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At least 793 records · Page 44
All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

SparseLU, A Novel Algorithm and Math Library for Sparse LU Factorization

Decomposing sparse matrices into lower and upper triangular matrices (sparse LU factorization) is a key operation in many computational scientific applications. We developed SparseLU, a sparse linear algebra library that implements a new algorithm for LU factorization on general sparse matrices. The new algorithm divides the input matrix into tiles to which OpenMP tasks are created for factorization computation, where only tiles that contain nonzero elements are computed. For comparative performance analysis, we used the reference library SuperLU. Testing was performed on synthetically generated matrices which replicate the conditions of the real-world matrices. SparseLU is able to reach a mean speedup of ~29× compared to SuperLU.

Valero Lara, Pedro↗
DOE OSTI2011

Constraints on SU(2) Circled-Times SU(2) invariant polynomials for a pair of entangled qubits

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) Circled-Plus SU(2) group on the space of density matrices P{sub +}. Since elements of P{sub +} are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, P{sub +} is an element of R{sup 15}. We define P{sub +} explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) Circled-Plus SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) Circled-Plus SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

A Class of Sparse Johnson–Lindenstrauss Transforms and Analysis of their Extreme Singular Values

The Johnson–Lindenstrauss (JL) lemma is a powerful tool for dimensionality reduction in modern algorithm design. The lemma states that any set of high-dimensional points in a Euclidean space can be projected into lower dimensions while approximately preserving pairwise Euclidean distances. Random matrices satisfying this lemma are called JL transforms (JLTs). Inspired by existing $s$-hashing JLTs with exactly $s$ nonzero elements on each column, the present work introduces an ensemble of sparse matrices encompassing so-called $s$-hashing-like matrices whose expected number of nonzero elements on each column is $s$. The independence of the sub-Gaussian entries of these matrices and the knowledge of their exact distribution play an important role in their analyses. Using properties of independent sub-Gaussian random variables, these matrices are demonstrated to be JLTs, and their smallest nontrivial singular values and largest singular values are estimated nonasymptotically using a technique from geometric functional analysis. As the dimensions of the matrix grow to infinity, these singular values are proved to converge almost surely to fixed quantities (by using the universal Bai–Yin law) and in distribution to the Gaussian orthogonal ensemble Tracy–Widom law after proper rescalings. Understanding the behaviors of extreme singular values is important in general because they are often used to define a measure of stability of matrix algorithms. For example, JLTs were recently used in derivative-free optimization algorithmic frameworks to select random subspaces in which are constructed random models or poll directions to achieve scalability, and hence estimating their smallest singular value in particular helps determine the dimension of these subspaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Wetting of Porous α-LiAlO 2 by Molten Carbonate

Cracking of α-LiAlO 2 matrices in molten carbonate fuel cells (MCFC) leads to reduction in the performance. It was demonstrated in this work that the mechanical strength of α-LiAlO2 matrices is improved by heat-treating at 800°C under ambient gas atmosphere. The mechanical strength (2.91MPa) of the heat-treated matrix was enhanced more than 5 fold compared to the non-heat-treated matrix (0.58MPa). The porosity and crystal structure of the α-LiAlO 2 matrices were not changed after the heat-treatment. The wetting behavior and distribution of carbonate in the heat-treated and the non-heat-treated matrix were investigated and compared. Both non-heat-treated and heat-treated α-LiAlO 2 matrices were completely wetted by molten carbonate. The molten carbonate distribution was concentrated right underneath the circular region formed by the molten carbonate drop in the matrix. Non-heat-treated α-LiAlO 2 samples cracked as a result of the complete absorption of molten carbonate, however, the heat-treated α-LiAlO 2 matrices did not crack, presumably due to their enhanced mechanical strength.

36 MATERIALS SCIENCE↗
DOE OSTI2015

Sandia Generated Matrix Tool (SGMT) v. 1.0

Provides a tool with which create and characterize a very large set of matrix-based visual analogy problems that have properties that are similar to Raven's Progressive Matrices (RPMs)™. The software uses the same underlying patterns found in RPMs to generate large numbers of unique matrix problems using parameters chosen by the researcher. Specifically, the software is designed so that researchers can choose the type, direction, and number of relations in a problem and then create any number of unique matrices that share the same underlying structure (e.g. changes in numerosity in a diagonal pattern) but have different surface features (e.g. shapes, colors).Raven's Progressive Matrices (RPMs) ™ are a widely-used test for assessing intelligence and reasoning ability. Since the test is non-verbal, it can be applied to many different populations and has been used all over the world. However, there are relatively few matrices in the sets developed by Raven, which limits their use in experiments requiring large numbers of stimuli. This tool creates a matrix set in a systematic way that allows researchers to have a great deal of control over the underlying structure, surface features, and difficulty of the matrix problems while providing a large set of novel matrices with which to conduct experiments.

