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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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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Sparse Cholesky factorization for solving nonlinear PDEs via Gaussian processes

In recent years, there has been widespread adoption of machine learning-based approaches to automate the solving of partial differential equations (PDEs). Among these approaches, Gaussian processes (GPs) and kernel methods have garnered considerable interest due to their flexibility, robust theoretical guarantees, and close ties to traditional methods. They can transform the solving of general nonlinear PDEs into solving quadratic optimization problems with nonlinear, PDE-induced constraints. However, the complexity bottleneck lies in computing with dense kernel matrices obtained from pointwise evaluations of the covariance kernel, and its partial derivatives, a result of the PDE constraint and for which fast algorithms are scarce. The primary goal of this paper is to provide a near-linear complexity algorithm for working with such kernel matrices. We present a sparse Cholesky factorization algorithm for these matrices based on the near-sparsity of the Cholesky factor under a novel ordering of pointwise and derivative measurements. The near-sparsity is rigorously justified by directly connecting the factor to GP regression and exponential decay of basis functions in numerical homogenization. We then employ the Vecchia approximation of GPs, which is optimal in the Kullback-Leibler divergence, to compute the approximate factor. This enables us to compute ϵ-approximate inverse Cholesky factors of the kernel matrices with complexity O(N log d (N/ϵ)) in space and O(N log 2d (N/ϵ)) in time. We integrate sparse Cholesky factorizations into optimization algorithms to obtain fast solvers of the nonlinear PDE. We numerically illustrate our algorithm’s near-linear space/time complexity for a broad class of nonlinear PDEs such as the nonlinear elliptic, Burgers, and Monge-Ampère equations. In summary, we provide a fast, scalable, and accurate method for solving general PDEs with GPs and kernel methods.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Multicomponent Cholesky Decomposition: Application to Nuclear–Electronic Orbital Theory

The Cholesky decomposition technique is commonly used to reduce the memory requirement for storing two-particle repulsion integrals in quantum chemistry calculations that use atomic orbital bases. However, when quantum methods use multicomponent bases, such as nuclear–electronic orbitals, additional challenges are introduced due to asymmetric two-particle integrals. This work proposes several multicomponent Cholesky decomposition methods for calculations using nuclear–electronic orbital density functional theory. To analyze the errors in different Cholesky decomposition components, benchmark calculations using water clusters are carried out. The largest benchmark calculation is a water cluster (H 2 O) 27 where all 54 protons are treated quantum mechanically. Furthermore, this study provides energetic and complexity analyses to demonstrate the accuracy and performance of the proposed multicomponent Cholesky decomposition method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Toward the Minimal Floating Operation Count Cholesky Decomposition of Electron Repulsion Integrals

As quantum chemistry calculations deal with molecular systems of increasing size, the memory requirement to store electron-repulsion integrals (ERIs) greatly outpaces the physical memory available in computing hardware. Here, the Cholesky decomposition of ERIs provides a convenient yet accurate technique to reduce the storage requirement of integrals. Recent developments of a two-step algorithm have drastically reduced the memory operation (MOP) count, leaving the floating operation (FLOP) count as the last frontier of cost reduction in the Cholesky ERI algorithm. In this report, we introduce a dynamic integral tracking, reusing, and compression/elimination protocol embedded in the two-step Cholesky ERI method. Benchmark studies suggest that this technique becomes particularly advantageous when the basis set consists of many computationally expensive high-angular-momentum basis functions. With this dynamic ERI improvement, the Cholesky ERI approach proves to be a highly efficient algorithm with minimal FLOP and MOP count.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
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↗
DOE OSTI2024

Cholesky Decomposition-Based Implementation of Relativistic Two-Component Coupled-Cluster Methods for Medium-Sized Molecules

A Cholesky decomposition (CD)-based implementation of relativistic two-component coupled-cluster (CC) and equation-of-motion CC (EOM-CC) methods using an exact two-component Hamiltonian augmented with atomic-mean-field integrals (the X2CAMF scheme) is reported. Furthermore, the present CD-based implementation of X2CAMF-CC and EOM-CC methods employs atomic-orbital-based algorithms to avoid the construction of two-electron integrals and intermediates involving three and four virtual indices. The CD-based implementation extends the applicability of X2CAMF-CC and EOMCC methods to medium-sized molecules with the correlation of around 1000 spinors. Benchmark calculations for uranium-containing small molecules have been performed to assess the dependence of CC results with respect to the Cholesky threshold. A Cholesky threshold of 10 –4 is shown to maintain chemical accuracy. Example calculations to illustrate the capability of the CD-based relativistic CC methods are reported for the bond dissociation energy of the uranium hexafluoride molecule, UF 6 , with up to quadruple-zeta basis sets and the lowest excitation energy in solvated uranyl ion [UO 2 2+ (H 2 O) 12 ].

