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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↗

DOE OSTI2024

Random projection using random quantum circuits

The random sampling task performed by Google's Sycamore processor gave us a glimpse of the “quantum supremacy era.” This has definitely shed some light on the power of random quantum circuits in this abstract task of sampling outputs from the (pseudo)random circuits. In this paper, we explore a practical near-term use of local random quantum circuits in dimensional reduction of large low-rank data sets. We make use of the well-studied dimensionality reduction technique called the random projection method. This method has been extensively used in various applications such as image processing, logistic regression, entropy computation of low-rank matrices, etc. We prove that the matrix representations of local random quantum circuits with sufficiently shorter depths [ ∼ O ( n ) ] serve as good candidates for random projection. We demonstrate numerically that their projection abilities are not far off from the computationally expensive classical principal components analysis on MNIST and CIFAR-100 image datasets. We also benchmark the performance of quantum random projection against the commonly used classical random projection in the tasks of dimensionality reduction of image data sets and computing von Neumann entropies of large low-rank density matrices. And finally, using variational quantum singular value decomposition, we demonstrate a near-term implementation of extracting the singular vectors with dominant singular values after quantum random projecting a large low-rank matrix to lower dimensions. All such numerical experiments unequivocally demonstrate the ability of local random circuits to randomize a large Hilbert space at sufficiently shorter depths with robust retention of properties of large data sets in reduced dimensions. Published by the American Physical Society 2024

Kumaran, Keerthi (ORCID:0009000949125721)↗

NASA NTRSDate not supplied

Use of Spectral Analysis of Singular Values as a Test Metric for IMMAT Trials

One of many challenges in the implementation of multiple exciter testing is establishing a reasonable set of test metrics to measure the quality of testing. This is especially true in the application of Impedance Matched Multi-Axis Testing; in that it is possible to have very large spectral density matrices that serve as reference criteria. While there exist plotting schemes to view a spectral density matrix, it is often necessary to break the overlay of reference and test results into subsections of the matrices to get sufficient resolution to interpret the data. In addition, as one attempts to control multiple locations on a structure, implementation of classical single degree-of freedom test tolerances across all channels and associated cross spectra is simply not feasible. Hence it is challenging to evaluate overall test quality. The use of spectral views of the dominant singular values from the singular value decomposition of the spectral density matrices and metrics based upon them is proposed for establishing a set of compact metrics for evaluating test quality. A laboratory experiment will be included to demonstrate this proposed technique.

Impedance Matched Multi-Axis Testing↗

NASA NTRSDate not supplied

Use of Spectral Analysis of Singular Values as a Test Metric for Impedance Matched Multi-Axis Test Trials

Vibration Testing↗

DOE OSTI2024

Parallel Randomized Tucker Decomposition Algorithms

The Tucker tensor decomposition is a natural extension of the singular value decomposition (SVD) to multiway data. Here, we propose to accelerate Tucker tensor decomposition algorithms by using randomization and parallelization. We present two algorithms that scale to large data and many processors, significantly reduce both computation and communication cost compared to previous deterministic and randomized approaches, and obtain nearly the same approximation errors. The key idea in our algorithms is to perform randomized sketches with Kronecker-structured random matrices, which reduces computation compared to unstructured matrices and can be implemented using a fundamental tensor computational kernel. We provide probabilistic error analysis of our algorithms and implement a new parallel algorithm for the structured randomized sketch. Our experimental results demonstrate that our combination of randomization and parallelization achieves accurate Tucker decompositions much faster than alternative approaches. We observe up to a 16X speedup over the fastest deterministic parallel implementation on 3D simulation data.

Tucker decompositions↗

NASA NTRS2003

Development of an Efficient Binaural Simulation for the Analysis of Structural Acoustic Data

Applying binaural simulation techniques to structural acoustic data can be very computationally intensive as the number of discrete noise sources can be very large. Typically, Head Related Transfer Functions (HRTFs) are used to individually filter the signals from each of the sources in the acoustic field. Therefore, creating a binaural simulation implies the use of potentially hundreds of real time filters. This paper details two methods of reducing the number of real-time computations required by: (i) using the singular value decomposition (SVD) to reduce the complexity of the HRTFs by breaking them into dominant singular values and vectors and (ii) by using equivalent source reduction (ESR) to reduce the number of sources to be analyzed in real-time by replacing sources on the scale of a structural wavelength with sources on the scale of an acoustic wavelength. The ESR and SVD reduction methods can be combined to provide an estimated computation time reduction of 99.4% for the structural acoustic data tested. In addition, preliminary tests have shown that there is a 97% correlation between the results of the combined reduction methods and the results found with the current binaural simulation techniques

Johnson, Marty E.↗

DOE OSTI2024

A flexible class of priors for orthonormal matrices with basis function-specific structure

Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1993

Ionospheric Imaging From a Low Earth Orbiter Tracking GPS

Tomographic imaging of the ionosphere is examined using singular value decomposition analysis. The interdependency of the obtainable resolution, the accuracy of the solution and the noise data is explained. A simulation is illustrated where a true 2-D ionospheric structure is generated, and a tomographic inversion of the structure is carried out. Inclusion of data taken in an occultation geometry reveals the strength that is added by these data. The effect of the ionosphere on a GPS signal as viewed by a user in space is examined. Under somewhat strong solar conditions, the absolute bending of the signal is of the order of 0.01 degrees for the L1 signal; the phase advance can be as large as 90 meters.

ionosphere↗

DOE OSTI2022

Nanoscale Mapping and Defect-Assisted Manipulation of Surface Plasmon Resonances in 2D Bi 2 Te 3 /Sb 2 Te 3 In-Plane Heterostructures

Here, the Bi 2 Te 3 /Sb 2 Te 3 in-plane heterostructure is reported as a low-dimensional tunable chalcogenide well suited as plasmonic building block for the visible-UV spectral range. Electron-driven plasmon excitations of low-dimensional Bi 2 Te 3 /Sb 2 Te 3 are investigated by monochromated electron energy loss spectroscopy spectrum imaging. To resolve the nanoscale spatial distribution of various local plasmonic resonances, singular value decomposition is used to disentangle the spectral data and identify the individual spectral contributions of various corner, edge, and face modes. Furthermore, defect-plasmon interactions are investigated both for nanoscale intrinsic and thermally induced extrinsic polygonal defects (in situ sublimation). Signature of defect-induced red shift ranging from a several hundreds of millielectronvolts to a few electronvolts, broadening of various plasmon response, together with selective enhancement and significant variations in their intensity are detected. This study highlights the presence of a heterointerface and identifies defects as physical tuning pathways to modulate the plasmonic response over a broad spectral range. Finally, the experimental observations are compared qualitatively and validated with numerical simulations using the electron-driven discrete dipole approximation. Low-dimensional Bi 2 Te 3 /Sb 2 Te 3 as a less explored plasmonic system holds great promises as emerging platform for integrated plasmonics. Furthermore, introducing controlled structural defects can open the door for nanoengineering of plasmonic properties in such systems.

36 MATERIALS SCIENCE↗

DOE OSTI2022

Parametric model-order reduction for radiation transport using multi-resolution proper orthogonal decomposition

For parametric high-fidelity simulations, it is often desirable to utilize a reduced-order model (ROM) to emulate, at a reduced computational cost, parametric solutions of the governing partial differential equations (PDEs) for unseen parameter values. One commonly employed option is to utilize a data-driven, projection-based ROM supplemented with subspace identification via proper orthogonal decomposition (POD). POD discovers the ROM subspace by computing the singular value decomposition (SVD) of a set of training data from the full-order model (FOM). In streaming-dominated radiation transport simulations with localized sources, solutions often greatly vary over the spatial domain by many orders of magnitude. In such cases, machine-precision arithmetic can be insufficient to obtain an accurate SVD, resulting in a poorly performing ROM. We present a method called multiresolution POD (mrPOD) that mitigates these inaccuracies. The mrPOD method works by decomposing the spatial domain into regions and performing proper orthogonal decomposition on the training dataset separately in each region. In conclusion, mrPOD is tested on single energy group and multigroup atmospheric shielding transport problems and is shown to outperform classic POD.

42 ENGINEERING↗

DOE OSTI2020

TuckerMPI: A Parallel C++/MPI Software Package for Large-scale Data Compression via the Tucker Tensor Decomposition

With this study, our goal is compression of massive-scale grid-structured data, such as the multi-terabyte output of a high-fidelity computational simulation. For such data sets, we have developed a new software package called TuckerMPI, a parallel C++/MPI software package for compressing distributed data. The approach is based on treating the data as a tensor, i.e., a multidimensional array, and computing its truncated Tucker decomposition, a higher-order analogue to the truncated singular value decomposition of a matrix. The result is a low-rank approximation of the original tensor-structured data. Compression efficiency is achieved by detecting latent global structure within the data, which we contrast to most compression methods that are focused on local structure. In this work, we describe TuckerMPI, our implementation of the truncated Tucker decomposition, including details of the data distribution and in-memory layouts, the parallel and serial implementations of the key kernels, and analysis of the storage, communication, and computational costs. We test the software on 4.5 and 6.7 terabyte data sets distributed across 100 s of nodes (1,000 s of MPI processes), achieving compression ratios between 100 and 200,000×, which equates to 99--99.999% compression (depending on the desired accuracy) in substantially less time than it would take to even read the same dataset from a parallel file system. Moreover, we show that our method also allows for reconstruction of partial or down-sampled data on a single node, without a parallel computer so long as the reconstructed portion is small enough to fit on a single machine, e.g., in the instance of reconstructing/visualizing a single down-sampled time step or computing summary statistics. The code is available at https://gitlab.com/tensors/TuckerMPI.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

