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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
NASA NTRS1987

The use of the QR factorization in the partial realization problem

The use of the QR factorization of the Hankel matrix in solving the partial realization problem is analyzed. Straightforward use of the QR factorization results in a realization scheme that possesses all of the computational advantages of Rissanen's realization scheme. These latter properties are computational efficiency, recursiveness, use of limited computer memory, and the realization of a system triplet having a condensed structure. Moreover, this scheme is robust when the order of the system corresponds to the rank of the Hankel matrix. When this latter condition is violated, an approximate realization could be determined via the QR factorization. In this second scheme, the given Hankel matrix is approximated by a low-rank non-Hankel matrix. Furthermore, it is demonstrated that column pivoting might be incorporated in this second scheme. The results presented are derived for a single input/single output system, but this does not seem to be a restriction.

Verhaegen, M. H.↗
NASA NTRS1987

The minimal residual QR-factorization algorithm for reliably solving subset regression problems

A new algorithm to solve test subset regression problems is described, called the minimal residual QR factorization algorithm (MRQR). This scheme performs a QR factorization with a new column pivoting strategy. Basically, this strategy is based on the change in the residual of the least squares problem. Furthermore, it is demonstrated that this basic scheme might be extended in a numerically efficient way to combine the advantages of existing numerical procedures, such as the singular value decomposition, with those of more classical statistical procedures, such as stepwise regression. This extension is presented as an advisory expert system that guides the user in solving the subset regression problem. The advantages of the new procedure are highlighted by a numerical example.

Verhaegen, M. H.↗
NASA NTRS1991

QR Factorization In The Partial-Realization Problem

Report discusses use of QR factorization in solving partial-realization problem. Subject matter relevant to characterization of system, details of which hidden (a "blackbox"), on basis of relationship between its inputs and outputs. When realization process successful, system shown linear with known number of degrees of freedom and representable by mathematical model providing basis for prediction of outputs and design of controllers.

Verhaegen, M. H.↗
DOE OSTI2023

Computing rank‐revealing factorizations of matrices stored out‐of‐core

This paper describes efficient algorithms for computing rank-revealing factorizations of matrices that are too large to fit in main memory (RAM), and must instead be stored on slow external memory devices such as disks (out-of-core or out-of-memory). Traditional algorithms for computing rank-revealing factorizations (such as the column pivoted QR factorization and the singular value decomposition) are very communication intensive as they require many vector-vector and matrix-vector operations, which become prohibitively expensive when data is not in RAM. Randomization allows to reformulate new methods so that large contiguous blocks of the matrix are processed in bulk. The paper describes two distinct methods. The first is a blocked version of column pivoted Householder QR, organized as a “left-looking” method to minimize the number of the expensive write operations. The second method results employs a UTV factorization. It is organized as an algorithm-by-blocks to overlap computations and I/O operations. As it incorporates power iterations, it is much better at revealing the numerical rank. Numerical experiments on several computers demonstrate that the new algorithms are almost as fast when processing data stored on slow memory devices as traditional algorithms are for data stored in RAM.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Large-Scale Optimization with Linear Equality Constraints Using Reduced Compact Representation

For optimization problems with linear equality constraints, we prove that the (1,1) block of the inverse KKT matrix remains unchanged when projected onto the nullspace of the constraint matrix. In this work, we develop reduced compact representations of the limited-memory inverse BFGS Hessian to compute search directions efficiently when the constraint Jacobian is sparse. Orthogonal projections are implemented by a sparse QR factorization or a preconditioned LSQR iteration. In numerical experiments two proposed trust-region algorithms improve in computation times, often significantly, compared to previous implementations of related algorithms and compared to IPOPT.

