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All sourcesNASA NTRSDOE OSTI
DOE OSTI2018

A Flexible-blocking Based Approach for Performance Tuning of Matrix Multiplication Routines for Large Matrices with Edge Cases

Efficient and scalable matrix operations are being highly demanding in the recent era of Machine Learning, Deep Learning, and Big Data Analytics. The two commonly used matrix-matrix operations in the Basic Linear Algebra Subprograms (BLAS) specification are General Matrix-Matrix multiplication (GEMM) and Symmetric Rank-k update (SYRK). The SYRK routine is a specialization of the GEMM routine, where half of the multiplications are skipped as the resultant matrix is known to be symmetric. Fortunately, several linear algebra libraries implement these BLAS routines quite efficiently. The libraries usually partition the input matrices into blocks and place them in processor caches, thus improving performance by leveraging the caches. However, the contemporary libraries are highly optimized for squarish matrices, but the performance degrades significantly for the matrices with edge case (strictly thin or strictly fat shapes) in the multicore machine. The primary reason is that the current state-of-the-art libraries make fixed block shapes based on a processor architecture, and do not consider the shape of the input matrices. In this paper, we propose a new blocking approach, we name it Flexible-blocking, to mitigate the scalability issues. In contrast to the contemporary libraries, our approach formulates the blocks of the input matrices based on the shapes of the matrices as well as the number of threads used in the implementation. Our proposed technique shows noticeable performance improvement on multicore shared-memory machines for the edge case matrices.

Hossain, Md Mosharaf↗
NASA NTRS1991

The asymptotic spectra of banded Toeplitz and quasi-Toeplitz matrices

Toeplitz matrices occur in many mathematical, as well as, scientific and engineering investigations. This paper considers the spectra of banded Toeplitz and quasi-Toeplitz matrices with emphasis on non-normal matrices of arbitrarily large order and relatively small bandwidth. These are the type of matrices that appear in the investigation of stability and convergence of difference approximations to partial differential equations. Quasi-Toeplitz matrices are the result of non-Dirichlet boundary conditions for the difference approximations. The eigenvalue problem for a banded Toeplitz or quasi-Toeplitz matrix of large order is, in general, analytically intractable and (for non-normal matrices) numerically unreliable. An asymptotic (matrix order approaches infinity) approach partitions the eigenvalue analysis of a quasi-Toeplitz matrix into two parts, namely the analysis for the boundary condition independent spectrum and the analysis for the boundary condition dependent spectrum. The boundary condition independent spectrum is the same as the pure Toeplitz matrix spectrum. Algorithms for computing both parts of the spectrum are presented. Examples are used to demonstrate the utility of the algorithms, to present some interesting spectra, and to point out some of the numerical difficulties encountered when conventional matrix eigenvalue routines are employed for non-normal matrices of large order. The analysis for the Toeplitz spectrum also leads to a diagonal similarity transformation that improves conventional numerical eigenvalue computations. Finally, the algorithm for the asymptotic spectrum is extended to the Toeplitz generalized eigenvalue problem which occurs, for example, in the stability of Pade type difference approximations to differential equations.

Beam, Richard M.↗
DOE OSTI2024

Symmetry-Based Structured Matrices for Efficient Approximately Equivariant Networks

