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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.↗

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↗

NASA NTRS1990

Improved solution for system identification equations by Epsilon-Decomposition

Matrix eigenvalue theory is used to examine the source of ill-conditioning in linear algebraic equations. This approach highlights the crucial role played by the zero and near-zero eigenvalues and corresponding eigenvectors of poorly conditioned systems. Insight gained from this approach is used to significantly improve a recently developed solution procedure called Epsilon-Decomposition (E-D). E-D is an efficient alternative to Singular Value Decomposition (SVD) for ill-conditioned systems arising in parameter estimation and system identification studies. The efficiency of the improved E-D over SVD resides in the need to only obtain the zero and near-zero eigenvalues of the coefficient matrix as opposed to all of its eigenvalues and vectors (as required by SVD). Thus, the efficiency of E-D is significant for large matrices with small rank deficiency.

Ojalvo, Irving U.↗

NASA NTRS2009

Regression Model Optimization for the Analysis of Experimental Data

A candidate math model search algorithm was developed at Ames Research Center that determines a recommended math model for the multivariate regression analysis of experimental data. The search algorithm is applicable to classical regression analysis problems as well as wind tunnel strain gage balance calibration analysis applications. The algorithm compares the predictive capability of different regression models using the standard deviation of the PRESS residuals of the responses as a search metric. This search metric is minimized during the search. Singular value decomposition is used during the search to reject math models that lead to a singular solution of the regression analysis problem. Two threshold dependent constraints are also applied. The first constraint rejects math models with insignificant terms. The second constraint rejects math models with near-linear dependencies between terms. The math term hierarchy rule may also be applied as an optional constraint during or after the candidate math model search. The final term selection of the recommended math model depends on the regressor and response values of the data set, the user s function class combination choice, the user s constraint selections, and the result of the search metric minimization. A frequently used regression analysis example from the literature is used to illustrate the application of the search algorithm to experimental data.

Ulbrich, N.↗

NASA NTRS1992

Large-scale sparse singular value computations

Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented. Lanczos and subspace iteration-based methods for determining several of the largest singular triplets (singular values and corresponding left and right-singular vectors) for sparse matrices arising from two practical applications: information retrieval and seismic reflection tomography are emphasized. The target architectures for implementations are the CRAY-2S/4-128 and Alliant FX/80. The sparse SVD problem is well motivated by recent information-retrieval techniques in which dominant singular values and their corresponding singular vectors of large sparse term-document matrices are desired, and by nonlinear inverse problems from seismic tomography applications which require approximate pseudo-inverses of large sparse Jacobian matrices.

Berry, Michael W.↗

NASA NTRS2003

An Efficient and Robust Singular Value Method for Star Pattern Recognition and Attitude Determination

A new star pattern recognition method is developed using singular value decomposition of a measured unit column vector matrix in a measurement frame and the corresponding cataloged vector matrix in a reference frame. It is shown that singular values and right singular vectors are invariant with respect to coordinate transformation and robust under uncertainty. One advantage of singular value comparison is that a pairing process for individual measured and cataloged stars is not necessary, and the attitude estimation and pattern recognition process are not separated. An associated method for mission catalog design is introduced and simulation results are presented.

Juang, Jer-Nan↗

DOE OSTI2021

Nonlinear Matrix Approximation with Radial Basis Function Components

We introduce and investigate matrix approximation by decomposition into a sum of radial basis function (RBF) components. An RBF component is a generalization of the outer product between a pair of vectors, where an RBF function replaces the scalar multiplication between individual vector elements. Even though the RBF functions are positive definite, the summation across components is not restricted to convex combinations and allows us to compute the decomposition for any real matrix that is not necessarily symmetric or positive definite. We formulate the problem of seeking such a decomposition as an optimization problem with a nonlinear and non-convex loss function. Several modern versions of the gradient descent method, including their scalable stochastic counterparts, are used to solve this problem. We provide extensive empirical evidence of the effectiveness of the RBF decomposition and that of the gradient-based fitting algorithm. While being conceptually motivated by singular value decomposition (SVD), our proposed nonlinear counterpart outperforms SVD by drastically reducing the memory required to approximate a data matrix with the same L2 error for a wide range of matrix types. For example, it leads to 2 to 6 times memory save for Gaussian noise, graph adjacency matrices, and kernel matrices. Moreover, this proximity-based decomposition can offer additional interpretability in applications that involve, e.g., capturing the inner low-dimensional structure of the data, retaining graph connectivity structure, and preserving the acutance of images.