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 OSTI2023

Iterated Gauss-Seidel GMRES

The GMRES algorithm of Saad and Schultz [SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856-869] is an iterative method for approximately solving linear systems Ax = b, with initial guess x0 and residual r0 = b Ax0. The algorithm employs the Arnoldi process to generate the Krylov basis vectors (the columns of Vk ). It is well known that this process can be viewed as a QR factorization of the matrix Bk = [r0, AVk] at each iteration. Despite an O (..epsilon..)..kappa.. (Bk ) loss of orthogonality, for unit roundoff ..epsilon..and condition number ..kappa.. , the modified Gram-Schmidt formulation was shown to be backward stable in the seminal paper by Paige et al. [SIAM J. Matrix Anal.Appl., 28 (2006), pp. 264-284]. We present an iterated Gauss-Seidel formulation of the GMRES algorithm (IGS-GMRES) based on the ideas of Ruhe [Linear Algebra Appl., 52 (1983), pp. 591-601] and Swirydowicz et al. [Numer. Linear Algebra Appl., 28 (2020), pp. 1-20]. IGS-GMRES maintains orthogonality to the level O (..epsilon..)..kappa.. (Bk ) or O (..epsilon..), depending on the choice of one or two iterations; for two Gauss-Seidel iterations, the computed Krylov basis vectors remain orthogonal to working accuracy and the smallest singular value of Vk remains close to one. The resulting GMRES method is thus backward stable. We show that IGS-GMRES can be implemented with only a single synchronization point per iteration, making it relevant to large-scale parallel computing environments. We also demonstrate that, unlike MGS-GMRES, in IGS-GMRES the relative Arnoldi residual corresponding to the computed approximate solution no longer stagnates above machine precision even for highly nonnormal systems.

Arnoldi-QR↗
DOE OSTI2020

Low synchronization Gram–Schmidt and generalized minimal residual algorithms

The Gram–Schmidt process uses orthogonal projection to construct the A = QR factorization of a matrix. When Q has linearly independent columns, the operator P = I - Q(QTQ)-1QT defines an orthogonal projection onto Q⊥. In finite precision, Q loses orthogonality as the factorization progresses. A family of approximate projections is derived with the form P = I - QTQT, with correction matrix T. When T = (QTQ)-1, and T is triangular, it is postulated that the best achievable orthogonality is $\mathcal{O}(ε)\mathcal{K}(A)$. We present new variants of modified (MGS) and classical Gram–Schmidt algorithms that require one global reduction step. An interesting form of the projector leads to a compact WY representation for MGS. In particular, the inverse compact WY MGS algorithm is equivalent to a lower triangular solve. Our main contribution is to introduce a backward normalization lag into the compact WY representation, resulting in a $\mathcal{O}(ε)\mathcal{K}[r_0, AV_m])$ stable Generalized Minimal Residual Method (GMRES) algorithm that requires only one global reduce per iteration. Finally, further improvements in performance are achieved by accelerating GMRES on GPUs.

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↗
NASA NTRS1990

Downdating a time-varying square root information filter

A new method to efficiently downdate an estimate and covariance generated by a discrete time Square Root Information Filter (SRIF) is presented. The method combines the QR factor downdating algorithm of Gill and the decentralized SRIF algorithm of Bierman. Efficient removal of either measurements or a priori information is possible without loss of numerical integrity. Moreover, the method includes features for detecting potential numerical degradation. Performance on a 300 parameter system with 5800 data points shows that the method can be used in real time and hence is a promising tool for interactive data analysis. Additionally, updating a time-varying SRIF filter with either additional measurements or a priori information proceeds analogously.

Muellerschoen, Ronald J.↗
NASA NTRS1992

Multivariable frequency domain identification via 2-norm minimization

The author develops a computational approach to multivariable frequency domain identification, based on 2-norm minimization. In particular, a Gauss-Newton (GN) iteration is developed to minimize the 2-norm of the error between frequency domain data and a matrix fraction transfer function estimate. To improve the global performance of the optimization algorithm, the GN iteration is initialized using the solution to a particular sequentially reweighted least squares problem, denoted as the SK iteration. The least squares problems which arise from both the SK and GN iterations are shown to involve sparse matrices with identical block structure. A sparse matrix QR factorization method is developed to exploit the special block structure, and to efficiently compute the least squares solution. A numerical example involving the identification of a multiple-input multiple-output (MIMO) plant having 286 unknown parameters is given to illustrate the effectiveness of the algorithm.

