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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 OSTI2018

Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation

In this paper, we develop a Bernstein dual-Petrov–Galerkin method for the numerical simulation of a two-dimensional fractional diffusion equation. A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones. We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix-based approach for the spatial discretization. It is shown that the method leads to banded linear systems that can be solved efficiently. The stability and convergence of the proposed method is discussed. Finally, some numerical examples are provided to support the theoretical claims and to show the accuracy and efficiency of the method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

An operational matrix method for solving variable-order fractional biharmonic equation

In this paper, we focus on the fourth-order biharmonic equation which also has been appeared with fractional derivative order. Then, we generalize the fractional biharmonic equation using the concept of variable-order (V-O) fractional derivative in the type of Caputo namely V-O fractional biharmonic equation (V-OFBE). To establish a method for solving V-OFBE, we firstly derive the operational matrix of V-O fractional derivatives for the shifted second kind Chebyshev polynomials. For obtaining this operational matrix with a general procedure, we implement the analytical form of these polynomials and some properties of the V-O fractional derivatives. This approach reduces the V-OFBE to a system of algebraic equations which considerably decreases the following computations. The applicability and reliability of the proposed method are investigated by solving some test problems. The experimental results show the spectral rate of convergence.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical solution of linear and nonlinear hyperbolic telegraph type equations with variable coefficients using shifted Jacobi collocation method

The telegraph equation is one of the important equations of mathematical physics. In this work, a spectral collocation scheme is proposed for the numerical solutions of one- and two-dimensional linear telegraph equations and telegraph equations with nonlinear forcing term. The homogeneous initial and boundary conditions are satisfied exactly by expanding the unknown variable using polynomial bases of functions which are built upon the Jacobi polynomials. The suggested scheme is successfully developed for the aforementioned problem with nonhomogeneous data. Extensive numerical experiments are presented to verify the efficiency of the proposed scheme.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

A Müntz wavelets collocation method for solving fractional differential equations

In this paper, we introduce a new set of wavelets, namely Müntz wavelets. These wavelets are defined based on Müntz Legendre polynomials which have the property of orthogonality. We stabilize the wavelet coefficients by utilizing Jacobi polynomials to construct the modified Müntz wavelets. Next, we present a collocation scheme based on Müntz wavelets for solving fractional differential equations. The fractional derivative is described in the Caputo sense. Some error estimates are given and numerical examples are included to demonstrate the applicability and accuracy of the proposed method. This includes the elastic-dissipative mathematical model of linear mechanical devices known as Boltzman operator.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2019

Efficiency of Uncertainty Propagation Methods for Estimating Output Moments

Uncertainty propagation methods are used to estimate the distribution of model outputs resulting from a set of uncertain model outputs. There are a number of uncertainty propagation methods available in literature. This paper compares six non-intrusive uncertainty propagation methods, Latin Hypercube Sampling, Full Factorial Integration, Univariate Dimension Reduction, Halton series, Sobol series, and Polynomial Chaos Expansion, in terms of their efficiency for estimating the first four moments of the output distribution using computational experiments. Here, the results suggest employing FFNI if there are few uncertain inputs, up to three. Uncertainty propagation methods that utilize Halton and Sobol series are found to be robust for estimating output moments as the number of uncertain inputs increases. In general, higher order polynomial chaos expansion approximations (3rd-5th order) obtained accurate estimates of model outputs with fewer model evaluations.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

Parameterized representation of macroscopic cross section for PWR reactor

Here we describe, by means of Chebyshev polynomials, a parameterized representation of the homogenized macroscopic cross section for PWR fuel element as a function of soluble boron concentration, moderator temperature, fuel temperature, moderator density and U-235(92) enrichment. The cross-section data analyzed are fission, scattering, total, transport, absorption and capture. The parameterization enables a quick and easy determination of problem-dependent cross-sections to be used in few group calculations. The methodology presented in this paper will allow generation of group cross-section data from stored polynomials to perform PWR core calculations without the need to generate them based on computer code calculations using standard steps. The results obtained by the proposed methodology when compared with results from the SCALE code calculations show very good agreement.