Rebrova, Elizaveta↗

NASA NTRS1983

Accuracy of earth albedo estimates from wide-angle radiation measurements

The instantaneous resolution and accuracy that can be expected from two inversion techniques to be used in the Earth Radiation Budget Experiment (ERBE) are examined. It is shown that measurement errors are magnified by the numerical filter and that this magnification is a function of the magnitude of the inversion factor. By using singular value decomposition, these magnitudes can be reduced to improve the estimates. A simulation of the estimation process using an albedo field derived from scanning radiometer data shows that retaining 6 of the 13 singular values gives the best results. If there are no bidirectional model errors the medium-field-of-view data give the best estimates. However, when bidirectional model errors are considered, the wide-field-of-view measurements give better estimates since they are less sensitive to these errors.

Green, R. N.↗

NASA NTRS1984

Optical systolic solutions of linear algebraic equations

The philosophy and data encoding possible in systolic array optical processor (SAOP) were reviewed. The multitude of linear algebraic operations achievable on this architecture is examined. These operations include such linear algebraic algorithms as: matrix-decomposition, direct and indirect solutions, implicit and explicit methods for partial differential equations, eigenvalue and eigenvector calculations, and singular value decomposition. This architecture can be utilized to realize general techniques for solving matrix linear and nonlinear algebraic equations, least mean square error solutions, FIR filters, and nested-loop algorithms for control engineering applications. The data flow and pipelining of operations, design of parallel algorithms and flexible architectures, application of these architectures to computationally intensive physical problems, error source modeling of optical processors, and matching of the computational needs of practical engineering problems to the capabilities of optical processors are emphasized.

Neuman, C. P.↗

DOE OSTI2020

Learning compact physics‐aware delayed photocurrent models using dynamic mode decomposition

Abstract Radiation‐induced photocurrent in semiconductor devices can be simulated using complex physics‐based models, which are accurate, but computationally expensive. This presents a challenge for implementing device characteristics in high‐level circuit simulations where it is computationally infeasible to evaluate detailed models for multiple individual circuit elements. In this work we demonstrate a procedure for learning compact delayed photocurrent models that are efficient enough to implement in large‐scale circuit simulations, but remain faithful to the underlying physics. Our approach utilizes dynamic mode decomposition (DMD), a system identification technique for learning reduced‐order discrete‐time dynamical systems from time series data based on singular value decomposition. To obtain physics‐aware device models, we simulate the excess carrier density induced by radiation pulses by solving numerically the ambipolar diffusion equation, then use the simulated internal state as training data for the DMD algorithm. Our results show that the significantly reduced‐order delayed photocurrent models obtained via this method accurately approximate the dynamics of the internal excess carrier density—which can be used to calculate the induced current at the device boundaries—while remaining compact enough to incorporate into larger circuit simulations.

Hanson, Joshua↗

DOE OSTI2025

Structural response reconstruction using a system-equivalent singular vector basis

Here, this paper develops a novel method for reconstructing the full-field response of structural dynamic systems using sparse measurements. The singular value decomposition is applied to a frequency response matrix relating the structural response to physical loads, base motion, or modal loads. The left singular vectors form a non-physical reduced basis that can be used for response reconstruction with far fewer sensors than existing methods. The contributions of the singular vectors to measured response are termed singular-vector loads (SVLs) and are used in a regularized Bayesian framework to generate full-field response estimates and confidence intervals. The reconstruction framework is applicable to the estimation of single data records and power spectral densities from multiple records. Reconstruction is successfully performed in configurations where the number of SVLs to identify is less than, equal to, and greater than the number of sensors used for reconstruction. In a simulation featuring a seismically excited shear structure, SVL reconstruction significantly outperforms modal FRF-based reconstruction and successfully estimates full-field responses with as few as two uniaxial accelerometers. SVL reconstruction is further verified in a simulation featuring an acoustically excited cylinder. Finally, response reconstruction and uncertainty quantification are performed on an experimental structure with three shaker inputs and 27 triaxial accelerometer outputs.

42 ENGINEERING↗

DOE OSTI2016

S-matrix decomposition, natural reaction channels, and the quantum transition state approach to reactive scattering

A new approach for the quantum-state resolved analysis of polyatomic reactions is introduced. Based on the singular value decomposition of the S-matrix, energy-dependent natural reaction channels and natural reaction probabilities are defined. It is shown that the natural reaction probabilities are equal to the eigenvalues of the reaction probability operator [U. Manthe and W. H. Miller, J. Chem. Phys. 99, 3411 (1993)]. Consequently, the natural reaction channels can be interpreted as uniquely defined pathways through the transition state of the reaction. The analysis can efficiently be combined with reactive scattering calculations based on the propagation of thermal flux eigenstates. In contrast to a decomposition based straightforwardly on thermal flux eigenstates, it does not depend on the choice of the dividing surface separating reactants from products. The new approach is illustrated studying a prototypical example, the H + CH{sub 4} → H{sub 2} + CH{sub 3} reaction. The natural reaction probabilities and the contributions of the different vibrational states of the methyl product to the natural reaction channels are calculated and discussed. The relation between the thermal flux eigenstates and the natural reaction channels is studied in detail.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

