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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Computationally Efficient Decompositions of Oblique Projection Matrices

Oblique projection matrices arise in problems in weighted least squares, signal processing, and optimization. While these matrices can be potentially very large, their low-rank structure can be exploited for efficient computation. Here, we propose fast and scalable algorithms for computing their eigendecomposition and singular value decomposition (SVD). Numerical experiments that compare our proposed approaches to existing methods, including randomized SVD, are presented. In addition, we test their accuracy on linear systems from equality constrained optimization problems.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1990

Input/output system identification - Learning from repeated experiments

The paper describes three approaches and possible variations for the determination of the Markov parameters for forced response data using general inputs. It is shown that, when the parameters in the solution procedure are bootstrapped, the results can be obtained very efficiently, but the errors propagate throughout all parameters. By arranging the data in a different form and using singular value decomposition, the resulting identified parameters are more accurate, in the least number of successive experiments, at the expense of a large matrix singular value decomposition. When a recursive procedure is employed, the calculations can be performed very efficiently, but the number of repetitions of the experiments is much greater for a given accuracy than for any of the previous approaches. An alternative formulation is proposed to combine the advantages of each of the approaches.

Juang, Jer-Nan↗
DOE OSTI2023

CP decomposition for tensors via alternating least squares with QR decomposition

The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Compression of tokamak boundary plasma simulation data using a maximum volume algorithm for matrix skeleton decomposition

This report demonstrates satisfactory data compression of SOLPS-ITER simulation output ranging from 2D fields, 1D profiles, and 0D scalar variables with a novel matrix decomposition approach. The singular value decomposition (SVD) scales poorly for large matrix sizes and is unsuited to the application on high dimensional data common to fusion plasma physics simulation. In this work, we employ the columns-submatrix-rows (CUR) matrix factorization technique in order to compute a low-rank approximation up to two orders of magnitude faster than the SVD, but within a nominal L2-norm relative error of ε = 10 –2 . In addition, the CUR approach maintains the original format of the data, in its extracted columns and rows, allowing for interpretable data storage at the original resolution of the simulation. We utilize an iterative algorithm to compute the CUR decomposition of simulation output by maximizing the volume, or linearly independent information content, of a low-rank submatrix contained within the data. Experiments over $\textit{n} × \textit{n}$ randomized test matrices with embedded rank-deficient features show that this maximum volume implementation of CUR matrix approximation has reduced asymptotic computational complexity on the order of n compared to the SVD, which scales approximately as $n^3$. These results show that the CUR technique can be used to effectively select time step snapshots (columns) of over 140 SOLPS-ITER output variables and the associated discretized coordinate timeseries (rows) allowing for reconstruction of the complete simulation dynamics.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1993

A new technique for deconvolution of data from instruments that make integral measurements, e.g. RIMS on DE-1

A general method for deconvolving an unknown function from integral measurements is described and applied to data from the instrument Retarding Ion Mass Spectrometer (RIMS) aboard the spacecraft Dynamics Explorer 1 (DE-1). The principal features of the method are: (1) it uses objective criteria based upon fundamental statistical principles, i.e. Bayesian statistics; (2) it provides for insertion of prior knowledge in a non-prejudicial, explicit manner through the choice of breakpoints that determine the bicubic spline expansion functions; (3) it prevents random fluctuations from controlling the fit to the data through the use of singular value decomposition and the elimination of small singular values; and (4) it guards agianst the introduction of spurious features into the result by including a penalty function and using the principle of generalized cross validation. Illustrative examples from RIMS data for H(+) and O(+) show that the method provides enhanced accuracy and detail in deconvolving the ion phase space density.