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗
DOE OSTI2013

Entanglement in a model for Hawking radiation: An application of quadratic algebras

Quadratic polynomially deformed su(1,1) and su(2) algebras are utilized in model Hamiltonians to show how the gravitational system consisting of a black hole, infalling radiation and outgoing (Hawking) radiation can be solved exactly. The models allow us to study the long-time behaviour of the black hole and its outgoing modes. In particular, we calculate the bipartite entanglement entropies of subsystems consisting of (a) infalling plus outgoing modes and (b) black hole modes plus the infalling modes, using the Janus-faced nature of the model. The long-time behaviour also gives us glimpses of modifications in the character of Hawking radiation. Finally, we study the phenomenon of superradiance in our model in analogy with atomic Dicke superradiance. - Highlights: Black-Right-Pointing-Pointer We examine a toy model for Hawking radiation with quantized black hole modes. Black-Right-Pointing-Pointer We use quadratic polynomially deformed su(1,1) algebras to study its entanglement properties. Black-Right-Pointing-Pointer We study the 'Dicke Superradiance' in black hole radiation using quadratically deformed su(2) algebras. Black-Right-Pointing-Pointer We study the modification of the thermal character of Hawking radiation due to quantized black hole modes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Cartan gravity, matter fields, and the gauge principle

Gravity is commonly thought of as one of the four force fields in nature. However, in standard formulations its mathematical structure is rather different from the Yang–Mills fields of particle physics that govern the electromagnetic, weak, and strong interactions. This paper explores this dissonance with particular focus on how gravity couples to matter from the perspective of the Cartan-geometric formulation of gravity. There the gravitational field is represented by a pair of variables: (1) a ‘contact vector’ V{sup A} which is geometrically visualized as the contact point between the spacetime manifold and a model spacetime being ‘rolled’ on top of it, and (2) a gauge connection A{sub μ}{sup AB}, here taken to be valued in the Lie algebra of SO(2,3) or SO(1,4), which mathematically determines how much the model spacetime is rotated when rolled. By insisting on two principles, the gauge principle and polynomial simplicity, we shall show how one can reformulate matter field actions in a way that is harmonious with Cartan’s geometric construction. This yields a formulation of all matter fields in terms of first order partial differential equations. We show in detail how the standard second order formulation can be recovered. In particular, the Hodge dual, which characterizes the structure of bosonic field equations, pops up automatically. Furthermore, the energy–momentum and spin-density three-forms are naturally combined into a single object here denoted the spin-energy–momentum three-form. Finally, we highlight a peculiarity in the mathematical structure of our first-order formulation of Yang–Mills fields. This suggests a way to unify a U(1) gauge field with gravity into a SO(1,5)-valued gauge field using a natural generalization of Cartan geometry in which the larger symmetry group is spontaneously broken down to SO(1,3)×U(1). The coupling of this unified theory to matter fields and possible extensions to non-Abelian gauge fields are left as open questions. -- Highlights: •Develops Cartan gravity to include matter fields. •Coupling to gravity is done using the standard gauge prescription. •Matter actions are manifestly polynomial in all field variables. •Standard equations recovered on-shell for scalar, spinor and Yang–Mills fields. •Unification of a U(1) field with gravity based on the orthogonal group SO(1,5)

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2013

The analytic solution for the power series expansion of Heun function

The Heun function generalizes all well-known special functions such as Spheroidal Wave, Lame, Mathieu, and hypergeometric {sub 2}F{sub 1}, {sub 1}F{sub 1} and {sub 0}F{sub 1} functions. Heun functions are applicable to diverse areas such as theory of black holes, lattice systems in statistical mechanics, solution of the Schrödinger equation of quantum mechanics, and addition of three quantum spins. In this paper I will apply three term recurrence formula (Y.S. Choun, (arXiv:1303.0806), 2013) to the power series expansion in closed forms of Heun function (infinite series and polynomial) including all higher terms of A{sub n}’s. Section 3 contains my analysis on applying the power series expansions of Heun function to a recent paper (R.S. Maier, Math. Comp. 33 (2007) 811–843). Due to space restriction final equations for the 192 Heun functions are not included in the paper, but feel free to contact me for the final solutions. Section 4 contains two additional examples using the power series expansions of Heun function. This paper is 3rd out of 10 in series “Special functions and three term recurrence formula (3TRF)”. See Section 5 for all the papers in the series. The previous paper in series deals with three term recurrence formula (3TRF). The next paper in the series describes the integral forms of Heun function and its asymptotic behaviors analytically. -- Highlights: •Power series expansion for infinite series of Heun function using 3 term rec. form. •Power series for polynomial which makes B{sub n} term terminated of Heun function. •Applicable to areas such as the Teukolsky equation in Kerr–Newman–de Sitter geometries.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2013