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2024

GPU Accelerated Sparse Cholesky Factorization

The solution of sparse symmetric positive definite linear systems is an important computational kernel in large-scale scientific and engineering modeling and simulation. We will solve the linear systems using a direct method, in which a Cholesky factorization of the coefficient matrix is performed using a right-looking approach and the resulting triangular factors are used to compute the solution. Sparse Cholesky factorization is compute intensive. In this work we investigate techniques for reducing the factorization time in sparse Cholesky factorization by offloading some of the dense matrix operations on a GPU. We will describe the techniques we have considered. We achieved up to 4x speedup compared to the CPU-only version.

Karsavuran, M Ozan↗
DOE OSTI2016

Implementation and Tuning of Batched Cholesky Factorization and Solve for NVIDIA GPUs

Many problems in engineering and scientific computing require the solution of a large number of small systems of linear equations. Due to their high processing power, Graphics Processing Units became an attractive target for this class of problems, and routines based on the LU and the QR factorization have been provided by NVIDIA in the cuBLAS library. This work addresses the situation where the systems of equations are symmetric positive definite. The paper describes the implementation and tuning of the kernels for the Cholesky factorization and the forward and backward substitution. Targeted workloads involve the solution of thousands of linear systems of the same size, where the focus is on matrix dimensions from 5 by 5 to 100 by 100. Due to the lack of a cuBLAS Cholesky factorization, execution rates of cuBLAS LU and cuBLAS QR are used for comparison against the proposed Cholesky factorization in this work. Execution rates of forward and backward substitution routines are compared to equivalent cuBLAS routines. Comparisons against optimized multicore implementations are also presented. Superior performance is reached in all cases.

Computer Science↗
NASA NTRS1992

A performance study of sparse Cholesky factorization on INTEL iPSC/860

The problem of Cholesky factorization of a sparse matrix has been very well investigated on sequential machines. A number of efficient codes exist for factorizing large unstructured sparse matrices. However, there is a lack of such efficient codes on parallel machines in general, and distributed machines in particular. Some of the issues that are critical to the implementation of sparse Cholesky factorization on a distributed memory parallel machine are ordering, partitioning and mapping, load balancing, and ordering of various tasks within a processor. Here, we focus on the effect of various partitioning schemes on the performance of sparse Cholesky factorization on the Intel iPSC/860. Also, a new partitioning heuristic for structured as well as unstructured sparse matrices is proposed, and its performance is compared with other schemes.

Zubair, M.↗
NASA NTRS2002

An Efficient Scheme for Updating Sparse Cholesky Factors

Raghavan had earlier developed the software package DCSPACK which can be used for solving sparse linear systems where the coefficient matrix is symmetric and positive definite (this project was not funded by NASA but by agencies such as NSF). DSCPACK-S is the serial code and DSCPACK-P is a parallel implementation suitable for multiprocessors or networks-of-workstations with message passing using MCI. The main algorithm used is the Cholesky factorization of a sparse symmetric positive positive definite matrix A = LL(T). The code can also compute the factorization A = LDL(T). The complexity of the software arises from several factors relating to the sparsity of the matrix A. A sparse N x N matrix A has typically less that cN nonzeroes where c is a small constant. If the matrix were dense, it would have O(N2) nonzeroes. The most complicated part of such sparse Cholesky factorization relates to fill-in, i.e., zeroes in the original matrix that become nonzeroes in the factor L. An efficient implementation depends to a large extent on complex data structures and on techniques from graph theory to reduce, identify, and manage fill. DSCPACK is based on an efficient multifrontal implementation with fill-managing algorithms and implementation arising from earlier research by Raghavan and others. Sparse Cholesky factorization is typically a four step process: (1) ordering to compute a fill-reducing numbering, (2) symbolic factorization to determine the nonzero structure of L, (3) numeric factorization to compute L, and, (4) triangular solution to solve L(T)x = y and Ly = b. The first two steps are symbolic and are performed using the graph of the matrix. The numeric factorization step is of dominant cost and there are several schemes for improving performance by exploiting the nested and dense structure of groups of columns in the factor. The latter are aimed at better utilization of the cache-memory hierarchy on modem processors to prevent cache-misses and provide execution rates (operations/second) that are close to the peak rates for dense matrix computations. Currently, EPISCOPACY is being used in an application at NASA directed by J. Newman and M. James. We propose the implementation of efficient schemes for updating the LL(T) or LDL(T) factors computed in DSCPACK-S to meet the computational requirements of their project. A brief description is provided in the next section.

