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DOE OSTI2017

Bayesian and variational Bayesian approaches for flows in heterogeneous random media

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in ) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2023

Exact Gaussian processes for massive datasets via non-stationary sparsity-discovering kernels

Abstract A Gaussian Process (GP) is a prominent mathematical framework for stochastic function approximation in science and engineering applications. Its success is largely attributed to the GP’s analytical tractability, robustness, and natural inclusion of uncertainty quantification. Unfortunately, the use of exact GPs is prohibitively expensive for large datasets due to their unfavorable numerical complexity of $$O(N^3)$$ O ( N 3 ) in computation and $$O(N^2)$$ O ( N 2 ) in storage. All existing methods addressing this issue utilize some form of approximation—usually considering subsets of the full dataset or finding representative pseudo-points that render the covariance matrix well-structured and sparse. These approximate methods can lead to inaccuracies in function approximations and often limit the user’s flexibility in designing expressive kernels. Instead of inducing sparsity via data-point geometry and structure, we propose to take advantage of naturally-occurring sparsity by allowing the kernel to discover—instead of induce—sparse structure. The premise of this paper is that the data sets and physical processes modeled by GPs often exhibit natural or implicit sparsities, but commonly-used kernels do not allow us to exploit such sparsity. The core concept of exact, and at the same time sparse GPs relies on kernel definitions that provide enough flexibility to learn and encode not only non-zero but also zero covariances. This principle of ultra-flexible, compactly-supported, and non-stationary kernels, combined with HPC and constrained optimization, lets us scale exact GPs well beyond 5 million data points.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1989

Domain decomposition methods for systems of conservation laws: Spectral collocation approximations

Hyperbolic systems of conversation laws are considered which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level a domain deposition method based on an iteration by subdomain procedure was introduced yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The method is set for a general nonlinear problem in several space variables. The convergence analysis, however, is carried out only for a linear one-dimensional system with continuous solutions. A precise form of the error reduction factor at each iteration is derived. Although the method is applied here to the case of spectral collocation approximation only, the idea is fairly general and can be used in a different context as well. For instance, its application to space discretization by finite differences is straight forward.

Quarteroni, Alfio↗

DOE OSTI2024

Griffin: A MOOSE-based reactor physics application for multiphysics simulation of advanced nuclear reactors

Griffin is a Multiphysics Object-Oriented Simulation Environment (MOOSE) based reactor physics application for multiphysics simulations of advanced reactor designs jointly developed by Idaho National Laboratory and Argonne National Laboratory. This paper summarizes the motivation, significance, architecture, design, and features of Griffin. Griffin offers flexible and extensible features to address the challenges associated with advanced reactor designs. These features range from fundamental particle transport to specific reactor physics tasks. The features cover a wide range including on-the-fly and traditional two-step cross-section generation methods, steady-state and transient transport solvers suitable for both heterogeneous and homogeneous models, high-fidelity depletion where thousands of isotopes can be tracked and low-fidelity depletion characterized by burnup, etc. The most fundamental aspect that sets Griffin apart from other reactor analysis codes is that it is developed based on the MOOSE framework. A modular development approach is strongly enforced, with multiphysics being an essential element considered since the beginning of Griffin’s development. Griffin links various MOOSE physics modules and couples to other MOOSE-based applications and non-MOOSE-based applications for multiphyiscs simulations. Griffin includes three modules: ISOXML for preparing and managing multigroup cross sections, radiation transport for solving the neutron transport equation, and reactor analysis for user-oriented reactor physics analysis functionalities. Griffin uses various finite element methods for spatial discretization, multigroup approximation for energy discretization and discrete ordinates method, spherical harmonics expansion method, and diffusion approximation for streaming direction discretization to solve the neutron transport equation. Griffin’s flexibility is evidenced through Griffin’s various applications to fast reactor, high-temperature reactor, pebble bed reactor, molten salt reactor, and microreactor designs. Griffin development follows the software quality assurance procedure for MOOSE-based applications and with software requirements consistent with the ASME NQA-1 standard. Griffin has been adopted into the reactor analysis system for the U.S. NRC and is in use at U.S. companies, universities and national laboratories.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2024