There has been much recent interest in designing symmetry-aware neural networks (NNs) exhibiting relaxed equivariance. Such NNs aim to interpolate between being exactly equivariant and being fully flexible, affording consistent performance benefits. In a separate line of work, certain structured parameter matrices -- those with displacement structure, characterized by low displacement rank (LDR) -- have been used to design small-footprint NNs. Displacement structure enables fast function and gradient evaluation, but permits accurate approximations via compression primarily to classical convolutional neural networks (CNNs). In this work, we propose a general framework -- based on a novel construction of symmetry-based structured matrices -- to build approximately equivariant NNs with significantly reduced parameter counts. Our framework integrates the two aforementioned lines of work via the use of so-called Group Matrices (GMs), a forgotten precursor to the modern notion of regular representations of finite groups. GMs allow the design of structured matrices -- resembling LDR matrices -- which generalize the linear operations of a classical CNN from cyclic groups to general finite groups and their homogeneous spaces. We show that GMs can be employed to extend all the elementary operations of CNNs to general discrete groups. Further, the theory of structured matrices based on GMs provides a generalization of LDR theory focussed on matrices with cyclic structure, providing a tool for implementing approximate equivariance for discrete groups. We test GM-based architectures on a variety of tasks in the presence of relaxed symmetry. We report that our framework consistently performs competitively compared to approximately equivariant NNs, and other structured matrix-based compression frameworks, sometimes with a one or two orders of magnitude lower parameter count.

Samudre, Ashwin↗
DOE OSTI2020

Measuring Intelligence with the Sandia Matrices: Psychometric Review and Recommendations for Free Raven-Like Item Sets

The Sandia Matrices are a free alternative to the Raven’s Progressive Matrices (RPMs). This study offers a psychometric review of Sandia Matrices items focused on two of the most commonly investigated issues regarding the RPMs: (a) dimensionality and (b) sex differences. Model-data fit of three alternative factor structures are compared using confirmatory multidimensional item response theory (IRT) analyses, and measurement equivalence analyses are conducted to evaluate potential sex bias. Although results are somewhat inconclusive regarding factor structure, results do not show evidence of bias or mean differences by sex. Finally, although the Sandia Matrices software can generate infinite items, editing and validating items may be infeasible for many researchers. Further, to aide implementation of the Sandia Matrices, we provide scoring materials for two brief static tests and a computer adaptive test. Implications and suggestions for future research using the Sandia Matrices are discussed.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2018

On matrices having equal spectral radius and some matrix norm

A class of matrices, called ∞-radial matrices (1-radial matrices) whose spectral radius equals their ∞-norm (1-norm), is proposed in this paper and some theoretical results are established to give several necessary and sufficient conditions of this class of matrices. Then, some properties of this class of matrices are presented. Finally, some applications of this class of matrices in the linear discrete dynamic systems are studied such that the zero solution of the linear discrete dynamic systems is asymptotically stable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Soft matrices downregulate FAK activity to promote growth of tumor-repopulating cells

Tumor-repopulating cells (TRCs) are a tumorigenic sub-population of cancer cells that drives tumorigenesis. We have recently reported that soft fibrin matrices maintain TRC growth by promoting histone 3 lysine 9 (H3K9) demethylation and Sox2 expression and that Cdc42 expression influences H3K9 methylation. However, the underlying mechanisms of how soft matrices induce H3K9 demethylation remain elusive. Here we find that TRCs exhibit lower focal adhesion kinase (FAK) and H3K9 methylation levels in soft fibrin matrices than control melanoma cells on 2D rigid substrates. Silencing FAK in control melanoma cells decreases H3K9 methylation, whereas overexpressing FAK in tumor-repopulating cells enhances H3K9 methylation. Overexpressing Cdc42 or RhoA in the presence of FAK knockdown restores H3K9 methylation levels. Importantly, silencing FAK, Cdc42, or RhoA promotes Sox2 expression and proliferation of control melanoma cells in stiff fibrin matrices, whereas overexpressing each gene suppresses Sox2 expression and reduces growth of TRCs in soft but not in stiff fibrin matrices. Our findings suggest that low FAK mediated by soft fibrin matrices downregulates H3K9 methylation through reduction of Cdc42 and RhoA and promotes TRC growth. - Highlights: • FAK activity, mRNA and protein are low in highly tumorigenic cells. • FAK is tightly associated with H3K9 methylation and negatively related to the growth of tumor-repopulating cells. • Low FAK promotes rather than blocks the growth of highly tumorigenic cells. • Matrix stiffness (rigidity) is a key player in regulating the response of the melanoma cells to FAK modulation.