Perez, J. D.↗
DOE OSTI2020

A parallel hierarchical blocked adaptive cross approximation algorithm

This article presents a low-rank decomposition algorithm based on subsampling of matrix entries. The proposed algorithm first computes rank-revealing decompositions of submatrices with a blocked adaptive cross approximation (BACA) algorithm, and then applies a hierarchical merge operation via truncated singular value decompositions (H-BACA). The proposed algorithm significantly improves the convergence of the baseline ACA algorithm and achieves reduced computational complexity compared to the traditional decompositions such as rank-revealing QR. Numerical results demonstrate the efficiency, accuracy, and parallel scalability of the proposed algorithm.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

CORRLA-RS

The CORRLA-RS package provides a suite of statistical methods for sampling multidimensional distributions and to conduct sensitivity and correlation analysis of large scale data in the Rust programming language. The software provides a unique solution to multidimensional constrained sampling problems utilizing a combination of parallelized Markov Chain Monte Carlo methods and traditional rejection sampling. The sensitivity and correlation analysis methods are backed by a high performance randomized singular value decomposition implementation which enables datasets larger than the random access memory (RAM) size to be analyzed. Additionally, CORRLA-RS implements the active subspace identification method using a KD-Tree and the randomized singular value decomposition acting in concert.

Gurecky, William [Oak Ridge National Laboratory (O↗
NASA NTRS1996

Design of Multi-Parameter Steerable Functions Using Cascade Basis Reduction

A new cascade basis reduction method of computing the optimal least-squares set of basis functions steering a given function is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches in such a way that their respective strengths complement each other. Since the Lie group-theoretic approach is used, the set of basis and steering functions computed can be expressed analytically. Because the singular value decomposition method is used, this set of basis and steering functions is optimal in the least-squares sense. Furthermore, the computational complexity in designing basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions that steers a Gabor function under the four-parameter linear transformation group.

Teo, P.↗
DOE OSTI2015

SU-E-T-610: Phosphor-Based Fiber Optic Probes for Proton Beam Characterization

Purpose: To investigate feasibility of using fiber optics probes with rare-earth-based phosphor tips for proton beam radiation dosimetry. We designed and fabricated a fiber probe with submillimeter resolution (<0.5 mm3) based on TbF3 phosphors and evaluated its performance for measurement of proton beam including profiles and range. Methods: The fiber optic probe with TbF3 phosphor tip, embedded in tissue-mimicking phantoms was irradiated with double scattering proton beam with energy of 180 MeV. Luminescence spectroscopy was performed by a CCD-coupled spectrograph to analyze the emission spectra of the fiber tip. In order to measure the spatial beam profile and percentage depth dose, we used singular value decomposition method to spectrally separate the phosphors ionoluminescence signal from the background Cerenkov radiation signal. Results: The spectra of the TbF3 fiber probe showed characteristic ionoluminescence emission peaks at 489, 542, 586, and 620 nm. By using singular value decomposition we found the contribution of the ionoluminescence signal to measure the percentage depth dose in phantoms and compared that with measurements performed with ion chamber. We observed quenching effect at the spread out Bragg peak region, manifested as under-responding of the signal, due to the high LET of the beam. However, the beam profiles were not dramatically affected by the quenching effect. Conclusion: We have evaluated the performance of a fiber optic probe with submillimeter resolution for proton beam dosimetry. We demonstrated feasibility of spectral separation of the Cerenkov radiation from the collected signal. Such fiber probes can be used for measurements of proton beams profile and range. The experimental apparatus and spectroscopy method developed in this work provide a robust platform for characterization of proton-irradiated nanophosphor particles for ultralow fluence photodynamic therapy or molecular imaging applications.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
NASA NTRSDate not supplied

Active Thermography Based on Tensor Rank Decomposition

Principal Component Thermography applies Singular Value Decomposition (SVD) to post-process data that are derived from active thermographic inspections. SVD provides useful compression of the data and allows for better understanding of substructure and indications of potential damage. In the standard approach, SVD is applied to a certain reshaping of a three-dimensional data stack into a two-dimensional array. This work applies the CANDECOMP-PARAFAC (CP) tensor rank decomposition directly to the three-dimensional data to avoid the initial reshaping step in order to begin to develop an inspection method that can more accurately detect defects in non-homogeneous and anisotropic materials. Tests against simulated data that compare the CP decomposition method with traditional Principal Component Thermography based on SVD are described. Finally, the method of Proper Generalized Decomposition (PGD) is used to derive the CP decomposition, and its performance against other algorithms is also discussed.