Simplified Approximations of Direct Cumulus Entrainment and Detrainment

Abstract In recent years, direct calculations of simulated cumulus entrainment and detrainment have facilitated new physical insights into these highly elusive but critically important processes. However, these calculations require substantial computational resources that may limit their widespread usage. To facilitate such calculations, two simplified approximations of direct cumulus entrainment and detrainment are examined herein. The first approximation, termed the “semidirect” method, follows a standard bulk approach but makes more realistic assumptions about the sources of entrained and detrained air near the cloud edges. In contrast, the second approximation (the “projection” method) uses the governing equations of motion to project whether grid points near the cloud edge will entrain or detrain as the mean cloud ascends by one grid point. Verification exercises using large-eddy simulations reveal that both methods generally agree better with corresponding direct entrainment/detrainment estimates than the traditional bulk formulation, with the projection method outperforming the semidirect method. The two methods can be used in a synergistic fashion, with the semidirect method helping to optimize the projection method, to suit a wide range of applications. Because the latter incorporates the essential dynamics of entrainment and detrainment at the local scale, it can be used to gain physical insight into the causal mechanisms regulating these complex processes.

Meteorology & Atmospheric Sciences↗

DOE OSTI2018

Design, analysis and numerical experiments for the virtual element p and hp approximations of elliptic eigenvalue problems

We discuss the p- and the hp-versions of the virtual element method for the approximation of the eigenvalues of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schr¨odinger equation with a pseudo-potential term. We present in details the analysis of the p-version of the method, proving the exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the hp-spaces.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2014

Benchmark calculations on the lowest-energy singlet, triplet, and quintet states of the four-electron harmonium atom

For a wide range of confinement strengths ω, explicitly-correlated calculations afford approximate energies E(ω) of the ground and low-lying excited states of the four-electron harmonium atom that are within few μhartree of the exact values, the errors in the respective energy components being only slightly higher. This level of accuracy constitutes an improvement of several orders of magnitude over the previously published data, establishing a set of benchmarks for stringent calibration and testing of approximate electronic structure methods. Its usefulness is further enhanced by the construction of differentiable approximants that allow for accurate computation of E(ω) and its components for arbitrary values of ω. The diversity of the electronic states in question, which involve both single- and multideterminantal first-order wavefunctions, and the availability of the relevant natural spinorbitals and their occupation numbers make the present results particularly useful in research on approximate density-matrix functionals. The four-electron harmonium atom is found to possess the {sup 3}P{sub +} triplet ground state at strong confinements and the {sup 5}S{sub −} quintet ground state at the weak ones, the energy crossing occurring at ω ≈ 0.0240919.

74 ATOMIC AND MOLECULAR PHYSICS↗

NASA NTRS1989

Linear, collisionless, bi-Maxwellian neutral-sheet tearing instability

The problem of linear bi-Maxwellian neutral-sheet tearing instability has been solved, using neither an approximation of the orbits nor the constant-psi approximation. The present method has a larger range of validity than the best previous approximation, the three-region approximation. It is found that as the temperature anisotropy is increased, the growth rate increases and the wavelength associated with the peak growth decreases. It is suggested that if small-island formation is experimentally observed, then temperature anisotropy may be present.