Benz, Zachary↗
NASA NTRS1973

Descriptions of the polarization states of vector processes - Applications to ULF magnetic fields

In recent years a wide variety of methods has been used to describe the polarization characteristics of ULF (.001 to 1 Hz) magnetic fields. This paper gives a detailed outline of some of the descriptions derived from the spectral matrices of n-variate stochastic processes. The matrices are expanded in three different, standard sets of matrices in order to add some simplification to the interpretation of the polarizations. One set is composed of n-squared trace-orthogonal, hermitean matrices and leads directly to a generalization of the Stokes parameters and the degree of polarization for n-variate processes. The second set is developed from the dyad expansion, which in particular cases is analogous to the spectral decomposition of the matrix. The third set is composed of n commuting idempotent matrices and proves to be the most useful set when the stochastic process is not strictly polarized.

Samson, J. C.↗
NASA NTRS1977

Matrix computations in MACSYMA

Facilities built into MACSYMA for manipulating matrices with numeric or symbolic entries are described. Computations will be done exactly, keeping symbols as symbols. Topics discussed include how to form a matrix and create other matrices by transforming existing matrices within MACSYMA; arithmetic and other computation with matrices; and user control of computational processes through the use of optional variables. Two algorithms designed for sparse matrices are given. The computing times of several different ways to compute the determinant of a matrix are compared.

Wang, P. S.↗
NASA NTRS1992

A robust quasi-simultaneous interaction method for a full potential flow with a boundary layer with application to wing/body configurations

The MATRICS flow solver calculates the inviscid transonic potential flow about a wing/body semi-configuration. At present, work is in progress to extend MATRICS to take viscous effects into account through coupling with a boundary layer solver. This solver, MATRICS-V, is based on robust calculation methods for the boundary layer, the outer wing flow and their interaction. MATRICS-V is intended for (inverse) design purposes. The boundary layer and wake are based on an integral formulation of the unsteady first order boundary layer equations, the inviscid method is the existing MATRICS potential flow solver, and the interaction algorithm is of the quasi-simultaneous type. The paper gives a progress report on the coupled potential-flow boundary-layer method for transonic wing/body configurations.

Vanderwees, A. J.↗
NASA NTRS2000

Model Refinement Using Eigensystem Assignment

IA novel approach for the refinement of finite-element-based analytical models of flexible structures is presented. The proposed approach models the possible refinements in the mass, damping, and stiffness matrices of the finite element model in the form of a constant gain feedback with acceleration, velocity, and displacement measurements, respectively. Once the free elements of the structural matrices have been defined, the problem of model refinement reduces to obtaining position, velocity, and acceleration gain matrices with appropriate sparsity that reassign a desired subset of the eigenvalues of the model, along with partial mode shapes, from their baseline values to those obtained from system identification test data. A sequential procedure is used to assign one conjugate pair of eigenvalues at each step using symmetric output feedback gain matrices, and the eigenvectors are partially assigned, while ensuring that the eigenvalues assigned in the previous steps are not disturbed. The procedure can also impose that gain matrices be dissipative to guarantee the stability of the refined model. A numerical example, involving finite element model refinement for a structural testbed at NASA Langley Research Center (Controls-Structures-Interaction Evolutionary model) is presented to demonstrate the feasibility of the proposed approach.

Maghami, Peiman G.↗
NASA NTRS1998

Model Refinement Using Eigensystem Assignment

This paper presented a novel approach for the refinement of finite-element-based analytical models of flexible structures is presented. The proposed approach models the possible refinements in the mass, damping, and stiffness matrices of the finite element model in the form of a constant gain feedback with acceleration, velocity, and displacement measurements, respectively. Once, the free elements of the structural matrices have been defined, the problem of model refinement reduces to obtaining position, velocity, and acceleration gain matrices, which reassign a desired subset of the eigenvalues of the model, along with partial mode shapes, from their baseline values to those obtained from system identification test data. A sequential procedure is used to assign one self-conjugate pair of closed-loop eigenvalues at each step using symmetric output feedback gain matrices, and the closed-loop eigenvectors are partially assigned, while ensuring that the eigenvalues assigned in the previous steps are not disturbed. The procedure can also impose that gain matrices be dissipative in order to guarantee the stability of the refined model. A numerical example, involving finite element model refinement for a structural testbed at NASA Langley (CSI Evolutionary Model) is presented to demonstrate the feasibility of the proposed approach.

