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All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Robust Tensor Hypercontraction of the Particle–Particle Ladder Term in Equation-of-Motion Coupled Cluster Theory

One method of representing a high-rank tensor as a (hyper-)product of lower-rank tensors is the tensor hypercontraction (THC) method of Hohenstein et al. This strategy has been found to be useful for reducing the polynomial scaling of coupled-cluster methods by representation of a four-dimensional tensor of electron-repulsion integrals in terms of five two-dimensional matrices. Pierce et al. have already shown that the application of a robust form of THC to the particle–particle ladder (PPL) term reduces the cost of this term in couple-cluster singles and doubles (CCSD) from O(N 6 ) to O(N 5 ) with negligible errors in energy with respect to the density-fitted variant. In this work, we have implemented the least-squares variant of THC (LS-THC) which does not require a nonlinear tensor factorization, including the robust form (R-LS-THC), for the calculation of the excitation and electron attachment energies using equation-of-motion coupled cluster methods EOMEE-CCSD and EOMEA-CCSD, respectively. We have benchmarked the effect of the R-LS-THC-PPL approximation on excitation energies using the comprehensive QUEST database and the accuracy of electron attachment energies using the NAB22 database. Here, we find that errors on the order of 1 meV are achievable with a reduction in total calculation time of approximately 5x.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2017

Highly Efficient and Scalable Compound Decomposition of Two-Electron Integral Tensor and Its Application in Coupled Cluster Calculations

The representation and storage of two-electron integral tensors are vital in large- scale applications of accurate electronic structure methods. Low-rank representation and efficient storage strategy of integral tensors can significantly reduce the numerical overhead and consequently time-to-solution of these methods. In this paper, by combining pivoted incomplete Cholesky decomposition (CD) with a follow-up truncated singular vector decomposition (SVD), we develop a decomposition strategy to approximately represent the two-electron integral tensor in terms of low-rank vectors. A systematic benchmark test on a series of 1-D, 2-D, and 3-D carbon-hydrogen systems demonstrates high efficiency and scalability of the compound two-step decomposition of the two-electron integral tensor in our implementation. For the size of atomic basis set N_b ranging from ~ 100 up to ~ 2, 000, the observed numerical scaling of our implementation shows O(N_b^{2.5~3}) versus O(N_b^{3~4}) of single CD in most of other implementations. More importantly, this decomposition strategy can significantly reduce the storage requirement of the atomic-orbital (AO) two-electron integral tensor from O(N_b^4) to O(N_b^2 log_{10}(N_b)) with moderate decomposition thresholds. The accuracy tests have been performed using ground- and excited-state formulations of coupled- cluster formalism employing single and double excitations (CCSD) on several bench- mark systems including the C_{60} molecule described by nearly 1,400 basis functions. The results show that the decomposition thresholds can be generally set to 10^{-4} to 10^{-3} to give acceptable compromise between efficiency and accuracy.

54 ENVIRONMENTAL SCIENCES↗
DOE OSTI2017

Tensor-Train Split-Operator Fourier Transform (TT-SOFT) Method: Multidimensional Nonadiabatic Quantum Dynamics

We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) method for simulations of multidimensional nonadiabatic quantum dynamics. TT-SOFT is essentially the grid-based SOFT method implemented in dynamically adaptive tensor-train representations. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the grid-based wavepacket tensor components, bypassing the need of actually computing the wave function in its full-rank tensor product grid space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the S 1 / S 2 interconversion dynamics of pyrazine after UV photoexcitation to the S 2 state. Furthermore, our results show that the TT-SOFT method is a powerful computational approach for simulations of quantum dynamics of polyatomic systems since it avoids the exponential scaling problem of full-rank grid-based representations.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2016

Experimental evaluation of electrical conductivity imaging of anisotropic brain tissues using a combination of diffusion tensor imaging and magnetic resonance electrical impedance tomography