Thermography↗
DOE OSTI2023

Fast truncated SVD of sparse and dense matrices on graphics processors

We investigate the solution of low-rank matrix approximation problems using the truncated singular value decomposition (SVD). For this purpose, we develop and optimize graphics processing unit (GPU) implementations for the randomized SVD and a blocked variant of the Lanczos approach. Our work takes advantage of the fact that the two methods are composed of very similar linear algebra building blocks, which can be assembled using numerical kernels from existing high-performance linear algebra libraries. Furthermore, the experiments with several sparse matrices arising in representative real-world applications and synthetic dense test matrices reveal a performance advantage of the block Lanczos algorithm when targeting the same approximation accuracy.

Computer Science↗
DOE OSTI2026

Accelerating multigrid with streaming chiral SVD for Wilson fermions in lattice QCD

A modification to the setup algorithm for the multigrid preconditioner of Wilson fermions in lattice QCD is presented. A larger basis of test vectors than that used in regular multigrid is calculated by the smoother and truncated by singular value decomposition on the chiral components of the test vectors. The truncated basis is used to form the prolongation and restriction matrices of the multigrid hierarchy. This modification of the setup method is demonstrated to increase the convergence of linear solvers on an anisotropic lattice with m π ≈ 239 MeV from the Hadron Spectrum Collaboration and an isotropic lattice with m π ≈ 220 MeV from the MILC Collaboration. The lattice volume dependence of the method is also examined. Increasing the number of test vectors improves speedup up to a point, but storing these vectors becomes impossible in limited memory resources such as GPUs. To address storage cost, we implement a streaming singular value decomposition of the basis of test vectors on the chiral components and demonstrate a decrease in the number of fine level iterations by a factor of 1.7 for m q ≈ m crit

Iterative methods↗
DOE OSTI2016

Efficient magnetic fields for supporting toroidal plasmas

The magnetic field that supports tokamak and stellarator plasmas must be produced by coils well separated from the plasma. However, the larger the separation, the more difficult it is to produce a given magnetic field in the plasma region, so plasma configurations should be chosen that can be supported as efficiently as possible by distant coils. The efficiency of an externally generated magnetic field is a measure of the field's shaping component magnitude at the plasma compared to the magnitude near the coils; the efficiency of a plasma equilibrium can be measured using the efficiency of the required external shaping field. Counterintuitively, plasma shapes with low curvature and spectral width may have low efficiency, whereas plasma shapes with sharp edges may have high efficiency. Two precise measures of magnetic field efficiency, which correctly identify such differences in difficulty, will be examined. These measures, which can be expressed as matrices, relate the externally produced normal magnetic field on the plasma surface to the either the normal field or current on a distant control surface. A singular value decomposition (SVD) of either matrix yields an efficiency ordered basis for the magnetic field distributions. Calculations are carried out for both tokamak and stellarator cases. For axisymmetric surfaces with circular cross-section, the SVD is calculated analytically, and the range of poloidal and toroidal mode numbers that can be controlled to a given desired level is determined. If formulated properly, these efficiency measures are independent of the coordinates used to parameterize the surfaces.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2015

Characterization of the inner knot of the Crab: The site of the gamma-ray flares?

A particularly intriguing recent result from γ-ray astronomy missions is the detection of powerful flares from the Crab Nebula, which challenges the current understanding of pulsar wind nebulae and acceleration mechanisms. To search for the production site(s) of these flares, we conducted a multi-wavelength observing campaign using Keck, the Hubble Space Telescope (HST), and the Chandra X-ray Observatory. As the short timescales of the γ-ray flares (≲1 day) suggest a small emitting region, the Crab’s inner knot (about 0.6 arcsec from the pulsar) is a candidate site for such flaring. This paper describes observations of the inner knot, seeking to understand its nature and possible relationship with γ-ray flares. Using singular-value decomposition, analysis of the HST images yielded results consistent with traditional methods while substantially reducing some uncertainties. These analyses show that the knot’s intrinsic properties (especially size and brightness) are correlated with its (projected) separation from the pulsar. This characterization of the inner knot helps in constraining standard shock model parameters, under the assumption that the knot lies near the shocked surface. While the standard shock model gives good agreement in several respects, two puzzles persist: (a) the observed angular size of the knot relative to the pulsar–knot separation is much smaller than expected; and (b) the variable high degree of polarization (reported by others) is difficult to reconcile with a highly relativistic downstream flow. However, the IR–optical flux of the inner knot is marginally consistent with the shock accelerating most of the Nebula’s optical-emitting particles.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2015