60 APPLIED LIFE SCIENCES↗
DOE OSTI2016

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

In this paper we propose versions of the associative Yang-Baxter equation and higher order R-matrix identities which can be applied to quantum dynamical R-matrices. As is known quantum non-dynamical R-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum Yang-Baxter equation and a set of identities useful for different applications in integrable systems. The dynamical R-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is described by the IRF (interaction-round-a-face)-Vertex transformation. An alternative approach to quantum (semi-)dynamical R-matrices and related quantum algebras was suggested by Arutyunov, Chekhov, and Frolov (ACF) in their study of the quantum Ruijsenaars-Schneider model. The purpose of this paper is twofold. First, we prove that the ACF elliptic R-matrix satisfies the associative Yang-Baxter equation with shifted spectral parameters. Second, we directly prove a simple relation of the IRF-Vertex type between the Baxter-Belavin and the ACF elliptic R-matrices predicted previously by Avan and Rollet. It provides the higher order R-matrix identities and an explanation of the obtained equations through those for non-dynamical R-matrices. As a by-product we also get an interpretation of the intertwining transformation as matrix extension of scalar theta function likewise R-matrix is interpreted as matrix extension of the Kronecker function. Relations to the Gervais-Neveu-Felder equation and identities for the Felder’s elliptic R-matrix are also discussed.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Raw tensiometer soil matric potential data, from Oct 2017-Jun 2020 in Manaus, Brazil

This data package contains raw output from two Campbell Scientific CR1000 data loggers each connected to three UMS T4e tensiometer sensors in Manaus, Brazil. Tensiometers at each data logger location were installed to depths of 10 cm, 20 cm and 50 cm below the ground surface for the purpose of measuring soil matric potential at each depth. “Station 3” tensiometers were installed at approximately 2°36'31.68"S, 60°12'34.70"W. “Station 4” tensiometers were installed at approximately 2°36'32.46"S, 60°12'34.92"W. The files “tensio3.edit.202109.csv” and “tensio4.edit.202109.csv” contain raw data (with invalid data removed). A text editor will be required to view the raw files in .csv (comma-separated values) format. Data processing was limited to: 1) Editing column headers for clarity; 2) Applying a correction that converts voltage read by the data loggers to pressure units (kPa); 3) removing all data greater than 50 kPa and less than -115 kPa; 4) Adding columns entitled “Caution.10cm”, “Caution.20cm” and “Caution.50cm” that contains values of either “NA” or “Caution”, the latter of which is applied when data exceeds the manufacturer reported data validity range of less than -85 kPa to suggest use of these data with caution; 5) Adding columns entitled “Invalid.10cm”, “Invalid.20cm”, and “Invalid.50cm” that contains values of either “NA” or “Invalid”, the latter of which is applied when data accuracy is deemed compromised due to the introduction of air bubbles into the instrument. Data marked “Invalid” may still indicate general trends of wet/high matric potential and dry/low matric potential, but values are inaccurate and dry/low matric potential values are likely not low enough. Wet/high matric potential values are likely to be more accurate than dry/low matric potential values; 6) Data format changed from .dat to .csv. This research was performed as a part of the NGEE Tropics project, which aims to advance model predictions of tropical forest carbon cycle responses to a changing climate over the 21st Century.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS2019

Development and Characterization of Hot-Pressed Matrices for Engineered Ceramic Matrix Composites (E-CMCs)