On solvability and integrability of the Rabi model

The quasi-exactly solvable Rabi model is investigated within the framework of the Bargmann Hilbert space of analytic functions B. On applying the theory of orthogonal polynomials, the eigenvalue equation and eigenfunctions are shown to be determined in terms of three systems of monic orthogonal polynomials. The formal Schweber quantization criterion for an energy variable x, originally expressed in terms of infinite continued fractions, can be recast in terms of a meromorphic function F(z)=a{sub 0}+∑{sub k=1}{sup ∞}M{sub k}/(z−ξ{sub k}) in the complex plane C with real simple poles ξ{sub k} and positive residues M{sub k}. The zeros of F(x) on the real axis determine the spectrum of the Rabi model. One obtains at once that, on the real axis, (i) F(x) monotonically decreases from +∞ to −∞ between any two of its subsequent poles ξ{sub k} and ξ{sub k+1}, (ii) there is exactly one zero of F(x) for x∈(ξ{sub k},ξ{sub k+1}), and (iii) the spectrum corresponding to the zeros of F(x) does not have any accumulation point. Additionally, one can provide a much simpler proof that the spectrum in each parity eigenspace B{sub ±} is necessarily nondegenerate. Thereby the calculation of spectra is greatly facilitated. Our results allow us to critically examine recent claims regarding solvability and integrability of the Rabi model. -- Highlights: •Schweber’s criterion shown equivalent to a meromorphic function F with real simple poles and positive residues. •Calculation of spectra determined as zeros of F greatly facilitated: one has exactly one zero between subsequent poles of F. •Spectrum in a given parity eigenspace is necessarily nondegenerate. •Recent claims regarding solvability and integrability of the Rabi model found to be largely unsubstantiated.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Low depth quantum circuits for Ising models

A scheme for measuring complex temperature partition functions of Ising models is introduced. Two applications of this scheme are presented. First, through appropriate Wick rotations, those amplitudes can be analytically continued to yield estimates for partition functions of Ising models. Bounds on the estimated error are provided through a central-limit theorem whose validity extends beyond the present context; it holds for example for estimations of the Jones polynomial. The kind of state preparations and measurements involved in this application can be made independent of the system size or the parameters of the system being simulated. Second, the scheme allows to accurately estimate non-trivial invariants of links. Another result concerns the computational power of estimations of partition functions for real temperature classical ferromagnetic Ising models. We provide conditions under which estimating such partition functions allows to reconstruct scattering amplitudes of quantum circuits, making the problem BQP-hard. We also show fidelity overlaps for ground states of quantum Hamiltonians, which serve as a witness to quantum phase transitions, can be estimated from classical Ising model partition functions. Finally, we discuss how accurate corner magnetisation measurements on thermal states of two-dimensional Ising models lead to fully polynomial random approximation schemes (FPRAS) for the partition function.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Haydock’s recursive solution of self-adjoint problems. Discrete spectrum

Haydock’s recursive solution is shown to underline a number of different concepts such as (i) quasi-exactly solvable models, (ii) exactly solvable models, (iii) three-term recurrence solutions based on Schweber’s quantization criterion in Hilbert spaces of entire analytic functions, and (iv) a discrete quantum mechanics of Odake and Sasaki. A recurrent theme of Haydock’s recursive solution is that the spectral properties of any self-adjoint problem can be mapped onto a corresponding sequence of polynomials (p{sub n}(E)) in energy variable E. The polynomials (p{sub n}(E)) are orthonormal with respect to the density of states n{sub 0}(E) and energy eigenstate |E〉 is the generating function of (p{sub n}(E)). The generality of Haydock’s recursive solution enables one to see the different concepts from a unified perspective and mutually benefiting from each other. Some results obtained within the particular framework of any of (i) to (iv) may have much broader significance.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference

Operator inference learns low-dimensional dynamical-system models with polynomial nonlinear terms from trajectories of high-dimensional physical systems (non-intrusive model reduction). Here, this work focuses on the large class of physical systems that can be well described by models with quadratic and cubic nonlinear terms and proposes a regularizer for operator inference that induces a stability bias onto learned models. The proposed regularizer is physics informed in the sense that it penalizes higher-order terms with large norms and so explicitly leverages the polynomial model form that is given by the underlying physics. This means that the proposed approach judiciously learns from data and physical insights combined, rather than from either data or physics alone. Additionally, a formulation of operator inference is proposed that enforces model constraints for preserving structure such as symmetry and definiteness in linear terms. Numerical results demonstrate that models learned with operator inference and the proposed regularizer and structure preservation are accurate and stable even in cases where using no regularization and Tikhonov regularization leads to models that are unstable.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: Adaptivity and generalizations

