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DOE OSTI2023

A New Semistructured Algebraic Multigrid Method

Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display good parallelization properties. Since current architecture trends are favoring regular compute patterns to achieve high performance, the ability to express structure has become much more important. The hypre software library provides high-performance multigrid preconditioners and solvers through conceptual interfaces, including a semistructured interface that describes matrices primarily in terms of stencils and logically structured grids. This paper presents a new semistructured algebraic multigrid (SSAMG) method built on this interface. The numerical convergence and performance of a CPU implementation of this method are evaluated for a set of semistructured problems. In conclusion, SSAMG achieves significantly better setup times than hypre’s unstructured AMG solvers and comparable convergence. In addition, the new method is capable of solving more complex problems than hypre’s structured solvers.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

A Linear-Complexity Tensor Butterfly Algorithm for Compressing High-Dimensional Oscillatory Integral Operators

This paper presents a multilevel tensor compression algorithm called tensor butterfly algorithm for efficiently representing large-scale and high-dimensional oscillatory integral operators, including Green's functions for wave equations and integral transforms such as Radon transforms and Fourier transforms. The proposed algorithm leverages a tensor extension of the so-called complementary low-rank property of existing matrix butterfly algorithms. The algorithm partitions the discretized integral operator tensor into subtensors of multiple levels and factorizes each subtensor at the middle level as a Tucker-type interpolative decomposition, whose factor matrices are formed in a multilevel fashion. For a d-dimensional (d > 1) integral operator discretized into a 2d-mode tensor with n2d entries, the overall CPU time and memory requirement scale as O(nd), in stark contrast to the O(nd log n) complexity of existing matrix algorithms such as matrix butterfly algorithms and fast Fourier transforms (FFTs), where n is the number of points per direction. When comparing with other tensor algorithms such as quantized tensor train (QTT), the proposed algorithm also shows superior CPU and memory performance for tensor contraction. Remarkably, the tensor butterfly algorithm can efficiently model high-frequency Green's function interactions between two unit cubes, each spanning 512 wavelengths per direction, which represents problems of scale over 512× larger than that existing butterfly algorithms can handle, with the same amount of computation resources. On the other hand, for a problem representing 64 wavelengths per direction, which is the largest size existing algebraic matrix algorithms can handle, our tensor butterfly algorithm exhibits 200x speedups and 30× memory reduction compared with existing ones. Moreover, the tensor butterfly algorithm also permits O(nd)-complexity FFTs and Radon transforms up to d = 6 dimensions.

Kielstra, P Michael↗
DOE OSTI2026

Laplace Transform–Based Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation

Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. Here, we propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, 𝐴 −𝑘 , and the exponential of the matrix inverse, 𝑒 −𝐴 −1 . The latter can be interpreted as the solution of a mass-matrix differential equation of the form form 𝐴⁢𝑢′⁡⁡(𝑡) =−𝑢⁡(𝑡). We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting 𝐴, thereby reducing the computational complexity.

Laplace transform↗
DOE OSTI2022

Data-directed search for new physics based on symmetries of the SM

We propose exploiting symmetries (exact or approximate) of the Standard Model (SM) to search for physics Beyond the Standard Model (BSM) using the data-directed paradigm (DDP). Symmetries are very powerful because they provide two samples that can be compared without requiring simulation. Focusing on the data, exclusive selections which exhibit significant asymmetry can be identified efficiently and marked for further study. Using a simple and generic test statistic which compares two matrices already provides good sensitivity, only slightly worse than that of the profile likelihood ratio test statistic which relies on the exact knowledge of the signal shape. This can be exploited for rapidly scanning large portions of the measured data, in an attempt to identify regions of interest. We also demonstrate that weakly supervised Neural Networks could be used for this purpose as well.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Search for low mass dark matter in DarkSide-50: the bayesian network approach

Abstract We present a novel approach for the search of dark matter in the DarkSide-50 experiment, relying on Bayesian Networks. This method incorporates the detector response model into the likelihood function, explicitly maintaining the connection with the quantity of interest. No assumptions about the linearity of the problem or the shape of the probability distribution functions are required, and there is no need to morph signal and background spectra as a function of nuisance parameters. By expressing the problem in terms of Bayesian Networks, we have developed an inference algorithm based on a Markov Chain Monte Carlo to calculate the posterior probability. A clever description of the detector response model in terms of parametric matrices allows us to study the impact of systematic variations of any parameter on the final results. Our approach not only provides the desired information on the parameter of interest, but also potential constraints on the response model. Our results are consistent with recent published analyses and further refine the parameters of the detector response model.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Improved semiclassical model for real-time evaporation of matrix black holes

In this paper, we study real-time classical matrix mechanics of a simplified 2 × 2 matrix model inspired by the black hole evaporation problem. This is a step towards making a quantitative model of real-time evaporation of a black hole, which is realized as a bound state of D0-branes in string theory. The model we study is the reduction of Yang–Mills in 2 + 1 dimension to 0 + 1 dimensions, which has been corrected with an additional potential that can be interpreted as a zero-point energy for fermions. Our goal is to understand the lifetime of such a classical bound state object in the classical regime. To do so, we pay particular attention to when D-particles separate to check that the “off-diagonal modes” of the matrices become adiabatic and use that information to improve on existing models of evaporation. It turns out that the naive expectation value of the lifetime with the fermionic correction is infinite. This is a logarithmic divergence that arises from very large excursions in the separation between the branes near the threshold for classical evaporation. The adiabatic behavior lets us get some analytic control of the dynamics in this regime to get this estimate. This divergence is cutoff in the quantum theory due to quantization of the adiabatic parameter, resulting in a long lifetime of the bound state, with a parametric dependence of order log(1/ℏ).