The present research effort was undertaken to develop a new generation of SiC fiber-reinforced engineered ceramic matrix composites (E-CMCs). In contrast to traditional CMCs with a brittle SiC matrix, an E-CMC is designed to consist of a matrix engineered to possess sufficient high temperature plasticity to minimize crack propagation, relatively high fracture toughness, and self-healing capabilities to prevent oxygen ingress to the BN-coated fibers through surface-connected cracks. The present paper discusses the bend strength, isothermal oxidation, microstructures and self-healing properties of several silicide-behaved engineered matrices. Based on the oxidation tests, where it was observed that some of the matrices exhibited either catastrophic oxidation ("pesting") or spalling of the oxide scale, two engineered matrices, CrSi2/SiC/Si3N4 and CrMoSi/SiC/Si3N4, were down-selected for further investigation. Four-point bend tests were conducted on these two engineered matrices between room temperature and 1698 degrees Kelvin. Although these matrices were brittle at low temperatures, it was observed that the bend strengths and bend ductility increased at high temperatures as the silicide particles became more ductile, which was qualitatively consistent with the theoretically expected behavior that crack blunting at these particles should increase the matrix strength. Additional studies were conducted to study the effects of different additives on the self-healing properties of the engineered matrices, which helped to identify the most effective additives.

Composite Materials↗
DOE OSTI2014

Time series, correlation matrices and random matrix models

In this set of five lectures the authors have presented techniques to analyze open classical and quantum systems using correlation matrices. For diverse reasons we shall see that random matrices play an important role to describe a null hypothesis or a minimum information hypothesis for the description of a quantum system or subsystem. In the former case various forms of correlation matrices of time series associated with the classical observables of some system. The fact that such series are necessarily finite, inevitably introduces noise and this finite time influence lead to a random or stochastic component in these time series. By consequence random correlation matrices have a random component, and corresponding ensembles are used. In the latter we use random matrices to describe high temperature environment or uncontrolled perturbations, ensembles of differing chaotic systems etc. The common theme of the lectures is thus the importance of random matrix theory in a wide range of fields in and around physics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Oligo p-Phenylenevinylene Derivatives as Electron Transfer Matrices for UV-MALDI

Phenylenevinylene oligomers (PVs) have outstanding photophysical characteristics for applications in the growing field of organic electronics. Yet, PVs are also versatile molecules, the optical and physicochemical properties of which can be tuned by manipulation of their structure. We report the synthesis, photophysical, and MS characterization of eight PV derivatives with potential value as electron transfer (ET) matrices for UV-MALDI. UV-vis analysis show the presence of strong characteristic absorption bands in the UV region and molar absorptivities at 355 nm similar or higher than those of traditional proton (CHCA) and ET (DCTB) MALDI matrices. Most of the PVs exhibit non-radiative quantum yields (φ) above 0.5, indicating favorable thermal decay. Ionization potential values (IP) for PVs, calculated by the Electron Propagator Theory (EPT), range from 6.88 to 7.96 eV, making these oligomers good candidates as matrices for ET ionization. LDI analysis of PVs shows only the presence of radical cations (M{sup +.}) in positive ion mode and absence of clusters, adducts, or protonated species; in addition, M{sup +.} threshold energies for PVs are lower than for DCTB. We also tested the performance of four selected PVs as ET MALDI matrices for analytes ranging from porphyrins and phthalocyanines to polyaromatic compounds. Two of the four PVs show S/N enhancement of 1961% to 304% in comparison to LDI, and laser energy thresholds from 0.17 μJ to 0.47 μJ compared to 0.58 μJ for DCTB. The use of PV matrices also results in lower LODs (low fmol range) whereas LDI LODs range from pmol to nmol. .

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2021

Selective penetration behavior of microgels in superpermeable channels and reservoir matrices