This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O((m/N{sub p}){sup 3}) for a quadratic nonlinear SPDE, where m is the number of mode pairs used in the DyBO method and N{sub p} is the number of elements in the polynomial basis in gPC. The effective dimensions of the stochastic solutions have been found to be small in many applications, so we can expect m is much smaller than N{sub p} and computational savings of our DyBO method against gPC are dramatic. The adaptive strategy plays an essential role for the DyBO method to be effective in solving some challenging problems. Another important contribution of this paper is the generalization of the DyBO formulation for a system of time-dependent SPDEs. Several numerical examples are provided to demonstrate the effectiveness of our method, including the Navier–Stokes equations and the Boussinesq approximation with Brownian forcing.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Minimal multi-element stochastic collocation for uncertainty quantification of discontinuous functions

We propose a multi-element stochastic collocation method that can be applied in high-dimensional parameter space for functions with discontinuities lying along manifolds of general geometries. The key feature of the method is that the parameter space is decomposed into multiple elements defined by the discontinuities and thus only the minimal number of elements are utilized. On each of the resulting elements the function is smooth and can be approximated using high-order methods with fast convergence properties. The decomposition strategy is in direct contrast to the traditional multi-element approaches which define the sub-domains by repeated splitting of the axes in the parameter space. Such methods are more prone to the curse-of-dimensionality because of the fast growth of the number of elements caused by the axis based splitting. The present method is a two-step approach. Firstly a discontinuity detector is used to partition parameter space into disjoint elements in each of which the function is smooth. The detector uses an efficient combination of the high-order polynomial annihilation technique along with adaptive sparse grids, and this allows resolution of general discontinuities with a smaller number of points when the discontinuity manifold is low-dimensional. After partitioning, an adaptive technique based on the least orthogonal interpolant is used to construct a generalized Polynomial Chaos surrogate on each element. The adaptive technique reuses all information from the partitioning and is variance-suppressing. We present numerous numerical examples that illustrate the accuracy, efficiency, and generality of the method. When compared against standard locally-adaptive sparse grid methods, the present method uses many fewer number of collocation samples and is more accurate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2014

A stochastic Galerkin method for the Euler equations with Roe variable transformation

The Euler equations subject to uncertainty in the initial and boundary conditions are investigated via the stochastic Galerkin approach. We present a new fully intrusive method based on a variable transformation of the continuous equations. Roe variables are employed to get quadratic dependence in the flux function and a well-defined Roe average matrix that can be determined without matrix inversion. In previous formulations based on generalized polynomial chaos expansion of the physical variables, the need to introduce stochastic expansions of inverse quantities, or square roots of stochastic quantities of interest, adds to the number of possible different ways to approximate the original stochastic problem. We present a method where the square roots occur in the choice of variables, resulting in an unambiguous problem formulation. The Roe formulation saves computational cost compared to the formulation based on expansion of conservative variables. Moreover, the Roe formulation is more robust and can handle cases of supersonic flow, for which the conservative variable formulation fails to produce a bounded solution. For certain stochastic basis functions, the proposed method can be made more effective and well-conditioned. This leads to increased robustness for both choices of variables. We use a multi-wavelet basis that can be chosen to include a large number of resolution levels to handle more extreme cases (e.g. strong discontinuities) in a robust way. For smooth cases, the order of the polynomial representation can be increased for increased accuracy.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2015

Mesh refinement for uncertainty quantification through model reduction

We present a novel way of deciding when and where to refine a mesh in probability space in order to facilitate uncertainty quantification in the presence of discontinuities in random space. A discontinuity in random space makes the application of generalized polynomial chaos expansion techniques prohibitively expensive. The reason is that for discontinuous problems, the expansion converges very slowly. An alternative to using higher terms in the expansion is to divide the random space in smaller elements where a lower degree polynomial is adequate to describe the randomness. In general, the partition of the random space is a dynamic process since some areas of the random space, particularly around the discontinuity, need more refinement than others as time evolves. In the current work we propose a way to decide when and where to refine the random space mesh based on the use of a reduced model. The idea is that a good reduced model can monitor accurately, within a random space element, the cascade of activity to higher degree terms in the chaos expansion. In turn, this facilitates the efficient allocation of computational sources to the areas of random space where they are more needed. For the Kraichnan–Orszag system, the prototypical system to study discontinuities in random space, we present theoretical results which show why the proposed method is sound and numerical results which corroborate the theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