Anisotropy of biological tissues is a low-frequency phenomenon that is associated with the function and structure of cell membranes. Imaging of anisotropic conductivity has potential for the analysis of interactions between electromagnetic fields and biological systems, such as the prediction of current pathways in electrical stimulation therapy. To improve application to the clinical environment, precise approaches are required to understand the exact responses inside the human body subjected to the stimulated currents. In this study, we experimentally evaluate the anisotropic conductivity tensor distribution of canine brain tissues, using a recently developed diffusion tensor-magnetic resonance electrical impedance tomography method. At low frequency, electrical conductivity of the biological tissues can be expressed as a product of the mobility and concentration of ions in the extracellular space. From diffusion tensor images of the brain, we can obtain directional information on diffusive movements of water molecules, which correspond to the mobility of ions. The position dependent scale factor, which provides information on ion concentration, was successfully calculated from the magnetic flux density, to obtain the equivalent conductivity tensor. By combining the information from both techniques, we can finally reconstruct the anisotropic conductivity tensor images of brain tissues. The reconstructed conductivity images better demonstrate the enhanced signal intensity in strongly anisotropic brain regions, compared with those resulting from previous methods using a global scale factor.

62 RADIOLOGY AND NUCLEAR MEDICINE↗
DOE OSTI2026

Quantifying the dislocation content of atomically resolved grain boundary line defects using the Nye tensor

The Nye tensor, which quantifies the density of Burgers vector for a given dislocation line direction, can be effectively used to characterize dislocation content in bulk crystals from atomic-resolution transmission electron microscopy images. The Nye tensor can be calculated from these images, in part because the reference state is simply defined by the lattice of the perfect crystal. The application of the Nye tensor to interfacial line defects, for which the natural reference state is the dichromatic pattern of the two grains in their reference orientation, poses additional challenges. In this work, we present a method that employs the Nye tensor to characterize the edge dislocation content of line defects at grain boundaries from atomic-resolution images. This approach enables us to rapidly characterize all edge dislocation content along a grain boundary. Additionally, the Nye tensor provides information about line defect core structure. Finally, we demonstrate this method on two exemplar defects: a twin boundary disconnection and a facet junction in face-centered cubic Au.

Crystallographic defects↗
DOE OSTI2017

Tensor perturbations during inflation in a spatially closed Universe

In a recent paper [1], we studied the evolution of the background geometry and scalar perturbations in an inflationary, spatially closed Friedmann-Lemaȋtre-Robertson-Walker (FLRW) model having constant positive spatial curvature and spatial topology S{sup 3}. Due to the spatial curvature, the early phase of slow-roll inflation is modified, leading to suppression of power in the scalar power spectrum at large angular scales. In this paper, we extend the analysis to include tensor perturbations. We find that, similarly to the scalar perturbations, the tensor power spectrum also shows suppression for long wavelength modes. The correction to the tensor spectrum is limited to the very long wavelength modes, therefore the resulting observable CMB B-mode polarization spectrum remains practically the same as in the standard scenario with flat spatial sections. However, since both the tensor and scalar power spectra are modified, there are scale dependent corrections to the tensor-to-scalar ratio that leads to violation of the standard slow-roll consistency relation.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2025

Monomer-dimer tensor-network basis for qubit-regularized lattice gauge theories

Traditional SU⁡(𝑁) lattice gauge theories (LGTs) can be formulated using an orthonormal basis constructed from the irreducible representations (irreps) 𝑉 𝜆 of the SU⁡(𝑁) gauge symmetry. On a lattice, the elements of this basis are tensor networks comprising dimer tensors on the links labeled by a set of irreps {𝜆 ℓ } and monomer tensors on sites labeled by {𝜆 𝑠 }. These tensors naturally define a local site Hilbert space, ℋ$^𝑔_𝑠$, on which gauge transformations act. Gauss’s law introduces an additional index 𝛼 𝑠 =1,2,…,𝒟⁡(ℋ$^𝑔_𝑠$) that labels an orthonormal basis of the gauge-invariant subspace of ℋ$^𝑔_𝑠$. This monomer-dimer tensor-network (MDTN) basis, |{𝜆 𝑠 },{𝜆 ℓ },{𝛼 𝑠 }⟩, of the physical Hilbert space enables the construction of new qubit-regularized SU⁡(𝑁) gauge theories that are free of sign problems while preserving key features of traditional LGTs. Here, we investigate finite-temperature confinement-deconfinement transitions in a simple qubit-regularized SU(2) and SU(3) gauge theory in 𝑑 =2 and 𝑑 =3 spatial dimensions, formulated using the MDTN basis, and show that they reproduce the universal results of traditional LGTs at these transitions. Additionally, in 𝑑 =1, we demonstrate using a plaquette chain that the string tension at zero temperature can be continuously tuned to zero by adjusting a model parameter that plays the role of the gauge coupling in traditional LGTs.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2018