Gel treatment is an effective way to attack excessive water production in many mature oilfields around the world. Selective penetration is desired for successful gel treatments. That is, gel materials should easily penetrate the target zones (i.e., channeling features such as superpermeable channels) without entering/damaging the nontarget zones (i.e., reservoir matrices or oil zones). This study revealed that presence of threshold penetration pressure (ΔP th ) was responsible for selective penetration behavior of tested microgels. The concept of ΔP th was utilized to figure out favorable working conditions for effective gel treatments. Microgel dispersions were injected into superpermeable (super-k) sandpacks (mimicking super-k channels in reservoirs, 60–221 darcies), heterogeneous models with super-k channels (79–230 darcies), and sandstone cores (mimicking reservoir matrices, 50–5000 md). The results demonstrated that a minimum differential driving pressure (i.e., threshold penetration pressure, ΔP th ) was required to push microgel particles to penetrate channels or matrices. The critical penetration behavior was closely related to the particle/pore size ratio. Low ΔP th at smaller particle/pore ratios was beneficial to allow easy penetration of gel materials into the channeling zones. On the contrary, high ΔPth at larger particle/pore ratios was desirable to prevent gel materials from massively invading and damaging the matrices. Instead, the gel particles accumulated at the inlet surface, and a gel cake was gradually formed. The cake further prevented the invasion of the gels. The cake could be removed by chemical breakers to resume the injectivity/productivity of the matrices. Correlations were developed to describe the relationship between ΔP th and particle/pore ratio. A distinct transition was identified at the particle/pore ratio of about 3. This research could help identify the favorable conditions to achieve successful gel treatments. In an effective conformance treatment, the particle/pore ratio in the channel should be sufficiently low to allow easy penetration of gel materials into the channel (e.g., particle/pore ratio<2 in this study). Meanwhile, the particle/pore ratio in the matrix should be large enough to support a high ΔP th and thus prevent massive gel invasion into the matrix. This study advances the current pore scale studies (a single particle passing through a single channel) to Darcy-scale characterization.

02 PETROLEUM↗
DOE OSTI2023

Generating accurate density matrices on the tangent space of a Grassmann manifold

Interpolating a density matrix from a set of known density matrices is not a trivial task. This is because a linear combination of density matrices does not necessarily correspond to another density matrix. In this Communication, density matrices are examined as objects of a Grassmann manifold. Although this manifold is not a vector space, its tangent space is a vector space. As a result, one can map the density matrices on this manifold to their corresponding vectors in the tangent space and then perform interpolations on that tangent space. The resulting interpolated vector can be mapped back to the Grassmann manifold, which can then be utilized (1) as an optimal initial guess for a self-consistent field (SCF) calculation or (2) to derive energy directly without time-consuming SCF iterations. Such a promising approach is denoted as Grassmann interpolation (G-Int). The hydrogen molecule has been used to illustrate that the described interpolated method in this work preserves the essential attributes of a density matrix. For phosphorus mononitride and ferrocene, it was demonstrated numerically that reference points for the definition of the corresponding tangent spaces can be chosen arbitrarily. In addition, the interpolated density matrices provide a superior and essentially converged initial guess for an SCF calculation to make the SCF procedure itself unnecessary. Finally, this accurate, efficient, robust, and systematically improved G-Int strategy has been used for the first time to generate highly accurate potential energy surfaces with fine details for the difficult case, ferrocene.

Chemistry↗
NASA NTRS1989

Efficient computer algebra algorithms for polynomial matrices in control design

The theory of polynomial matrices plays a key role in the design and analysis of multi-input multi-output control and communications systems using frequency domain methods. Examples include coprime factorizations of transfer functions, cannonical realizations from matrix fraction descriptions, and the transfer function design of feedback compensators. Typically, such problems abstract in a natural way to the need to solve systems of Diophantine equations or systems of linear equations over polynomials. These and other problems involving polynomial matrices can in turn be reduced to polynomial matrix triangularization procedures, a result which is not surprising given the importance of matrix triangularization techniques in numerical linear algebra. Matrices with entries from a field and Gaussian elimination play a fundamental role in understanding the triangularization process. In the case of polynomial matrices, matrices with entries from a ring for which Gaussian elimination is not defined and triangularization is accomplished by what is quite properly called Euclidean elimination. Unfortunately, the numerical stability and sensitivity issues which accompany floating point approaches to Euclidean elimination are not very well understood. New algorithms are presented which circumvent entirely such numerical issues through the use of exact, symbolic methods in computer algebra. The use of such error-free algorithms guarantees that the results are accurate to within the precision of the model data--the best that can be hoped for. Care must be taken in the design of such algorithms due to the phenomenon of intermediate expressions swell.