Maghami, Pieman G.↗
NASA NTRS2011

Graph Theory Roots of Spatial Operators for Kinematics and Dynamics

Spatial operators have been used to analyze the dynamics of robotic multibody systems and to develop novel computational dynamics algorithms. Mass matrix factorization, inversion, diagonalization, and linearization are among several new insights obtained using such operators. While initially developed for serial rigid body manipulators, the spatial operators and the related mathematical analysis have been shown to extend very broadly including to tree and closed topology systems, to systems with flexible joints, links, etc. This work uses concepts from graph theory to explore the mathematical foundations of spatial operators. The goal is to study and characterize the properties of the spatial operators at an abstract level so that they can be applied to a broader range of dynamics problems. The rich mathematical properties of the kinematics and dynamics of robotic multibody systems has been an area of strong research interest for several decades. These properties are important to understand the inherent physical behavior of systems, for stability and control analysis, for the development of computational algorithms, and for model development of faithful models. Recurring patterns in spatial operators leads one to ask the more abstract question about the properties and characteristics of spatial operators that make them so broadly applicable. The idea is to step back from the specific application systems, and understand more deeply the generic requirements and properties of spatial operators, so that the insights and techniques are readily available across different kinematics and dynamics problems. In this work, techniques from graph theory were used to explore the abstract basis for the spatial operators. The close relationship between the mathematical properties of adjacency matrices for graphs and those of spatial operators and their kernels were established. The connections hold across very basic requirements on the system topology, the nature of the component bodies, the indexing schemes, etc. The relationship of the underlying structure is intimately connected with efficient, recursive computational algorithms. The results provide the foundational groundwork for a much broader look at the key problems in kinematics and dynamics. The properties of general graphs and trees of nodes and edge were examined, as well as the properties of adjacency matrices that are used to describe graph connectivity. The nilpotency property of such matrices for directed trees was reviewed, and the adjacency matrices were generalized to the notion of block weighted adjacency matrices that support block matrix elements. This leads us to the development of the notion of Spatial Kernel Operator SKO kernels. These kernels provide the basis for the development of SKO resolvent operators.

Jain, Abhinandan↗
DOE OSTI2014

On boundary fusion and functional relations in the Baxterized affine Hecke algebra

We construct boundary type operators satisfying fused reflection equation for arbitrary representations of the Baxterized affine Hecke algebra. These operators are analogues of the fused reflection matrices in solvable half-line spin chain models. We show that these operators lead to a family of commuting transfer matrices of Sklyanin type. We derive fusion type functional relations for these operators for two families of representations.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

Topological insulators and C*-algebras: Theory and numerical practice

Research Highlights: > We classify topological insulators using C* algebras. > We present new K-theory invariants. > We develop efficient numerical algorithms based on this technique. > We observe unexpected quantum phase transitions using our algorithm. - Abstract: We apply ideas from C*-algebra to the study of disordered topological insulators. We extract certain almost commuting matrices from the free Fermi Hamiltonian, describing band projected coordinate matrices. By considering topological obstructions to approximating these matrices by exactly commuting matrices, we are able to compute invariants quantifying different topological phases. We generalize previous two dimensional results to higher dimensions; we give a general expression for the topological invariants for arbitrary dimension and several symmetry classes, including chiral symmetry classes, and we present a detailed K-theory treatment of this expression for time reversal invariant three dimensional systems. We can use these results to show non-existence of localized Wannier functions for these systems. We use this approach to calculate the index for time-reversal invariant systems with spin-orbit scattering in three dimensions, on sizes up to 12{sup 3}, averaging over a large number of samples. The results show an interesting separation between the localization transition and the point at which the average index (which can be viewed as an 'order parameter' for the topological insulator) begins to fluctuate from sample to sample, implying the existence of an unsuspected quantum phase transition separating two different delocalized phases in this system. One of the particular advantages of the C*-algebraic technique that we present is that it is significantly faster in practice than other methods of computing the index, allowing the study of larger systems. In this paper, we present a detailed discussion of numerical implementation of our method.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2011

