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All sourcesNASA NTRSDOE OSTI
DOE OSTI2021

Randomized Algorithms for Scientific Computing (RASC)

Randomized algorithms have propelled advances in artificial intelligence (AI) and represent a foundational research area in advancing AI for Science. Future advancements in DOE Office of Science priority areas such as climate science, astrophysics, fusion, advanced materials, combustion, and quantum computing all require randomized algorithms for surmounting challenges of complexity, robustness, and scalability. Advances in data collection and numerical simulation have changed the dynamics of scientific research and motivate the need for randomized algorithms. For instance, advances in imaging technologies such as X-ray ptychography, electron microscopy, electron energy loss spectroscopy, or adaptive optics lattice light-sheet microscopy collect hyperspectral imaging and scattering data in terabytes, at breakneck speed enabled by state-of-the-art detectors. The data collection is exceptionally fast compared with its analysis. Likewise, advances in high-performance architectures have made exascale computing a reality and changed the economies of scientific computing in the process. Floating-point operations that create data are essentially free in comparison with data movement. Thus far, most approaches have focused on creating faster hardware. Ironically, this faster hardware has exacerbated the problem by making data still easier to create. Under such an onslaught, scientists often resort to heuristic deterministic sampling schemes (e.g., low-precision arithmetic, sampling every nth element) and sacrifice potentially valuable accuracy. Dramatically better results can be achieved via randomized algorithms, reducing the data size as much as or more than naive deterministic subsampling can achieve, while retaining the high accuracy of computing on the full data set. By randomized algorithms we mean those algorithms that employ some form of randomness in internal algorithmic decisions to accelerate time to solution, increase scalability, or improve reliability. Examples include matrix sketching for solving large-scale least-squares problems (see Figure 1) and stochastic gradient descent for training machine learning models. We are not recommending heuristic methods but rather randomized algorithms that have certificates of correctness and probabilistic guarantees of optimality and near-optimality. Such approaches can be useful beyond acceleration, for example, in understanding how to avoid measure zero worst-case scenarios that plague methods such as QR matrix factorization.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Randomized Algorithms for Linear Solvers

Recently, randomized algorithms in numerical linear algebra, specifically those centered around random sketching, have gained traction in primarily theoretical research due to their potential to significantly reduce problem dimensionality at the cost of an O(1) multiplicative distortion factor. It has been assumed that this sketching can be done efficiently, but thorough investigation into how precisely to do it has been neglected. Moreover, the theory-based community has argued for sketching’s ability to reduce computational cost via complexity analysis, but has not researched how it affects the stability of the algorithms. At Sandia, efficient linear solvers that scale well on modern HPC architectures while maintaining stability are imperative for practical applications. In this LDRD, we developed a random sketching strategy that is substantially faster than existing ones, and demonstrate its superior performance in practice on a NVIDIA H100 GPU. Moreover, we show how this can be used to significantly outperform existing linear least squares solvers while improving the solver’s stability as well. Additionally, we demonstrate how this sketching strategy can be used to make a fast, stable QR factorization that can subsequently be used in s-step and block Krylov solvers. Finally, we incorporate a sketching-based block orthogonalization scheme into s-step GMRES, which is stable and faster than existing approaches on the Perlmutter supercomputer.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Randomized Algorithms for Low-Rank Matrix and Tensor Decompositions

This paper surveys randomized algorithms in numerical linear algebra for low-rank decompositions of matrices and tensors. The survey begins with a review of classical matrix algorithms that can be accelerated by randomized dimensionality reduction, such as the singular value decomposition (SVD) or interpolative (ID) and CUR decompositions. Recent advances in randomized dimensionality reduction are discussed, including new methods of fast matrix sketching and sampling techniques, which are incorporated into classical matrix algorithms for fast low-rank matrix approximations. The extension of randomized matrix algorithms to tensors is then explored for several low-rank tensor decompositions in the CP and Tucker formats, including the higher-order SVD, ID, and CUR decomposition.

Pearce, Katherine J. [The University of Texas at A↗
DOE OSTI2024

Bringing randomized algorithms to mainstream numerical linear algebra

Numerical linear algebra (NLA) underpins huge swaths of computational science and engineering. For scientists and engineers to make the most of the DOE’s computing resources, it is essential that they have access to high-performance implementations of algorithms with best-in-class scalability and reliability. Despite this, prevailing NLA libraries have little to no support for breakthrough algorithms from the field of randomized numerical linear algebra (RandNLA) that have been developed over the past twenty years. The goal of this LDRD was to break a log-jam that had prevented broad adoption of RandNLA. Our work had two thrusts. The first was to develop RandBLAS: a trustworthy and high-performance C++ library for randomized dimension reduction (an operation widely known as sketching). The second was the development of a novel randomized algorithm for computing a challenging type of matrix decomposition known as Householder QR with column pivoting (Householder QRCP). In this one-year late-start LDRD we successfully delivered RandBLAS 1.0 and new CPU and GPU codes for Householder QRCP. RandBLAS has extensive documentation at https://randblas.readthedocs.io/en/stable/. Papers on RandBLAS and and our high-performance QRCP codes are forthcoming.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Randomized Algorithms for Symmetric Nonnegative Matrix Factorization

