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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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DOE OSTI2020

Estimating Higher-Order Moments Using Symmetric Tensor Decomposition

In this paper, we consider the problem of decomposing higher-order moment tensors, i.e., the sum of symmetric outer products of data vectors. Such a decomposition can be used to estimate the means in a Gaussian mixture model and for other applications in machine learning. The dth-order empirical moment tensor of a set of p observations of n variables is a symmetric d-way tensor. Our goal is to nd a low-rank tensor approximation comprising r $\ll$ p symmetric outer products. The challenge is that forming the empirical moment tensor costs O(pn d ) operations and O(n d ) storage, which may be prohibitively expensive; additionally, the algorithm to compute the low-rank approximation costs O(n d ) per iteration. Our contribution is avoiding formation of the moment tensor, computing the low-rank tensor approximation of the moment tensor implicitly using O(pnr) operations per iteration and no extra memory. This advance opens the door to more applications of higher-order moments since they can now be efficiently computed. We present numerical evidence of the computational savings and show an example of estimating the means for higher-order moments.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Scalable Tensor Methods for Nonuniform Hypergraphs

While multilinear algebra appears natural for studying the multiway interactions modeled by hypergraphs, tensor methods for general hypergraphs have been stymied by theoretical and practical barriers. A recently proposed adjacency tensor is applicable to nonuniform hypergraphs, but is prohibitively costly to form and analyze in practice. We develop tensor times same vector (TTSV) algorithms for this tensor which improve complexity from $O(n^r)$ to a low-degree polynomial in $r$, where $n$ is the number of vertices and $r$ is the maximum hyperedge size. Our algorithms are implicit, avoiding formation of the order $r$ adjacency tensor. Here, we demonstrate the flexibility and utility of our approach in practice by developing tensor-based hypergraph centrality and clustering algorithms. We also show these tensor measures offer complementary information to analogous graph-reduction approaches on data, and are also able to detect higher-order structure that many existing matrix-based approaches provably cannot.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Sylvester-preconditioned adaptive-rank implicit time integrators for advection-diffusion equations with variable coefficients

Here, we consider the adaptive-rank integration of multi-dimensional time-dependent advection-diffusion partial differential equations (PDEs) with variable coefficients. We employ a standard finite-difference method for spatial discretization coupled with high-order diagonally implicit Runge-Kutta temporal schemes. The discrete equation is a generalized Sylvester equation (GSE), which we solve with a projection-based adaptive-rank algorithm structured around two key strategies: (i) constructing dimension-wise subspaces using a novel atypical extended Krylov strategy, and (ii) efficiently solving the basis coefficient matrix with a preconditioned GMRES solver. The low-rank decomposition is performed in 2D using SVD and with high-order SVD (HOSVD) in 3D to represent the tensor in a compressed Tucker format. For d-dimensional problems (here, d = 2 or 3), the computational complexity and memory storage of the approach are found numerically to scale as and $\mathscr{O}(Nr^2) + \mathscr{O} (r^{d+1})$ and $\mathscr{O}(Nr) + \mathscr{O} (r^{d})$, respectively, with the one-dimensional resolution and the maximal rank during the Krylov iteration (which we find to be largely independent of on our numerical examples). We present numerical examples that illustrate the advertised properties of the algorithm.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Density functional theory calculations of the thermodynamic and kinetic properties of point defects in β -U

Density functional theory (DFT) calculations of the thermodynamic and kinetic properties of point defects in the β phase of uranium are reported. Defect energies and entropies were calculated using 2 x 2 x 2 supercells and the Generalized Gradient Approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) for the exchange-correlation potential. Due to computational cost, calculations of the vibrational properties governing entropies were performed by only displacing atoms within (roughly) the 3rd or 4th nearest neighbor shell of the defect, which implicitly assumes that atoms beyond this distance are unaffected by the defect. Migration barriers were estimated by nudged elastic band (NEB) calculations. The low symmetry of the β-U phase (the unit cell is tetragonal and contains 30 atoms) results in many point defect configurations and even more migration pathways. A connectivity map, starting from the most stable point defects, was developed in order to identify the rate-limiting step controlling the net diffusion rate in each crystallographic direction. The uranium self-diffusivity tensor was calculated by combining the defect formation energies, entropies, migration barriers and attempt frequencies. The fastest diffusion rate was determined to be a vacancy mechanism in the z crystallographic direction. The predicted uranium self-diffusivity for this mechanism agrees well with available experimental data. Finally, the diffusion mechanisms and rates identified in this study will inform fuel performance models of swelling and gas evolution.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