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Parallel Embeddings: a Visualization Technique for Contrasting Learned Representations

We introduce ``Parallel Embeddings'', a new technique that generalizes the classical Parallel Coordinates visualization technique from a sequence of features within a single matrix to a sequence of multiple feature matrices. This is accomplished by separately embedding each matrix in the sequence into a 2-D frame, juxtaposing these 2-D representations, then connecting points between frames. We outline how this visualization can be used for several ``model comparison'' tasks in machine learning. We compare user performance with Parallel Embeddings to a baseline using the TensorFlow Embedding Projector for estimating model accuracy and understanding conceptual model differences. We found that users were more accurate with Parallel Embeddings for model comparison and that users learned how to use Parallel Embeddings more quickly than TensorFlow Embedding Projector. Furthermore, users' analytical process using Embedding Projector was positively affected by using Parallel Embeddings beforehand.

Model comparison, machine learing, machine learnin↗
DOE OSTI2020

A Shift Selection Strategy for Parallel Shift-invert Spectrum Slicing in Symmetric Self-consistent Eigenvalue Computation

The central importance of large-scale eigenvalue problems in scientific computation necessitates the development of massively parallel algorithms for their solution. Recent advances in dense numerical linear algebra have enabled the routine treatment of eigenvalue problems with dimensions on the order of hundreds of thousands on the world’s largest supercomputers. In cases where dense treatments are not feasible, Krylov subspace methods offer an attractive alternative due to the fact that they do not require storage of the problem matrices. However, demonstration of scalability of either of these classes of eigenvalue algorithms on computing architectures capable of expressing massive parallelism is non-trivial due to communication requirements and serial bottlenecks, respectively. In this work, we introduce the SISLICE method: a parallel shift-invert algorithm for the solution of the symmetric self-consistent field (SCF) eigenvalue problem. The SISLICE method drastically reduces the communication requirement of current parallel shift-invert eigenvalue algorithms through various shift selection and migration techniques based on density of states estimation and k-means clustering, respectively. This work demonstrates the robustness and parallel performance of the SISLICE method on a representative set of SCF eigenvalue problems and outlines research directions that will be explored in future work.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

PLANC: Parallel Low-rank Approximation with Nonnegativity Constraints

In this work, we consider the problem of low-rank approximation of massive dense nonnegative tensor data, for example, to discover latent patterns in video and imaging applications. As the size of data sets grows, single workstations are hitting bottlenecks in both computation time and available memory. We propose a distributed-memory parallel computing solution to handle massive data sets, loading the input data across the memories of multiple nodes, and performing efficient and scalable parallel algorithms to compute the low-rank approximation. We present a software package called Parallel Low-rank Approximation with Nonnegativity Constraints, which implements our solution and allows for extension in terms of data (dense or sparse, matrices or tensors of any order), algorithm (e.g., from multiplicative updating techniques to alternating direction method of multipliers), and architecture (we exploit GPUs to accelerate the computation in this work). We describe our parallel distributions and algorithms, which are careful to avoid unnecessary communication and computation, show how to extend the software to include new algorithms and/or constraints, and report efficiency and scalability results for both synthetic and real-world data sets.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Sampling-based Sublinear Low-rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning

We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.

Computer Science↗
DOE OSTI2023

Improved Water Sound Synthesis using Coupled Bubbles

We introduce a practical framework for synthesizing bubble-based water sounds that captures the rich inter-bubble coupling effects responsible for low-frequency acoustic emissions from bubble clouds. We propose coupled-bubble oscillator models with regularized singularities, and techniques to reduce the computational cost of time stepping with dense, time-varying mass matrices. Airborne acoustic emissions are estimated using finite-difference time-domain (FDTD) methods. We propose a simple, analytical surface-acceleration model, and a sample-and-hold GPU wavesolver that is simple and faster than prior CPU wavesolvers. Sound synthesis results are demonstrated using bubbly flows from incompressible, two-phase simulations, as well as procedurally generated examples using single-phase FLIP fluid animations. Our results demonstrate sound simulations with hundreds of thousands of bubbles, and perceptually significant frequency transformations with fuller low-frequency content.