Symmetric Nonnegative Matrix Factorization (SymNMF) is a technique in data analysis and machine learning that approximates a matrix with a product of a nonnegative, low-rank matrix and it transpose. To design faster and more scalable algorithms for SymNMF we develop two randomized algorithms for its computation. The first method uses randomized matrix sketching to compute an initial low-rank approximation to the input matrix and proceeds to uses this as a low-rank input to rapidly compute a SymNMF. The second methods uses randomized leverage score sampling to approximately solve constrained least squares problems. Many successful methods for SymNMF rely on (approximately) solving sequences of constrained least squares problems. Here, we prove theoretically that leverage score sampling can approximately solve constrained least squares problems to e-accuracy. Finally we demonstrate both methods work in practice by applying them to graph clustering tasks on large real world data sets. These experiments show that our methods approximately maintain solution quality and achieve significant speed ups for both large dense and large sparse problems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large-scale problems is challenging. Motivated by applications in hyper-differential sensitivity analysis (HDSA), in this work we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large-scale model problem where HDSA is used to study subsurface flow.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Ensemble Simulation Techniques and Fast Randomized Algorithms

The major goals of the project were to develop and analyze new ensemble simulation techniques, including trajectory stratification and preconditioned MCMC techniques, as well as develop fast numerical linear algebra techniques closely related to ensemble simulation ideas. The trajectory stratification techniques involve simulating in parallel short trajectory fragments of a Markov process confined to a specific region of space‐time and then patching together the statistics gathered to assemble estimates of very general dynamical properties. We have also developed this approach for rare event simulation and extended the techniques to applications requiring a more general framework (such as electronic structure calculations). The preconditioned MCMC techniques involve simulating multiple Markov chains in parallel and then using information from the ensemble to speed the mixing of each individual chain. The fast randomized linear algebra methods are motivated by the diffusion Monte Carlo technique, but are applicable to finding the dominant eigenvalue of (almost) general matrices. For most non‐negative matrices, the schemes result in an error (compared to the power method) that is constant in the dimension of the problem. For more general matrices, we see a very clear sublinear cost trend in computational tests.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Randomized algorithms for accelerating linear algebraic computations

The project supported the development of new methodologies for performing matrix computations that form key building blocks in modern scientific computing, such as low rank approximation of matrices, and efficient representations of global operators that arise in simulations of physical phenomena.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Surrogate-Based Autotuning for Randomized Sketching Algorithms in Regression Problems

Algorithms from Randomized Numerical Linear Algebra (RandNLA) are known to be effective in handling high-dimensional computational problems, providing high-quality empirical performance as well as strong probabilistic guarantees. However, their practical application is complicated by the fact that the user needs to set various algorithm-specific tuning parameters which are different from those used in traditional NLA. This paper demonstrates how a surrogate-based autotuning approach can be used to address fundamental problems of parameter selection in RandNLA algorithms. In particular, we provide a detailed investigation of surrogate-based autotuning for sketch-and-precondition (SAP)-based randomized least squares methods, which have been one of the great success stories in modern RandNLA. Empirical results show that our surrogate-based autotuning approach can achieve near-optimal performance with much less tuning cost than a random search (up to about 7.6x fewer trials of different parameter configurations). Moreover, while our experiments focus on least squares, our results demonstrate a general-purpose autotuning pipeline applicable to any kind of RandNLA algorithm.

Cho, Younghyun↗
DOE OSTI2024

Stochastic Trust-Region Algorithm in Random Subspaces with Convergence and Expected Complexity Analyses

Here, this work proposes a framework for large-scale stochastic derivative-free optimization (DFO) by introducing STARS, a trust-region method based on iterative minimization in random subspaces. This framework is both an algorithmic and theoretical extension of a random subspace derivative-free optimization (RSDFO) framework, and an algorithm for stochastic optimization with random models (STORM). Moreover, like RSDFO, STARS achieves scalability by minimizing interpolation models that approximate the objective in low-dimensional affine subspaces, thus significantly reducing per-iteration costs in terms of function evaluations and yielding strong performance on largescale stochastic DFO problems. The user-determined dimension of these subspaces, when the latter are defined, for example, by the columns of so-called Johnson-Lindenstrauss transforms, turns out to be independent of the dimension of the problem. For convergence purposes, inspired by the analyses of RSDFO and STORM, both a particular quality of the subspace and the accuracies of random function estimates and models are required to hold with sufficiently high, but fixed, probabilities. Using martingale theory under the latter assumptions, an almost sure global convergence of STARS to a first-order stationary point is shown, and the expected number of iterations required to reach a desired first-order accuracy is proved to be similar to that of STORM and other stochastic DFO algorithms, up to constants.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Reducing Randomized Quantum Algorithm Cost [Slides]

We derive the optimal sampling strategy for minimizing total resource cost in randomized quantum algorithms. Our framework is completely general, allowing for resources as diverse as gate counts circuit depth, runtime, or even dissipated energy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2026

Solovay-Kitaev Algorithm and Randomized Compilation Data Availability

This zipped folder contains simulation notebooks, simulated data, and experimental data from the QSCOUT trapped-ion device that were used in the publication "Solovay-Kitaev Algorithm and Randomized Compilation" (https://doi.org/10.1103/ll6m-dbl7). The raw data is in the form of measurement outcomes of simple tomographic quantum circuits that were executed on the QSCOUT device and simulated using JAQALPAQ. These data are used to create plots within the jupyter notebooks that were included in the publication.