Burkhart, G. R.↗

NASA NTRS1995

Sensitivity analysis in multipole-accelerated panel methods for potential flow

In the design of an airframe, the effect of changing the geometry on resulting computations is necessary for design optimization. The geometry is defined in terms of a series of design variables, including design variables to define the wing planform, tail, canard, pylon, and nacelle. Design optimization in this research is based on how these design variable affect the potential flow. The potential flow is computed as a function of the geometry and location of a series of panels describing the airframe, which are in turn a function of the design variables. Multipole accelerated panel methods improve the computational complexity of the problem and thus are an attractive approach. To utilize the methods in design optimization, it was necessary to define the appropriate sensitivity derivatives. The overhead incurred from finding the sensitivity derivatives in conjunction with the original computation should be small. This research developed the background for multipole-accelerated panel methods and the framework for finding sensitivity derivatives in the methods. Potential flow panel codes are commonly used for powered-lift aerodynamic predictions for three dimensional geometries. Given an airframe which has been discretized into a series of panels to define the airframe geometry, potential is computed as a function of the influence of all panels on all other panels. This is a computationally intensive problem for which efficient solutions are desired to improve the computational time and to allow greater resolution by use of more panels. One such solution is the use of hierarchical multipole methods which entail approximations of the effects of far-field terms. Hierarchical multipole methods have become prevalent in molecular dynamics and gravitational physics, and have been introduced into the fields of capacitance calculations, computational fluid dynamics, and electromagnetics. The methods utilize multipole expansions to describe the effect of bodies (i.e. particles, astrophysical bodies, panels, etc.) within a sphere on points distant from the sphere, where the influence diminishes as a function of distance. The expansions are exact with infinite series, however, for practical computations, the series are truncated and accuracy is selected based on the number of terms retained in the expansions. A hierarchical tree structure groups bodies together based on proximity to allow definition of multipole expansions for each group. The multipole expansions are then used to compute the effect of the bodies in a group on distant bodies.

Leathrum, James F., Jr.↗

DOE OSTI2021

Design of double- and multi-bend achromat lattices with large dynamic aperture and approximate invariants

A numerical method to design nonlinear double- and multi-bend achromat (DBA and MBA) lattices with approximate invariants of motion is investigated. The search for such nonlinear lattices is motivated by Fermilab’s Integrable Optics Test Accelerator (IOTA), whose design is based on an integrable Hamiltonian system with two invariants of motion. While it may not be possible to design an achromatic lattice for a dedicated synchrotron light source storage ring with one or more exact invariants of motion, it is possible to tune the sextupoles and octupoles in existing DBA and MBA lattices to produce approximate invariants. In our procedure, the lattice is tuned while minimizing the turn-by-turn fluctuations of the Courant-Snyder actions J x and J y at several distinct amplitudes, while simultaneously minimizing diffusion of the on-energy betatron tunes. The resulting lattices share some important features with integrable ones, such as a large dynamic aperture, trajectories confined to invariant tori, robustness to resonances and errors, and a large amplitude-dependent tune-spread. Compared to the nominal NSLS-II lattice, the single- and multi-bunch instability thresholds are increased and the bunch-by-bunch feedback gain can be reduced.

43 PARTICLE ACCELERATORS↗

NASA NTRS1997

Efficient algorithms for robust feature matching

One of the basic building blocks in any point-based registration scheme involves matching feature points that are extracted from the sensed image to their counterparts in the reference image. This leads to the fundamental problem of point matching: given two sets of points, find the affine transformation that transforms one point set so that its distance from the other point set is minimized. Because of measurement errors and the presence of outlying data points, it is important that the distance measure between two point sets be robust to these effects. We measure distances using the generalized Hausdorff distance. Point matching can be a computationally intensive task, and there have been a number of algorithms and approaches proposed for solving this problem both theoretical and applied. We present two approaches to the point matching problem, in an attempt to reduce the computational complexity of the problem, while still providing guarantees on the quality of the final match. Our first method is an approximation algorithm, which is loosely based on a branch-and-bound approach due to Huttenlocher and Rucklidge. We show that by varying the approximation error bounds, it is possible to achieve a tradeoff between the quality of the match and the running time of the algorithm. Our second method involves a Monte Carlo method for accelerating the search process used in the first algorithm. With high probability this method succeeds in finding an approximately optimal match. We establish the efficiency of our approaches empirically.

