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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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At least 577 records · Page 32
All sourcesNASA NTRSDOE OSTI
DOE OSTI2019

A Framework for Distributed Approximation of Moments with Higher-Order Derivatives through Automatic Differentiation

We present a framework for the distributed approximation of moments, enabling the evaluation of the uncertainty in a dynamical system. The first and second moment, mean, and variance are computed with up to third-order Taylor series expansion. The required derivatives for the expansion are generated automatically by automatic differentiation and propagated through an implicit time stepper. The computational kernels are the accumulation of the derivatives (Jacobian, Hessian, tensor) and the covariance matrix. We apply distributed parallelism to the Hessian or third-order tensor, and the user merely has to provide a function for the differential equation, thus achieving similar ease of use as Monte Carlo-based methods. We demonstrate our approach using with benchmarks on Theta, a KNL-based system at the Argonne Leadership Computing Facility.

Automatic Differentiation↗
DOE OSTI2023

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2017

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative \(\mathbf {d}\) , Hodge star \(\star \) , and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator \(\overline{\mathbf {d}}\) and Hodge star operator \(\overline{\star }\) showing each converge spectrally to \(\mathbf {d}\) and \(\star \) . We demonstrate how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We introduce findings for the Laplace–Beltrami equations demonstrating our approach.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2025

Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations

In this work, we develop implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) (Ceruti et al. in BIT Numer Math 62(4):1149–1174, 2022) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using an implicit solves for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. Here, we prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Representation of Solutions of Systems of Linear Differential Equations with Multiple Delays and Linear Parts Given by Nonpermutable Matrices

In recent results on representation of the solutions of systems of delayed differential equations, the condition that the linear parts are given by pairwise permutable matrices was assumed. We show how this strong condition can be avoided and derive a representation of solutions of systems of differential equations with nonconstant coefficients and variable delays. The results are applied to a system with two constant delays. In addition, the nonexistence of blow-up solutions is proved for nonlinear systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Differential-Invariant Solutions

We construct differential-invariant solutions to automorphic systems and propose a method for classifying differential-invariant submodels.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On Periodic Boundary-Value Problems for Systems of Functional-Differential Equations

We obtain necessary and sufficient conditions for the existence of a unique solution to a periodic boundary-value problem for all systems of first-order functional-differential equations from a given family of systems. Families of systems of functional-differential equations are detrmined by the norms of positive functional operators of equations of the system. The verification of necessary and sufficient conditions of the existence of a unique periodic solution for all systems from a given family consists of the verification of positivity of a finite number of a real-valued functions defined on a finite-dimensional set.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On a Certain First-Order Differential Equation with Delay

We consider the Cauchy problem for a first-order quasilinear differential equation with delayed argument of neutral type, and obtain sufficient conditions of existence and uniqueness of its solutions. Proofs of the solvability of nonlinear problems, estimates of solutions, and constructions of approximate methods are based on Chaplygin-type theorems on differential inequalities.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Stochastic differential equation in a random environment

Solutions of the Itô stochastic differential equation in a random environment are considered. The random environment is formed by the generalized telegraph process. It is proved that the initial problem is equivalent to a system of two stochastic differential equations with nonrandom coefficients. The first equation is the Itô equation, and the initial process is its solution. The second equation is an equation with Poisson process, and its solution is a generalized telegraph process. The theorems of existence and uniqueness of strong and weak solutions are proved.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Nonlocal Boundary-Value Problem for a Differential-Operator Equation with Weak Nonlinearity in the Spaces of Dirichlet–Taylor Series with Fixed Spectrum

We study the nonlocal boundary-value problem for a differential-operator equation with nonlinear right-hand side and an operator B = (B{sub 1},…B{sub p}), where its components B{sub j} ≡ z{sub j}∂/∂z{sub j}, j = 1,…,p, are the operators of generalized differentiation with respect to the complex variable z{sub j}. By using the Nash–Moser iterative scheme, we establish the conditions of solvability of this problem in the scale of spaces of functions of several complex variables, which are Dirichlet–Taylor series with fixed spectrum.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

On the Solutions of Oscillating-Type Countable Differential Systems with Slowly Varying Parameters

For a countable quasilinear system of differential equations whose coefficients have the form of absolutely and uniformly convergent Fourier series with slowly varying coefficients and frequency, we obtain conditions for the existence of a partial solution of a similar structure. As a result, we establish conditions for the possibility of complete decomposition of a countable linear homogeneous differential system with coefficients of the same structure.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Fitting Matérn smoothness parameters using automatic differentiation

The Mat$\acute{e}$rn covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter $\nu$ gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Mat$\acute{e}$rn covariance function with respect to $\nu$ require derivatives of the modified second-kind Bessel function $K$ $\nu$ with respect to $\nu$. While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing $\nu$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $\nu$ use finite difference estimates for $\partial$ $\nu$ $K$ $\nu$ . In this work, we introduce a new implementation of $K$$\nu$ that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. Here, we provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Time-parallel solutions to differential equations via functional optimization