Raghavan, Padma↗
DOE OSTI2024

MatRIS: Addressing the Challenges for Portability and Heterogeneity Using Tasking for Matrix Decomposition (Cholesky)

The ubiquitous in-node heterogeneity of HPC and cloud computing platforms makes software portability and performance optimization extremely challenging. Described here, the MatRIS multilevel math library abstraction framework employs tasking to alleviate these difficulties. MatRIS includes the IRIS task-based runtime on the bottom level and exposes different layers of abstraction to render algorithms architecturally agnostic. MatRIS ensures the decomposition and creation of tasks that represent the necessary encapsulation of the optimized kernels from both vendor and open-source math libraries. Once built, MatRIS can select different combinations of accelerators at runtime, making it portable even on diverse heterogeneous architectures. By leveraging the IRIS runtime’s features for managing heterogeneity, MatRIS deploys algorithms that remove the need to specify orchestration and data transfer. This study describes how the serial task abstraction of a tiled Cholesky factorization is made portable and scalable in the case of multi-device and multi-vendor heterogeneity on a node with NVIDIA and AMD GPUs by using MatRIS. First, we demonstrate that Cholesky in MatRIS provides multi-GPU scalability that offers competitive performance versus cuSolverMG. Then, we present the challenges and opportunities for heterogeneous execution.

Monil, M. A. H.↗
DOE OSTI2018

An evaluation of reordering algorithms to reduce the computational cost of the incomplete Cholesky-conjugate gradient method

This paper is concerned with applying bandwidth and profile reduction reordering algorithms prior to computing an incomplete Cholesky factorization and using this as a preconditioner for the conjugate gradient method. Hundreds of reordering algorithms have been proposed to solve the problems of bandwidth and profile reductions since the mid-1960s. In previous publications, a large range of heuristics for bandwidth and/or profile reductions was reviewed. Based on this experience, 13 heuristics were selected as the most promising methods. These are evaluated in this paper along with a variant of the breadth-first search procedure that is proposed. Numerical results confirm the effectiveness of this modified reordering algorithm for linear systems derived from specific application areas. Moreover, the most promising heuristics for several application areas are identified when reducing the computational cost of the incomplete Cholesky-conjugate gradient method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Calculation of excitation energies from the CC2 linear response theory using Cholesky decomposition

A new implementation of the approximate coupled cluster singles and doubles CC2 linear response model is reported. It employs a Cholesky decomposition of the two-electron integrals that significantly reduces the computational cost and the storage requirements of the method compared to standard implementations. Our algorithm also exploits a partitioning form of the CC2 equations which reduces the dimension of the problem and avoids the storage of doubles amplitudes. We present calculation of excitation energies of benzene using a hierarchy of basis sets and compare the results with conventional CC2 calculations. The reduction of the scaling is evaluated as well as the effect of the Cholesky decomposition parameter on the quality of the results. The new algorithm is used to perform an extrapolation to complete basis set investigation on the spectroscopically interesting benzylallene conformers. A set of calculations on medium-sized molecules is carried out to check the dependence of the accuracy of the results on the decomposition thresholds. Moreover, CC2 singlet excitation energies of the free base porphin are also presented.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Relativistic resolution-of-the-identity with Cholesky integral decomposition

In this study, we present an efficient integral decomposition approach called the restricted-kinetic-balance resolution-of-the-identity (RKB-RI) algorithm, which utilizes a tunable RI method based on the Cholesky integral decomposition for in-core relativistic quantum chemistry calculations. The RKB-RI algorithm incorporates the restricted-kinetic-balance condition and offers a versatile framework for accurate computations. Notably, the Cholesky integral decomposition is employed not only to approximate symmetric large-component electron repulsion integrals but also those involving small-component basis functions. In addition to comprehensive error analysis, we investigate crucial conditions, such as the kinetic balance condition and variational stability, which underlie the applicability of Dirac relativistic electronic structure theory. Here we compare the computational cost of the RKB-RI approach with the full in-core method to assess its efficiency. To evaluate the accuracy and reliability of the RKB-RI method proposed in this work, we employ actinyl oxides as benchmark systems, leveraging their properties for validation purposes. This investigation provides valuable insights into the capabilities and performance of the RKB-RI algorithm and establishes its potential as a powerful tool in the field of relativistic quantum chemistry.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Task Parallel Incomplete Cholesky Factorization using 2D Partitioned-Block Layout