Mount, David M.↗

NASA NTRS1994

Vibration suppression with approximate finite dimensional compensators for distributed systems: Computational methods and experimental results

Based on a distributed parameter model for vibrations, an approximate finite dimensional dynamic compensator is designed to suppress vibrations (multiple modes with a broad band of frequencies) of a circular plate with Kelvin-Voigt damping and clamped boundary conditions. The control is realized via piezoceramic patches bonded to the plate and is calculated from information available from several pointwise observed state variables. Examples from computational studies as well as use in laboratory experiments are presented to demonstrate the effectiveness of this design.

Banks, H. T.↗

DOE OSTI2015

A NEW APPROACH TO DETERMINE OPTICALLY THICK H{sub 2} COOLING AND ITS EFFECT ON PRIMORDIAL STAR FORMATION

We present a new method for estimating the H{sub 2} cooling rate in the optically thick regime in simulations of primordial star formation. Our new approach is based on the TreeCol algorithm, which projects matter distributions onto a spherical grid to create maps of column densities for each fluid element in the computational domain. We have improved this algorithm by using the relative gas velocities to weight the individual matter contributions with the relative spectral line overlaps, in order to properly account for the Doppler effect. We compare our new method to the widely used Sobolev approximation, which yields an estimate for the column density based on the local velocity gradient and the thermal velocity. This approach generally underestimates the photon escape probability because it neglects the density gradient and the actual shape of the cloud. We present a correction factor for the true line overlap in the Sobolev approximation and a new method based on local quantities, which fits the exact results reasonably well during the collapse of the cloud, with the error in the cooling rates always being less than 10%. Analytical fitting formulae fail at determining the photon escape probability after formation of the first protostar (error of ∼40%) because they are based on the assumption of spherical symmetry and therefore break down once a protostellar accretion disk has formed. Our method yields lower temperatures and hence promotes fragmentation for densities above ∼10{sup 10} cm{sup –3} at a distance of ∼200 AU from the first protostar. Since the overall accretion rates are hardly affected by the cooling implementation, we expect Pop III stars to have lower masses in our simulations, compared to the results of previous simulations that used the Sobolev approximation.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2019

Recommended Nuclear Data for the Production of Selected Therapeutic Radionuclides

Increasing interest in the treatment of human disease by means of targeted radionuclide-based therapies requires an accurate understanding of achievable yields and quantification of realisable degrees of purity. Important studies within a more extensive Coordinated Research Project of the International Atomic Energy Agency (IAEA CRP) have included in-depth assessments and evaluations of all available measured cross-section data for ten nuclear reactions leading to the formation of 131 Cs, 131 Ba, 225 Ac, 225 Ra, 227 Th, 227 Ac, 230 U and 230 Pa for various on-going and proposed therapeutic applications in nuclear medicine. Equivalent studies were also undertaken of possible reactions to produce the 178W /low-spin 178m Ta generator system that has been clinically applied in diagnosis by means of SPECT - these particular cross-section studies and recommendations are included in this paper for completeness. All available measured data have been critically reviewed and assessed to produce statistically consistent datasets. Selected datasets have been fitted by a least-squares method with Padé approximations of variable order to evaluate the production cross sections and corresponding uncertainties. New measurements have been made in some instances, and reported data have been adjusted to accommodate new nuclear decay or monitor reaction data which altered the original reported quantities adversely. Insufficient measured data are available in many cases to permit a reliable evaluation of cross sections, and these instances are discussed. Finally, recommended therapeutic radionuclide production reaction data and their uncertainties are available on the Web page of the IAEA Nuclear Data Section at www-nds.iaea.org/radionuclides/ and also at the IAEA medical portal www-nds.iaea.org/medportal/.