Limited-memory adaptive snapshot selection for proper orthogonal decomposition

Reduced order models are useful for accelerating simulations in many-query contexts, such as optimization, uncertainty quantification, and sensitivity analysis. However, offline training of reduced order models can have prohibitively expensive memory and floating-point operation costs in high-performance computing applications, where memory per core is limited. To overcome this limitation for proper orthogonal decomposition, we propose a novel adaptive selection method for snapshots in time that limits offline training costs by selecting snapshots according an error control mechanism similar to that found in adaptive time-stepping ordinary differential equation solvers. The error estimator used in this work is related to theory bounding the approximation error in time of proper orthogonal decomposition-based reduced order models, and memory usage is minimized by computing the singular value decomposition using a single-pass incremental algorithm. Results for a viscous Burgers’ test problem demonstrate convergence in the limit as the algorithm error tolerances go to zero; in this limit, the full order model is recovered to within discretization error. The resulting method can be used on supercomputers to generate proper orthogonal decomposition-based reduced order models, or as a subroutine within hyperreduction algorithms that require taking snapshots in time, or within greedy algorithms for sampling parameter space.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Neutron diffusion calculation in heterogeneous geometry based on local/global iteration using proper orthogonal decomposition

This study newly proposes a heterogeneous core calculation method based on local/global iteration using proper orthogonal decomposition (POD). By using the singular value decomposition (SVD) and the low-rank approximation, appropriate POD bases for expanding the neutron flux can be obtained from snapshot data of the neutron flux obtained by fine mesh calculations. By projection using the POD bases, the dimension of the target equation (e.g., discretized neutron diffusion equation) can be dramatically reduced. In the proposed method, POD is effectively applied to each single assembly calculation (local calculation). Furthermore, using the local/global iteration, the effective neutron multiplication factor and the neutron flux distribution in the whole core geometry can be obtained by combining the numerical results of the local calculation for each fuel assembly and the global calculation for the whole core. As a feasibility study, the proposed method is applied to a one-dimensional heterogeneous core analysis, and the accuracy is investigated by changing the total number of POD bases. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2023

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

(U) A Linear Response Model Predicts Reactivity From a Density Profile

We tested the ability to predict the system reactivity, described by alpha, given a density profile using a simple linear system response. We generated a suite of 1-dimensional density profiles that consisted of nominal density, a discontinuity, and a decay. These profiles were prescribed a functional form and the mass was conserved in all cases. From these density profiles, we calculated the alpha value of the 3-dimensional system.We calculated a linear response function given a training set of the 1-dimensional density profiles, and the system reactivity described by alpha. We tested the robustness of the response function using the remaining test data. Our results showed very good agreement between the predicted and calculated test values, where the distribution of alpha differences was centered about zero and had a standard deviation of 0.005 gens/shake. The predicted and calculated alpha values did not significantly differ (t=-0.0009 p<0.99). We used Singular Value Decomposition (SVD) to reduce the matrix rank by retaining95% of the cumulative singular value contributions. This reduced the matrix rank by 91.7%. We generated the linear response matrix and calculated the difference between the predicted and calculated alpha values. Using the reduced order matrix, we showed good agreement between the predicted and calculated alpha values where the distribution of differences was centered near zero, the standard deviation was 0.006 gens/shake, and the statistical t-test showed good agreement (t=0.02, p<0.98). These results show a linear relationship between a series of 1-dimensional density profiles,where the mass was conserved, and the system reactivity. The next steps of this work will be to investigate the linear response using 2-dimensional density profiles.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