Quantum benchmarking↗
DOE OSTI2022

Improving convergence of the Matrix Power Control Algorithm for random vibration testing

Herein, this paper describes modifications to the Matrix Power Control Algorithm (MPCA) to improve convergence for Random Vibration Control (RVC) testing. In particular, this paper presents Multiple-Input Multiple-Output (MIMO) implementations of MPCA in simulation and experiment. An Euler–Bernoulli beam model was simulated with applied base excitations and the Box Assembly with Removable Component (BARC) was used in experiment to validate results. The Bayesian optimization package Dragonfly was used to optimize control parameters. Additionally, a moving-average was employed and optimized to improve the measured response feedback for MPCA, reduce the number of averages needed to be taken between control updates, and further improve convergence. The key results of this paper show that the performance of MPCA can be improved by tuning the control parameters and by applying an optimized moving-average. Furthermore, it is demonstrated that convergence can be achieved within 12 drive-frames, which greatly enhances vibration control capability.

42 ENGINEERING↗
DOE OSTI2022

Online peak-aware energy scheduling with untrusted advice

This paper studies the online energy scheduling problem in a hybrid model where the cost of energy is proportional to both the volume and peak usage, and where energy can be either locally generated or drawn from the grid. Inspired by recent advances in online algorithms with Machine Learned (ML) advice, we develop parameterized deterministic and randomized algorithms for this problem such that the level of reliance on the advice can be adjusted by a trust parameter. We then analyze the performance of the proposed algorithms using two performance metrics: robustness that measures the competitive ratio as a function of the trust parameter when the advice is inaccurate, and consistency for competitive ratio when the advice is accurate. Since the competitive ratio is analyzed in two different regimes, we further investigate the Pareto optimality of the proposed algorithms. Our results show that the proposed deterministic algorithm is Pareto-optimal, in the sense that no other online deterministic algorithms can dominate the robustness and consistency of our algorithm. Furthermore, we show that the proposed randomized algorithm dominates the Pareto-optimal deterministic algorithm. Our large-scale empirical evaluations using real traces of energy demand, energy prices, and renewable energy generations highlight that the proposed algorithms outperform worst-case optimized algorithms and fully data-driven algorithms.

Lee, Russell↗
DOE OSTI2024

A Provably Accurate Randomized Sampling Algorithm for Logistic Regression

In statistics and machine learning, logistic regression is a widely-used supervised learning technique primarily employed for binary classification tasks. When the number of observations greatly exceeds the number of predictor variables, we present a simple, randomized sampling-based algorithm for logistic regression problem that guarantees high-quality approximations to both the estimated probabilities and the overall discrepancy of the model. Our analysis builds upon two simple structural conditions that boil down to randomized matrix multiplication, a fundamental and well-understood primitive of randomized numerical linear algebra. We analyze the properties of estimated probabilities of logistic regression when leverage scores are used to sample observations, and prove that accurate approximations can be achieved with a sample whose size is much smaller than the total number of observations. To further validate our theoretical findings, we conduct comprehensive empirical evaluations. Overall, our work sheds light on the potential of using randomized sampling approaches to efficiently approximate the estimated probabilities in logistic regression, offering a practical and computationally efficient solution for large-scale datasets.

Chowdhury, Agniva↗
DOE OSTI2021

Randomized Sketching Algorithms for Low-Memory Dynamic Optimization

This paper develops a novel limited-memory method to solve dynamic optimization problems. The memory requirements for such problems often present a major obstacle, particularly for problems with PDE constraints such as optimal flow control, full waveform inversion, and optical tomography. In these problems, PDE constraints uniquely determine the state of a physical system for a given control; the goal is to find the value of the control that minimizes an objective. While the control is often low dimensional, the state is typically more expensive to store. This paper suggests using randomized matrix approximation to compress the state as it is generated and shows how to use the compressed state to reliably solve the original dynamic optimization problem. Concretely, the compressed state is used to compute approximate gradients and to apply the Hessian to vectors. The approximation error in these quantities is controlled by the target rank of the sketch. This approximate first- and second-order information can readily be used in any optimization algorithm. As an example, we develop a sketched trust-region method that adaptively chooses the target rank using a posteriori error information and provably converges to a stationary point of the original problem. Numerical experiments with the sketched trust-region method show promising performance on challenging problems such as the optimal control of an advection-reaction-diffusion equation and the optimal control of fluid flow past a cylinder.

97 MATHEMATICS AND COMPUTING↗