We introduce a task-parallel algorithm for sparse incomplete Cholesky factorization that utilizes a 2D sparse partitioned-block layout of a matrix. Our factorization algorithm follows the idea of algorithms-by-blocks by using the block layout. The algorithm-byblocks approach induces a task graph for the factorization. These tasks are inter-related to each other through their data dependences in the factorization algorithm. To process the tasks on various manycore architectures in a portable manner, we also present a portable tasking API that incorporates different tasking backends and device-specific features using an open-source framework for manycore platforms i.e., Kokkos. A performance evaluation is presented on both Intel Sandybridge and Xeon Phi platforms for matrices from the University of Florida sparse matrix collection to illustrate merits of the proposed task-based factorization. Experimental results demonstrate that our task-parallel implementation delivers about 26.6x speedup (geometric mean) over single-threaded incomplete Choleskyby- blocks and 19.2x speedup over serial Cholesky performance which does not carry tasking overhead using 56 threads on the Intel Xeon Phi processor for sparse matrices arising from various application problems.

2D layout↗
DOE OSTI2022

Randomized Cholesky Preconditioning for Graph Partitioning Applications

A graph is a mathematical representation of a network; we say it consists of a set of vertices, which are connected by edges. Graphs have numerous applications in various fields, as they can model all sorts of connections, processes, or relations. For example, graphs can model intricate transit systems or the human nervous system. However, graphs that are large or complicated become difficult to analyze. This is why there is an increased interest in the area of graph partitioning, reducing the size of the graph into multiple partitions. For example, partitions of a graph representing a social network might help identify clusters of friends or colleagues. Graph partitioning is also a widely used approach to load balancing in parallel computing. The partitioning of a graph is extremely useful to decompose the graph into smaller parts and allow for easier analysis. There are different ways to solve graph partitioning problems. For this work, we focus on a spectral partitioning method which forms a partition based upon the eigenvectors of the graph Laplacian (details presented in Acer, et. al.). This method uses the LOBPCG algorithm to compute these eigenvectors. LOBPCG can be accelerated by an operator called a preconditioner. For this internship, we evaluate a randomized Cholesky (rchol) preconditioner for its effectiveness on graph partitioning problems with LOBPCG. We compare it with two standard preconditioners: Jacobi and Incomplete Cholesky (ichol). This research was conducted from August to December 2021 in conjunction with Sandia National Laboratories.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1991

Parallel variable-band Choleski solvers for computational structural analysis applications on vector multiprocessor supercomputers

A Choleski method used to solve linear systems of equations that arise in large scale structural analyses is described. The method uses a novel variable-band storage scheme and is structured to exploit fast local memory caches while minimizing data access delays between main memory and vector registers. Several parallel implementations of this method are described for the CRAY-2 and CRAY Y-MP computers demonstrating the use of microtasking and autotasking directives. A portable parallel language, FORCE, is also used for two different parallel implementations, demonstrating the use of CRAY macrotasking. Results are presented comparing the matrix factorization times for three representative structural analysis problems from runs made in both dedicated and multi-user modes on both the CRAY-2 and CRAY Y-MP computers. CPU and wall clock timings are given for the various parallel methods and are compared to single processor timings of the same algorithm. Computation rates over 1 GIGAFLOP (1 billion floating point operations per second) on a four processor CRAY-2 and over 2 GIGAFLOPS on an eight processor CRAY Y-MP are demonstrated as measured by wall clock time in a dedicated environment. Reduced wall clock times for the parallel methods relative to the single processor implementation of the same Choleski algorithm are also demonstrated for runs made in multi-user mode.

Poole, E. L.↗
DOE OSTI2013

Conditioning Analysis of Incomplete Cholesky Factorizations with Orthogonal Dropping

The analysis of preconditioners based on incomplete Cholesky factorization in which the neglected (dropped) components are orthogonal to the approximations being kept is presented. General estimate for the condition number of the preconditioned system is given which only depends on the accuracy of individual approximations. The estimate is further improved if, for instance, only the newly computed rows of the factor are modified during each approximation step. In this latter case it is further shown to be sharp. The analysis is illustrated with some existing factorizations in the context of discretized elliptic partial differential equations.

97 MATHEMATICS AND COMPUTING↗