Baras, J. S.↗
DOE OSTI2022

On the effect of scalar flux weighing of linearly anisotropic scattering matrices in few-group transport calculations

The majority of the codes available for homogenized group constant generation for deterministic transport calculations apply the approximation of scalar flux weighting during energy group condensation of higher-order anisotropic scattering matrices. In this paper, we discuss the effect of scalar flux weighting of the linearly anisotropic scattering matrices in the frame of S P{sub 3} and S{sub 12} calculations performed for a two-dimensional VVER-440 reactor benchmark. To compare group constants generated for two neutron energy groups, an infinite pin cell was homogenized with Serpent 2 and ERANOS ECCO. Serpent 2 applies scalar flux, while ERANOS ECCO performs current weighting of the linearly anisotropic scattering matrices during energy group condensation. For analyzing the effect of the various weighting options, three simple reactor models were built assuming different core sizes using standard rectangular assemblies with 15*15 fuel pins. Diffusion, SP{sub 3} and S{sub 12} calculations were performed for the 3 models using group constants generated with Serpent 2 and ERANOS ECCO. The effect of scalar flux weighting of linearly anisotropic scattering matrices is shown by comparing the decrease in reactivity due to the decreased reactor size, as well as the assembly power distribution to reference results obtained with Serpent 2 Monte Carlo calculations. Neglecting higher than linearly anisotropic scattering and indirect application of diffusion coefficients in higher-order transport calculations is advised if angular flux-moment spectra weighted higher-order scattering matrices cannot be generated. (authors)

22 GENERAL STUDIES OF NUCLEAR REACTORS↗
DOE OSTI2018

Factorization of generalized γ-generating matrices

The class of γ-generating matrices and its subclasses of regular and singular γ-generating matrices were introduced by D. Z. Arov in [8], where it was shown that every γ-generating matrix admits an essentially unique regular–singular factorization. The class of generalized γ-generating matrices was introduced in [14, 20]. In the present paper, subclasses of singular and regular generalized –generating matrices are introduced and studied. As the main result, a theorem of existence of the regular–singular factorization for a rational generalized γ-generating matrix is proved.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2016

Synchronous correlation matrices and Connes’ embedding conjecture

In the work of Paulsen et al. [J. Funct. Anal. (in press); preprint arXiv:1407.6918], the concept of synchronous quantum correlation matrices was introduced and these were shown to correspond to traces on certain C*-algebras. In particular, synchronous correlation matrices arose in their study of various versions of quantum chromatic numbers of graphs and other quantum versions of graph theoretic parameters. In this paper, we develop these ideas further, focusing on the relations between synchronous correlation matrices and microstates. We prove that Connes’ embedding conjecture is equivalent to the equality of two families of synchronous quantum correlation matrices. We prove that if Connes’ embedding conjecture has a positive answer, then the tracial rank and projective rank are equal for every graph. We then apply these results to more general non-local games.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

Invariant functions for the Lyapunov exponents of random matrices

A new approach to the study of Lyapunov exponents of random matrices is presented. We prove that any family of nonnegative (dxd)-matrices has a continuous concave invariant functional on R{sup d}{sub +}. Under some standard assumptions on the matrices, this functional is strictly positive, and the coefficient corresponding to it is equal to the largest Lyapunov exponent. As a corollary we obtain asymptotics for the expected value of the logarithm of norms of matrix products and of their spectral radii. Another corollary gives new upper and lower bounds for the Lyapunov exponent, and an algorithm for computing it for families of nonnegative matrices. We consider possible extensions of our results to general nonnegative matrix families and present several applications and examples. Bibliography: 29 titles.

97 MATHEMATICS AND COMPUTING↗