Pinch point singularities of tensor spin liquids

Recently, a new class of three-dimensional spin liquid models have been theoretically discovered, which feature generalized Coulomb phases of emergent symmetric tensor U(1) gauge theories. These “higher rank” tensor models are particularly intriguing due to the presence of quasiparticles with restricted mobility, such as fractons. We investigate universal experimental signatures of tensor Coulomb phases. Most notably, we show that tensor Coulomb spin liquids (both quantum and classical) feature characteristic pinch point singularities in their spin-spin correlation functions, accessible via neutron scattering, which can be readily distinguished from pinch points in conventional U(1) spin liquids. These pinch points can thus serve as a crisp experimental diagnostic for such phases. Here, we also tabulate the low-temperature heat capacity of various tensor Coulomb phases, which serves as a useful additional diagnostic in certain cases.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2021

Gauge-symmetrization method for energy-momentum tensors in high-order electromagnetic field theories

For electromagnetic field theories, canonical energy-momentum conservation laws can be derived from the underpinning spacetime translation symmetry according to the Noether procedure. However, the canonical energy-momentum tensors (EMTs) are neither symmetric nor gauge-symmetric (gauge invariant). The Belinfante-Rosenfeld (BR) method is a well-known procedure to symmetrize the EMTs, which also renders them gauge symmetric for first-order field theories. High-order electromagnetic field theories appear in the study of gyrokinetic systems for magnetized plasmas and the Podolsky system for the radiation reaction of classical charged particles. For these high-order field theories, gauge-symmetric EMTs are not necessarily symmetric and vice versa. In the present study, we develop a new gauge-symmetrization method for EMTs in high-order electromagnetic field theories. The Noether procedure is carried out using the Faraday tensor $F_{μν}$, instead of the 4-potential $A_{μ}$, to derive a canonical EMT $T^{μν}_{N}$. We show that the gauge-dependent part of $T^{μν}_{N}$ can be removed using the displacement-potential tensor $F^{σμν}$ ≡ $D^{σμ}A^{ν}/4π$, where $D^{σμ}$ is the antisymmetric electric displacement tensor. This method gauge-symmetrizes the EMT without necessarily making it symmetric, which is adequate for applications not involving general relativity. For first-order electromagnetic field theories, such as the standard Maxwell system, $F^{σμν}$ reduces to the familiar BR superpotential $S^{σμν}$, and the method developed can be used as a simpler procedure to calculate $S^{σμν}$ without employing the angular momentum tensor in 4D spacetime. When the electromagnetic system is coupled to classical charged particles, the gauge-symmetrization method for EMTs is shown to be effective as well.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2017