hdsullivan/ResSR

This is the official implementation of ResSR [1]. ResSR is a computationally efficient MSI-SR method that achieves high-quality reconstructions by using a closed-form spectral decomposition along with a spatial residual correction. ResSR applies singular value decomposition to identify correlations across spectral bands, uses pixel-wise computation to upsample the MSI, and then applies a residual correction process to correct the high-spatial frequency components of the upsampled bands. While ResSR is formulated as the solution to a spatially-coupled optimization problem, we use pixel-wise regularization and derive an approximate closed-form solution, resulting in a pixel-wise algorithm with a dramatic reduction in computation that achieves state-of-the-art reconstructions. [1] Duba-Sullivan, H., Reid, E. J., Voisin, S., Bouman, C. A., & Buzzard, G. T. (2024). ResSR: A Computationally Efficient Residual Approach to Super-Resolving Multispectral Images. arXiv preprint arXiv:2408.13225.

Duba-Sullivan, Haley [Oak Ridge National Laborator↗

DOE OSTI2024

Extended dynamic mode decomposition for model reduction in fluid dynamics simulations

High computational cost and storage/memory requirements of fluid dynamics simulations constrain their usefulness as a predictive tool. Reduced-order models (ROMs) provide a viable solution to this challenge by extracting the key underlying dynamics of a complex system directly from data. We investigate the efficacy and robustness of an extended dynamic mode decomposition (xDMD) algorithm in constructing ROMs of three-dimensional cardiovascular computations. Focusing on the ROMs' accuracy in representation and interpolation, we relate these metrics to the truncation rank of singular value decomposition, which underpins xDMD and other approaches to ROM construction. Our key innovation is to relate the truncation rank to the singular values of the original flow problem. This result establishes a priori guidelines for the xDMD deployment and its likely success as a means of data compression and reconstruction of the system's dynamics from dominant spatiotemporal structures present in the data.

Mechanics↗

DOE OSTI2023

Smoothing Lexis diagrams using kernel functions: A contemporary approach

Lexis diagrams are rectangular arrays of event rates indexed by age and period. Analysis of Lexis diagrams is a cornerstone of cancer surveillance research. Typically, population-based descriptive studies analyze multiple Lexis diagrams defined by sex, tumor characteristics, race/ethnicity, geographic region, etc. Inevitably the amount of information per Lexis diminishes with increasing stratification. Several methods have been proposed to smooth observed Lexis diagrams up front to clarify salient patterns and improve summary estimates of averages, gradients, and trends. In this article, we develop a novel bivariate kernel-based smoother that incorporates two key innovations. First, for any given kernel, we calculate its singular values decomposition, and select an optimal truncation point—the number of leading singular vectors to retain—based on the bias-corrected Akaike information criterion. Second, we model-average over a panel of candidate kernels with diverse shapes and bandwidths. The truncated model averaging approach is fast, automatic, has excellent performance, and provides a variance-covariance matrix that takes model selection into account. We present an in-depth case study (invasive estrogen receptor-negative breast cancer incidence among non-Hispanic white women in the United States) and simulate operating characteristics for 20 representative cancers. The truncated model averaging approach consistently outperforms any fixed kernel. Our results support the routine use of the truncated model averaging approach in descriptive studies of cancer.

60 APPLIED LIFE SCIENCES↗

NASA NTRS1988

Statistical analysis of effective singular values in matrix rank determination

A major problem in using SVD (singular-value decomposition) as a tool in determining the effective rank of a perturbed matrix is that of distinguishing between significantly small and significantly large singular values to the end, conference regions are derived for the perturbed singular values of matrices with noisy observation data. The analysis is based on the theories of perturbations of singular values and statistical significance test. Threshold bounds for perturbation due to finite-precision and i.i.d. random models are evaluated. In random models, the threshold bounds depend on the dimension of the matrix, the noisy variance, and predefined statistical level of significance. Results applied to the problem of determining the effective order of a linear autoregressive system from the approximate rank of a sample autocorrelation matrix are considered. Various numerical examples illustrating the usefulness of these bounds and comparisons to other previously known approaches are given.

Konstantinides, Konstantinos↗

NASA NTRS2002

Development of an Efficient Binaural Simulation for the Analysis of Structural Acoustic Data

Binaural or "virtual acoustic" representation has been proposed as a method of analyzing acoustic and vibroacoustic data. Unfortunately, this binaural representation can require extensive computer power to apply the Head Related Transfer Functions (HRTFs) to a large number of sources, as with a vibrating structure. This work focuses on reducing the number of real-time computations required in this binaural analysis through the use of Singular Value Decomposition (SVD) and Equivalent Source Reduction (ESR). The SVD method reduces the complexity of the HRTF computations by breaking the HRTFs into dominant singular values (and vectors). The ESR method reduces the number of sources to be analyzed in real-time computation by replacing sources on the scale of a structural wavelength with sources on the scale of an acoustic wavelength. It is shown that the effectiveness of the SVD and ESR methods improves as the complexity of the source increases. In addition, preliminary auralization tests have shown that the results from both the SVD and ESR methods are indistinguishable from the results found with the exhaustive method.

Lalime, Aimee L.↗