Nonlinear Bogolyubov-Valatin transformations: Two modes

Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper, we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n=2 fermionic modes, which can be implemented by means of unitary SU(2{sup n}=4) transformations, is isomorphic to SO(6;R)/Z{sub 2}. The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0,4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonians. Relying on Jordan-Wigner transformations for two-spin-1/2 and single-spin-3/2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1/2 and single-spin-3/2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU(4) and SO(6;R) matrices (of respective sizes 4x4 and 6x6) and their mutual relation. - Highlights: > Reveals the structure of nonlinear canonical transformations for two fermionic modes. > Studies nonlinear basis transformations in a Clifford algebra. > Focuses on methodological and structural aspects. > Presents a new approach to the diagonalization of 4x4 matrix Hamiltonians. > Presents the relation between explicit parametrizations of SU(4) and SO(6;R) matrices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Macromolecular gelatin properties affect fibrin microarchitecture and tumor spheroid behavior in fibrin-gelatin gels

The biophysical properties of extracellular matrices (ECM) are known to regulate cell behavior, however decoupling cell behavior changes due to the relative contributions of material microstructure versus biomechanics or nutrient permeability remains challenging, especially within complex, multi-material matrices. We developed four gelatin-fibrin interpenetrating network (IPN) formulations which are identical in composition but possess variable gelatin molecular weight distributions, and display differences in microstructure, biomechanics, and diffusivity. In this work we interrogate the response of multicellular tumor spheroids to these IPN formulations and found that a high stiffness, gelatin-network dominated IPNs impeded remodeling and invasion of multicellular tumor spheroids; whereas relatively lower stiffness, fibrin-network dominated IPNs permitted protease-dependent remodeling and spheroid invasion. Cell proliferation correlated to nutrient diffusivity across tested IPN formulations. These findings demonstrate the complexity of ECM IPNs, relative to single polymer matrices, and highlight that cell response does not derive from a single aspect of the ECM, but rather from the interplay of multiple biomechanical properties. The methodology developed in this study represents a framework for future studies which aim to characterize cellular phenotypic responses to biophysical cues present within complex, multi-material matrices.

36 MATERIALS SCIENCE↗
DOE OSTI2014

Noiseless Vlasov–Poisson simulations with linearly transformed particles

We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Formation of Ge nanoparticles in SiO{sub x}N{sub y} by ion implantation and thermal annealing

Germanium nanoparticles embedded within dielectric matrices hold much promise for applications in optoelectronic and electronic devices. Here we investigate the formation of Ge nanoparticles in amorphous SiO{sub 1.67}N{sub 0.14} as a function of implanted atom concentration and thermal annealing temperature. Using x-ray absorption spectroscopy and other complementary techniques, we show Ge nanoparticles exhibit significant finite-size effects such that the coordination number decreases and structural disorder increases as the nanoparticle size decreases. While the composition of SiO{sub 1.67}N{sub 0.14} is close to that of SiO{sub 2}, we demonstrate that the addition of this small fraction of N yields a much reduced nanoparticle size relative to those formed in SiO{sub 2} under comparable implantation and annealing conditions. We attribute this difference to an increase in an atomic density and a much reduced diffusivity of Ge in the oxynitride matrix. These results demonstrate the potential for tailoring Ge nanoparticle sizes and structural properties in the SiO{sub x}N{sub y} matrices by controlling the oxynitride stoichiometry.

36 MATERIALS SCIENCE↗
DOE OSTI2010

Efficient Sparse Matrix-Matrix Multiplication on Heterogeneous High Performance Systems

The efficient implementation of sparse matrix-matrix multiplications on high performance parallel machines poses several challenges: large size of input matrices, compressed representation, density of the output matrices, partitioning and load balancing of matrices that present parts with large differences in density and, thus, in computation times. In this paper we show how, starting from the requirements of such application, we developed a framework that allows its efficient implementation on heterogeneous clusters. We introduce a task based programming model and a runtime supported execution model which provides dynamic load balancing on clusters composed by CPUs and GPUs, allowing better utilization of the system while easing the handling of sparse matrices. Our results show that our solution, which co-designs the application together with the programming model and the runtime system, is able to obtain significant speed-ups due to a virtually perfect load balancing with respect to other programming approaches.

97 MATHEMATICS AND COMPUTING↗