Dispersive transport and symmetry of the dispersion tensor in porous media

The macroscopic laws controlling the advection and diffusion of solute at the scale of the porous continuum are derived in a general manner that does not place limitations on the geometry and time evolution of the pore space. Special focus is given to the definition and symmetry of the dispersion tensor that is controlling how a solute plume spreads out. Here, we show that the dispersion tensor is not symmetric and that the asymmetry derives from the advective derivative in the pore-scale advection-diffusion equation. When flow is spatially variable across a voxel, such as in the presence of a permeability gradient, the amount of asymmetry can be large. As first shown by Auriault [J.-L. Auriault Transp. Porous Med. 85, 771 (2010)TPMEEI0169-391310.1007/s11242-010-9591-y] in the limit of low Péclet number, we show that at any Péclet number, the dispersion tensor D ij satisfies the flow-reversal symmetry D ij (+q)=D ji (-q) where q is the mean flow in the voxel under analysis; however, Reynold's number must be sufficiently small that the flow is reversible when the force driving the flow changes sign. We also demonstrate these symmetries using lattice-Boltzmann simulations and discuss some subtle aspects of how to measure the dispersion tensor numerically. Particularly, the numerical experiments demonstrate that the off-diagonal components of the dispersion tensor are antisymmetric which is consistent with the analytical dependence on the average flow gradients that we propose for these off-diagonal components.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Calculated electron paramagnetic resonance g-tensor and hyperfine parameters for zinc vacancy and N related defects in ZnO

Various defects in ZnO, focused on substitutional N O and N 2 in various sites, O-site, interstitial and Zn-site are studied using first-principles calculations with the goal of understanding the electron paramagnetic resonance (EPR) center reported for N 2 in ZnO and substitutional N on the O-site. The g tensors are calculated using the gauge including projector augmented wave (GIPAW) method and compared with experiments. The g-tensor of the free $N^{+}_{2}$ and $N^{–}_{2}$ radicals and their various contributions within the GIPAW theory are analyzed first to provide a baseline reference for the accuracy of the method and for understanding the N 2 behavior in ZnO. Previous controversies on the site location of N 2 in ZnO for this EPR center and on the shallow or deep nature and donor or acceptor nature of this center are resolved. Here, we find that the N 2 on the Zn site is mostly zinc-vacancy like in its spin density and g-tensor, while for the O-site, a model with the N 2 axis lying in-the basal plane and the singly occupied π g -orbital along the c axis provides good agreement with experiment. For the interstitial location, if the N 2 is not strongly interacting with the surroundings, no levels in the gap are found and hence also no possible EPR center. The calculated g-tensors for N O and V Zn are also found to be in good agreement with experiment. The effects of different functionals affecting the localization of the spin density are shown to affect the g-tensor values.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Precision Reconstruction of Rational Conformal Field Theory from Exact Fixed-Point Tensor Network

The novel concept of entanglement renormalization and its corresponding tensor network renormalization technique have been highly successful in developing a controlled real-space renormalization group (RG) scheme. Numerically approximate fixed-point (FP) tensors are widely used to extract the conformal data of the underlying conformal field theory (CFT) describing critical phenomena. In this paper, we present an explicit analytical construction of the FP tensor for 2D rational CFT. We define it as a correlation function between the “boundary-changing operators” (BCO) on triangles. Our construction fully captures all the real-space RG conditions. We also provide concrete examples, such as Ising, Yang-Lee, and tricritical Ising models, to compute the scaling dimensions explicitly based on the corresponding FP tensor. The BCO descendants turn out to be an optimal basis such that truncation in bond dimensions naturally produces comparable accuracies with the leading existing FP algorithms. Interestingly, our construction of FP tensors is closely related to a strange correlator, where the holographic picture naturally emerges. Our results also open a new door toward understanding CFT in higher dimensions. Published by the American Physical Society 2025

Cheng, Gong (ORCID:0009000891587404)↗
DOE OSTI2026

Loop series expansions for tensor networks

Belief propagation (BP) can be a useful tool to approximately contract a tensor network, provided that the contributions from any closed loops in the network are sufficiently weak. In this article, we describe how a loop series expansion can be applied to systematically improve the accuracy of a BP approximation to a tensor network contraction, in principle converging arbitrarily close to the exact result. More generally, our result provides a framework for expanding a tensor network as a sum of component networks in a hierarchy of increasing complexity. We benchmark this proposal for the contraction of infinite projected entangled pair states, either representing the ground state of an Affleck-Kennedy-Lieb-Tasaki (AKLT) model or with randomly defined tensors, where it is shown to improve in accuracy over standard BP by several orders of magnitude while incurring only a minor increase in computational cost. These results indicate that the proposed series expansions could be a useful tool to accurately evaluate tensor networks in cases that otherwise exceed the limits of established contraction routines.

