core parallel linear algebra techniques along with data parallel batch algorithms form the foundation of the least squares application structure. A secondary topic includes the adoption of object oriented programming techniques to enhance modularity and reusability of code. Applications implementing the parallel and object oriented methods successfully calculate the degree variance for a degree and order 110 geopotential solution on 32 processors of the Cray T3E. The memory resident gradiometer application exhibits an overall application performance of 5.4 Gflops, and the out-of-core linear solver exhibits an overall performance of 2.4 Gflops. The combination solution derived from a sun synchronous gradiometer orbit produce average geoid height variances of 17 millimeters.
Improving signal strength in serial crystallography with DIALS geometry refinement
The DIALS diffraction-modeling software package has been applied to serial crystallography data. Diffraction modeling is an exercise in determining the experimental parameters, such as incident beam wavelength, crystal unit cell and orientation, and detector geometry, that are most consistent with the observed positions of Bragg spots. These parameters can be refined by nonlinear least-squares fitting. In previous work, it has been challenging to refine both the positions of the sensors (metrology) on multipanel imaging detectors such as the CSPAD and the orientations of all of the crystals studied. Since the optimal models for metrology and crystal orientation are interdependent, alternate cycles of panel refinement and crystal refinement have been required. To simplify the process, a sparse linear algebra technique for solving the normal equations was implemented, allowing the detector panels to be refined simultaneously against the diffraction from thousands of crystals with excellent computational performance. Separately, it is shown how to refine the metrology of a second CSPAD detector, positioned at a distance of 2.5 m from the crystal, used for recording low-angle reflections. With the ability to jointly refine the detector position against the ensemble of all crystals used for structure determination, it is shown that ensemble refinement greatly reduces the apparent nonisomorphism that is often observed in the unit-cell distributions from still-shot serial crystallography. In addition, it is shown that batching the images by timestamp and re-refining the detector position can realistically model small, time-dependent variations in detector position relative to the sample, and thereby improve the integrated structure-factor intensity signal and heavy-atom anomalous peak heights.
Online Primal-Dual Methods With Measurement Feedback for Time-Varying Convex Optimization
This paper addresses the design and analysis of feedback-based online algorithms to control systems or networked systems based on performance objectives and engineering constraints that may evolve over time. The emerging timevarying convex optimization formalism is leveraged to model optimal operational trajectories of the systems, as well as explicit local and network-level operational constraints. Departing from existing batch and feed-forward optimization approaches, the design of the algorithms capitalizes on an online implementation of primal-dual projected-gradient methods; the gradient steps are, however, suitably modified to accommodate feedback from the system in the form of measurements - hence, the term 'online optimization with feedback.' By virtue of this approach, the resultant algorithms can cope with model mismatches in the algebraic representation of the system states and outputs, they avoid pervasive measurements of exogenous inputs, and they naturally lend themselves to a distributed implementation. Under suitable assumptions, analytical convergence claims are established in terms of dynamic regret. Furthermore, when the synthesis of the feedback-based online algorithms is based on a regularized Lagrangian function, Q-linear convergence to solutions of the time-varying optimization problem is shown.
Coupling flux balance analysis with reactive transport modeling through machine learning for rapid and stable simulation of microbial metabolic switching
Integrating genome-scale metabolic networks with reactive transport models (RTMs) provides a detailed description of the dynamic changes in microbial growth and metabolism. Despite promising demonstrations in the past, computational inefficiency has been pointed out as a critical issue to overcome because it requires repeated application of linear programming (LP) to obtain flux balance analysis (FBA) solutions in every time step and spatial grid. To address this challenge, we propose a new simulation method where we train and validate artificial neural networks (ANNs) using randomly sampled FBA solutions and incorporate the resulting surrogate FBA model (represented as algebraic equations) into RTMs as source/sink terms. We demonstrate the efficiency of our method via a case study of Shewanella oneidensis MR-1. During aerobic growth on lactate, S. oneidensis produces metabolic byproducts (such as pyruvate and acetate), which are subsequently consumed as alternative carbon sources when the preferred nutrients are depleted. To effectively simulate these complex dynamics, we used a cybernetic approach that models metabolic switches as the outcome of dynamic competition among multiple growth options. In both zero-dimensional batch and one-dimensional column configurations, the ANN-based surrogate models achieved substantial reduction of computational time by several orders of magnitude compared to the original LP-based FBA models. Moreover, the ANN models produced robust solutions without any special measures to prevent numerical instability. These developments significantly promote our ability to utilize genome-scale networks in complex, multi-physics, and multi-dimensional ecosystem modeling.