Computer Science↗
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 OSTI2024

Accelerated Constrained Sparse Tensor Factorization on Massively Parallel Architectures

This study presents the first constrained sparse tensor factorization (cSTF) framework that optimizes and fully offloads computation to massively parallel GPU architectures, and the first performance characterization of cSTF on GPU architectures. In contrast to prior work on tensor factorization, where the matricized tensor times Khatri-Rao product (MTTKRP) is the primary performance bottleneck, our systematic analysis of the cSTF algorithm on GPUs reveals that adding constraints creates an additional bottleneck in the update operation for many real-world sparse tensors. While executing the update operation on the GPU brings significant speedup over its CPU counterpart, it remains a significant bottleneck. To further accelerate the update operation, we propose cuADMM, a new update algorithm that leverages algorithmic and code optimization strategies to minimize both computation and data movement on GPUs. As a result, our framework delivers significantly improved performance compared to prior state-of-the-art. On 10 real-world sparse tensors, our framework achieves geometric mean speedup of 5.1 × (max 41.59 ×) and 7.01 × (max 58.05 ×) on the NIVIDA A100 and H100 GPUs, respectively, over the state-of-the-art SPLATT library running on a 26-core Intel Ice Lake Xeon CPU.

Soh, Yongseok↗
DOE OSTI2025

Communication Lower Bounds and Optimal Algorithms for Symmetric Matrix Computations

In this article, we focus on the communication costs of three symmetric matrix computations: (i) multiplying a matrix with its transpose, known as a symmetric rank-k update (SYRK) (ii) adding the result of the multiplication of a matrix with the transpose of another matrix and the transpose of that result, known as a symmetric rank-2k update (SYR2K) (iii) performing matrix multiplication with a symmetric input matrix (SYMM). All three computations appear in the Level 3 Basic Linear Algebra Subroutines (BLAS) and have wide use in applications involving symmetric matrices. We establish communication lower bounds for these kernels using sequential and distributed-memory parallel computational models, and we show that our bounds are tight by presenting communication-optimal algorithms for each setting. Our lower bound proofs rely on applying a geometric inequality for symmetric computations and analytically solving constrained nonlinear optimization problems. As a result, the symmetric matrix and its corresponding computations are accessed and performed according to a triangular block partitioning scheme in the optimal algorithms.

Al Daas, Hussam [Rutherford Appleton Laboratory, D↗
DOE OSTI2024

Extracting and Interpreting Electrochemical Impedance Spectra (EIS) from Physics-Based Models of Lithium-Ion Batteries

This paper implements a highly efficient algorithm to extract electrochemical impedance spectra (EIS) from physics-based battery models (e.g., a P2D model). The mathematical approach is different from how EIS is practiced experimentally. Experimentally, the voltage (current) is harmonically perturbed over a wide range of frequencies and the amplitude and phase shift of the corresponding current (voltage) is measured. The experimental approach can be implemented in simulation software, but is computationally expensive. The approach here is to determine locally linear state-space models from the full physical model. The four Jacobian matrices that are the basis of the state-space models can be derived by numerical differentiation of the physical model. The EIS is then extracted from the state-space model using computationally efficient matrix-manipulation techniques. The algorithm can evaluate the full EIS at an instant in time during a transient, independent of whether the battery is in a stationary state. The approach is also able to separate the full-cell impedance to evaluate partial EIS, such as for a battery anode alone. Although such partial EIS is difficult to measure experimentally, the partial EIS provides valuable insights in interpreting the full-cell EIS.

25 ENERGY STORAGE↗
DOE OSTI2020

Synchronization-Based Guaranteed-Performance Formation Design for Swarm Systems

Guaranteed-performance formation control for swarm systems with the second-order dynamics is investigated based on the synchronization control strategy. Firstly, a new formation protocol is presented, where the weights of connected edges are adaptively regulated and the performance constraint is imposed. Then, on the basis of the Riccati inequality, sufficient conditions for synchronization-based guaranteed-performance formation are proposed, and an explicit expression of the guaranteed-performance cost is shown, where it is fully distributed to design gain matrices of the formation protocol in the sense that it is independent of global information of swarm systems. Moreover, the whole motion of a swarm system is determined, which is associated with initial states of all agents and formation control vectors. Finally, two numerical examples are shown to demonstrate theoretical conclusions, where the static whole motion and the dynamic whole motion are considered, respectively.

Dang, Hongtao↗
DOE OSTI2020

Computation of graph hitting time moments; Chapel code implementation.

The project that developed this is unclassified, with the mandate to produce open source code. This code computes the hitting time moments of a graph using a linear algebra configuration. The goal of this work is to explore the performance capabilities of the Chapel programming language. Toward that end, we generate random adjacency matrices which represent a random graph. The code can also read in an adjacency matrix from a file. The main computation is the Conjugate Gradient method.SAND2020-12651 M. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525.

Barrett, Richard↗
DOE OSTI2020

Multivariate Curve Resolution (MCR) using Principal Components Inputs and Rigorous Equality and Inequality Constraints

This MATLAB pseudocode perform multivariate curve resolution (MCR) using PCA scores & loadings of data as inputs. It employs rigorous least squares equality and inequality constraints for all elements in the solution factor matrices. It also can be used to perform nonnegative matrix factorization. SAND2020-12650 M Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525.

Van Benthem, Mark↗