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 OSTI2014

Classification of subsurface objects using singular values derived from signal frames

The classification system represents a detected object with a feature vector derived from the return signals acquired by an array of N transceivers operating in multistatic mode. The classification system generates the feature vector by transforming the real-valued return signals into complex-valued spectra, using, for example, a Fast Fourier Transform. The classification system then generates a feature vector of singular values for each user-designated spectral sub-band by applying a singular value decomposition (SVD) to the N.times.N square complex-valued matrix formed from sub-band samples associated with all possible transmitter-receiver pairs. The resulting feature vector of singular values may be transformed into a feature vector of singular value likelihoods and then subjected to a multi-category linear or neural network classifier for object classification.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2022

Parametric dynamic mode decomposition for reduced order modeling

Dynamic Mode Decomposition (DMD) is a model-order reduction approach, whereby spatial modes of fixed temporal frequencies are extracted from numerical or experimental data sets. The DMD low-rank or reduced operator is typically obtained by singular value decomposition of the temporal data sets. For parameter-dependent models, as found in many multi-query applications such as uncertainty quantification or design optimization, the only parametric DMD technique developed was a stacked approach, with data sets at multiple parameter values were aggregated together, increasing the computational work needed to devise low-rank dynamical reduced-order models. Here in this paper, we present two novel approach to carry out parametric DMD: one based on the interpolation of the reduced-order DMD eigen-pair and the other based on the interpolation of the reduced DMD (Koopman) operator. Numerical results are presented for diffusion-dominated nonlinear dynamical problems, including a multiphysics radiative transfer example. All three parametric DMD approaches are compared.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1998

Microwave Imaging on Metal Objects

This final report for the project discusses the attempts to model, using different methods, microwave image reconstruction. Maximum Entropy Method was not successful. Attempts to use Singular Value Decomposition (SVD) got some good results after initial failure. SVD is based upon a theory of linear algebra, to the effect that any M X N Matrix A whose number of rows M is greater than or equal to its number of columns, N can be written as the product of an M X N column-orthogonal matrix U, an N X N diagonal Matrix, W, with m positive or zero elements (the singular values) and the transposition of an N X N orthogonal matrix V. In microwave imaging, the scattered fields can be expressed by the induced current distribution. The SVD method required more contiguous computer memory than was available. Work was also done on the Conjugate Gradient Method (CGM), which didn't work well when tried earlier. It was found that separation of the imaginary part and the real part during calculation may work. This work was considered incomplete as of the end of the grant period.

Tolliver, C. L.↗

DOE OSTI2017

Using dynamic mode decomposition for real-time background/foreground separation in video

The technique of dynamic mode decomposition (DMD) is disclosed herein for the purpose of robustly separating video frames into background (low-rank) and foreground (sparse) components in real-time. Foreground/background separation is achieved at the computational cost of just one singular value decomposition (SVD) and one linear equation solve, thus producing results orders of magnitude faster than robust principal component analysis (RPCA). Additional techniques, including techniques for analyzing the video for multi-resolution time-scale components, and techniques for reusing computations to allow processing of streaming video in real time, are also described herein.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

A Parametric, Data-Driven, Non-Intrusive Reduced-Order Model Framework for Crystal Plasticity Simulations of Voids

The influence of the internal structure at micrometer length scales on the deformation of polycrystalline materials can be effectively captured using crystal plasticity finite element methods (CPFEM). However, the complexity and nonlinearity of the deformation equations CPFEM solves demand significant computational power and resources to achieve accurate predictions, limiting its broader application. To address this challenge, we have identified a reduced-order representation of the complex data in order to establish a computationally efficient reduced-order models (ROM) and drastically reduce the computational expense of CPFEM. Specifically, in this work, we developed a parametric, data-driven, and non-intrusive ROM framework for CPFEM using proper orthogonal decomposition (POD) and sparse variational Gaussian process (SVGP) regression for single-crystal microstructures under tensile loading conditions. The developed protocol enables one to compress field into a latent/low-dimensional space described by principal component analysis (PCA) via the singular value decomposition (SVD) algorithm. As a result, the high-dimensional data are reduced to a significantly smaller amount of dimensions with POD bases and POD coefficients. Furthermore, we deployed an ensemble of SVGPs—extended from the classical Gaussian process (GP) regression for scalability and handling big data—in a massively parallel manner to train and predict latent POD coefficients using known POD bases from a set of previously obtained simulations results. Lastly, using the predicted POD coefficients, we reconstructed the full-field results and showed reasonable agreement compared with the true values obtained from running CPFEM. The developed framework is validated with a set of CPFEM simulations of a single embedded void in single-crystal aluminum alloy. While the framework is broadly applicable, this work specifically focuses on single-crystal microstructures, a single load case (e.g., tensile), and a specific void geometry (spherical).

Anisotropy↗