63 RADIATION, THERMAL, AND OTHER ENVIRON. POLLUTAN↗

DOE OSTI2017

Efficient Scalable Parallel Higher Order Direct MoM-SIE Method With Hierarchically Semiseparable Structures for 3-D Scattering

A novel fast scalable parallel algorithm is proposed for the solution of large 3-D scattering problems based on: 1) the double (geometrical and current-approximation) higher order (DHO) method of moments (MoM) in the surface integral equation (SIE) formulation and 2) a direct solver for dense linear systems utilizing hierarchically semiseparable (HSS) structures. Namely, an HSS matrix representation is used for compression, factorization, and solution of the system matrix. In addition, a rank-revealing QR decomposition for memory compression is used, with a stopping criterion in terms of the relative rank tolerance value. One method for geometrical preprocessing of the scatterers based on the cobblestone distance sorting technique is employed in order to enhance the HSS algorithm accuracy and parallelization. Numerical examples show how the accuracy of the DHO HSS-MoM-SIE method is easily controllable by using the relative tolerance for the matrix compression. Furthermore, the examples demonstrate low memory consumption, as well as much faster simulation time, when compared to the direct LU decomposition. The method enables dramatically faster monostatic scattering computations than iterative solvers and reduced number of unknowns when compared to low-order discretizations. Finally, great scalability of the algorithm is demonstrated on more than one thousand processes.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2016

Efficient linear-scaling second-order Møller-Plesset perturbation theory: The divide–expand–consolidate RI-MP2 model

The Resolution of the Identity second-order Møller-Plesset perturbation theory (RI-MP2) method is implemented within the linear-scaling Divide-Expand-Consolidate (DEC) framework. In a DEC calculation, the full molecular correlated calculation is replaced by a set of independent fragment calculations each using a subset of the total orbital space. The number of independent fragment calculations scales linearly with the system size, rendering the method linear-scaling and massively parallel. The DEC-RI-MP2 method can be viewed as an approximation to the DEC-MP2 method where the RI approximation is utilized in each fragment calculation. The individual fragment calculations scale with the fifth power of the fragment size for both methods. However, the DEC-RI-MP2 method has a reduced prefactor compared to DEC-MP2 and is well-suited for implementation on massively parallel supercomputers, as demonstrated by test calculations on a set of medium-sized molecules. The DEC error control ensures that the standard RI-MP2 energy can be obtained to the predefined precision. The errors associated with the RI and DEC approximations are compared, and it is shown that the DEC-RI-MP2 method can be applied to systems far beyond the ones that can be treated with a conventional RI-MP2 implementation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1987

Approximate Analysis of Semiconductor Laser Arrays

Simplified equation yields useful information on gains and output patterns. Theoretical method based on approximate waveguide equation enables prediction of lateral modes of gain-guided planar array of parallel semiconductor lasers. Equation for entire array solved directly using piecewise approximation of index of refraction by simple functions without customary approximation based on coupled waveguid modes of individual lasers. Improved results yield better understanding of laser-array modes and help in development of well-behaved high-power semiconductor laser arrays.

Marshall, William K.↗

NASA NTRS1974

Multiple zeros of polynomials

For polynomials of higher degree, iterative numerical methods must be used. Four iterative methods are presented for approximating the zeros of a polynomial using a digital computer. Newton's method and Muller's method are two well known iterative methods which are presented. They extract the zeros of a polynomial by generating a sequence of approximations converging to each zero. However, both of these methods are very unstable when used on a polynomial which has multiple zeros. That is, either they fail to converge to some or all of the zeros, or they converge to very bad approximations of the polynomial's zeros. This material introduces two new methods, the greatest common divisor (G.C.D.) method and the repeated greatest common divisor (repeated G.C.D.) method, which are superior methods for numerically approximating the zeros of a polynomial having multiple zeros. These methods were programmed in FORTRAN 4 and comparisons in time and accuracy are given.

Wood, C. A.↗