Evenbly, Glen [AWS Center for Quantum Computing, P↗
DOE OSTI2021

Low-rank Tensor Completion for PMU Data Recovery

This paper proposes a tensor completion method for the recovery of missing phasor measurement unit (PMU) measurements. Tensor completion as the general case of matrix completion has attracted increasing attention in recent years. The imputation accuracy for the existing matrix completion methods may be significantly reduced when there are consecutive data losses across multiple data channels. To tackle this issue, we explore the multi-way characteristics of PMU measurements by using a tensor model. We leverage the low-rank property of the PMU measurements and formulate the missing PMU data recovery problem as a low-rank tensor completion problem. An efficient algorithm based on alternating direction method of multipliers (ADMM) is developed to solve the tensor completion problem. The experiments using the real PMU dataset show that the proposed method exhibits better imputation accuracy compared with the conventional data recovery methods.

Ghasemkhani, Amir↗
DOE OSTI2019

Toward Generalized Tensor Algebra for ab initio Quantum Chemistry Methods

The widespread use of tensor operations in describing electronic structure calculations has motivated the design of software frameworks for productive development of scalable optimized tensor-based electronic structure methods. Whereas prior work focused on Cartesian abstractions for dense tensors, we present an algebra to specify and perform tensor operations on a larger class of block-sparse tensors. We illustrate the use of this framework in expressing real-world computational chemistry calculations beyond the reach of existing frameworks.

Mutlu, Erdal↗
DOE OSTI2024

Accelerated Constrained Sparse Tensor Factorization on Massively Parallel Architectures

This study presents the first constrained sparse tensor factorization (cSTF) framework that optimizes and fully offloads computation to massively parallel GPU architectures, and the first performance characterization of cSTF on GPU architectures. In contrast to prior work on tensor factorization, where the matricized tensor times Khatri-Rao product (MTTKRP) is the primary performance bottleneck, our systematic analysis of the cSTF algorithm on GPUs reveals that adding constraints creates an additional bottleneck in the update operation for many real-world sparse tensors. While executing the update operation on the GPU brings significant speedup over its CPU counterpart, it remains a significant bottleneck. To further accelerate the update operation, we propose cuADMM, a new update algorithm that leverages algorithmic and code optimization strategies to minimize both computation and data movement on GPUs. As a result, our framework delivers significantly improved performance compared to prior state-of-the-art. On 10 real-world sparse tensors, our framework achieves geometric mean speedup of 5.1 × (max 41.59 ×) and 7.01 × (max 58.05 ×) on the NIVIDA A100 and H100 GPUs, respectively, over the state-of-the-art SPLATT library running on a 26-core Intel Ice Lake Xeon CPU.

Soh, Yongseok↗
DOE OSTI2017

Tensor Toolbox for MATLAB v. 3.0

Tensors (also known as multidimensional arrays or N-way arrays) are used in a variety of applications ranging from chemometrics to network analysis. The Tensor Toolbox provides classes for manipulating dense, sparse, and structured tensors using MATLAB's object-oriented features. It also provides algorithms for tensor decomposition and factorization, algorithms for computing tensor eigenvalues, and methods for visualization of results.

Kolda, Tamara↗
DOE OSTI2017

Genten: Software for Generalized Tensor Decompositions v. 1.0.0

Tensors, or multidimensional arrays, are a powerful mathematical means of describing multiway data. This software provides computational means for decomposing or approximating a given tensor in terms of smaller tensors of lower dimension, focusing on decomposition of large, sparse tensors. These techniques have applications in many scientific areas, including signal processing, linear algebra, computer vision, numerical analysis, data mining, graph analysis, neuroscience and more. The software is designed to take advantage of parallelism present emerging computer architectures such has multi-core CPUs, many-core accelerators such as the Intel Xeon Phi, and computation-oriented GPUs to enable efficient processing of large tensors.

Phipps, EricT.↗