Bayard, David S.↗
DOE OSTI2020

Parsimonious Inference Information-Theoretic Foundations for a Complete Theory of Machine Learning (CIS-LDRD Project 218313 Final Technical Report)

This work examines how we may cast machine learning within a complete Bayesian framework to quantify and suppress explanatory complexity from first principles. Our investigation into both the philosophy and mathematics of rational belief leads us to emphasize the critical role of Bayesian inference in learning well-justified predictions within a rigorous and complete extended logic. The Bayesian framework allows us to coherently account for evidence in the learned plausibility of potential explanations. As an extended logic, the Bayesian paradigm regards probability as a notion of degrees of truth. In order to satisfy critical properties of probability as a coherent measure, as well as maintain consistency with binary propositional logic, we arrive at Bayes' Theorem as the only justifiable mechanism to update our beliefs to account for empiracle evidence. Yet, in the machine learning paradigm, where explanations are unconstrained algorithmic abstractions, we arrive at a critical challenge: Bayesian inference requires prior belief. Conventional approaches fail to yield a consistent framework in which we could compare prior plausibility among the infinities of potential choices in learning architectures. The difficulty of articulating well-justified prior belief over abstract models is the provinence of memorization in traditional machine learning training practices. This becomes exceptionally problematic in the context of limited datasets, when we wish to learn justifiable predictions from only a small amount of data.

97 MATHEMATICS AND COMPUTING↗
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 OSTI2020

Multifrontal Non-negative Matrix Factorization

Non-negative matrix factorization (Nmf) is an important tool in high-performance large scale data analytics with applications ranging from community detection, recommender system, feature detection and linear and non-linear unmixing. While traditional Nmf works well when the data set is relatively dense, however, it may not extract sufficient structure when the data is extremely sparse. Specifically, traditional Nmf fails to exploit the structured sparsity of the large and sparse data sets resulting in dense factors. We propose a new algorithm for performing Nmf on sparse data that we call multifrontal Nmf (Mf-Nmf) since it borrows several ideas from the multifrontal method for unconstrained factorization (e.g. LU and QR). We also present an efficient shared memory parallel implementation of Mf-Nmf and discuss its performance and scalability. We conduct several experiments on synthetic and realworld datasets and demonstrate the usefulness of the algorithm by comparing it against standard baselines. We obtain a speedup of 1.2x to 19.5x on 24 cores with an average speed up of 10.3x across all the real world datasets.

Sao, Piyush↗
DOE OSTI2023

Reconstruction of tokamak plasma safety factor profile using deep learning

Abstract The motional Stark effect (MSE) diagnostic has been a standard measurement for the magnetic field line pitch angle in tokamaks that are equipped with neutral beams. However, the MSE data are not always available due to experimental constraints, especially in future devices without neutral beams. Here we develop a deep-learning based model (SGTC-QR) that can reconstruct the safety factor profile without the MSE diagnostic to mimic the traditional equilibrium reconstruction with the MSE constraint. The model demonstrates promising performance, and the sub-millisecond inference time is compatible with the real-time plasma control system.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations

Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices, the difference in performance is due to increased communication between the processor and slow memory, which QRCP needs in order to choose pivots during decomposition. Our main algorithm, Randomized QR with Column Pivoting (RQRCP), uses randomized projection to make pivot decisions from a much smaller sample matrix, which we can construct to reside in a faster level of memory than the original matrix. This technique may be understood as trading vastly reduced communication for a controlled increase in uncertainty during the decision process. Furthermore, for rank-revealing purposes, the selection mechanism in RQRCP produces results that are the same quality as the standard algorithm, but with performance near that of unpivoted QR (often an order of magnitude faster for large matrices). Additionally, we also propose two formulas that facilitate further performance improvements. The first efficiently updates sample matrices to avoid computing new randomized projections. The second avoids large trailing updates during the decomposition in truncated low-rank approximations. Our truncated version of RQRCP also provides a key initial step in our truncated SVD approximation, TUXV. These advances open up a new performance domain for large matrix factorizations that will support efficient problem-solving techniques for challenging applications in science, engineering, and data analysis.

97 MATHEMATICS AND COMPUTING↗