We describe an approach to solving a generic time-dependent differential equation (DE) that recasts the problem as a functional optimization one. The techniques employed to solve for the functional minimum, which we relate to the Sobolev Gradient method, allow for large-scale parallelization in time, and therefore potential faster “wall-clock” time computing on machines with significant parallel computing capacity. We are able to come up with numerous different discretizations and approximations for our optimization-derived equations, each of which either puts an existing approach, the Parareal method, in an optimization context, or provides a new time-parallel (TP) scheme with potentially faster convergence to the DE solution. We describe how the approach is particularly effective for solving multiscale DEs and present TP schemes that incorporate two different solution scales. Sample results are provided for three differential equations, solved with TP schemes, and we discuss how the choice of TP scheme can have an orders of magnitude effect on the accuracy or convergence rate.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Fast iterative refinement method for mixed systems of integral and fractional integro-differential equations

The authors’ new method of iterative refinement has been successfully applied to the linear fractional Fredholm integro-differential equation. In this work, we adapt our method to obtain approximate analytical solutions for linear Volterra equations of both the first and the second kind of fractional type. A detailed convergence analysis is presented for each kind. The authors also apply their method to the challenging first kind linear Fredholm integral equation. Three different cases are considered to test the efficacy of our method. These include mixed systems of various forms of linear integral and fractional integro-differential equations. We compare our method with six well-established methods: Picard’s successive approximations, an accelerated form of the latter, the Adomian decomposition, the variational iteration, the homotopy perturbation and the homotopy analysis. The results show the versatility of the new method in solving a wide variety of equations, its high convergence speed and accuracy and its capability of directly dealing with equations of the first kind. We also prove that Picard’s method for linear equations is a special case of our method.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Numerical approach based on fractional-order Lagrange polynomials for solving a class of fractional differential equations

In this study, we propose a new set of fractional functions based on the Lagrange polynomials to solve a class of fractional differential equations. Fractional differential equations are the best tools for modelling natural phenomenon that are elaborated by fractional calculus. Therefore, we need an accurate and efficient technique for solving them. The main purpose of this article is to generalize new functions based on Lagrange polynomials to the fractional calculus. At first, we present a new representation of Lagrange polynomials and in continue, we propose a new set of fractional-order functions which are called fractional-order Lagrange polynomials (FLPs). Besides, a general formulation for operational matrices of fractional integration and derivative of FLPs on arbitrary nodal points are extracted. These matrices are obtained using Laplace transform. The initial value problems is reduced to the system of algebraic equations using the operational matrix of fractional integration and collocation method. Also, we find the upper bound of error vector for the fractional integration operational matrix and we indicate convergence of approximations of FLPs. Illustrative examples are included to demonstrate the validity and applicability of the proposed technique.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Differential and total cross sections and astrophysical S-factors for {sup 2}H(d,n){sup 3}He and {sup 2}H(d,p){sup 3}H reactions in a wide energy range

Differential cross sections of deuteron–deuteron fusion reactions are needed in astrophysics, precision cosmology, nuclear physics, and controlled nuclear fusion research. New parametrizations for center of mass differential cross sections, total cross sections and astrophysical S-factors in a wide range of specific energy from 0.5 keV/u to about 0.5 GeV/u have been obtained. Straightforward approximating formulae readily suitable for multithreaded algorithmization are given here for {sup 2}H(d,n){sup 3}He and {sup 2}H(d,p){sup 3}H reactions. The energy range is about two orders of magnitude broader than that in the tables published earlier for neutron branch.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2018

The role of Zhx2 transcription factor in bipolar cell differentiation during mouse retinal development

Highlights: • Zhx2 is expressed in developing retina. • Zhx2 is essential for subsets of bipolar cell proper differentiation. • Overexpression of Zhx2 showed perturbation of bipolar differentiation. We found that the Zhx2 gene (whose product is known to act as a tumor suppressor in hepatocellular carcinoma) is expressed in embryonic retinal progenitors and in developing cone bipolar cells in the postnatal retina, as well as in Müller glia in the mature retina. To examine the functions of Zhx2 protein during retinal development, we performed loss- and gain-of-function analyses using a retinal explant culture system. We introduced a plasmid encoding Zhx2 shRNA into isolated mouse retinas at E17.5, and the retinas were cultured as explants. After 3 days of culture, proliferation was slightly enhanced, leading to retinas thicker than in the control, but this phenomenon was observed only transiently. After 14 days of the culture, the thickness and gross morphology of retinas expressing sh-Zhx2 were indistinguishable from those of the control. The numbers of rod cells, amacrine cells, and Müller glia were the same in both groups. However, although the total number of bipolar cells was the same, the experimental group saw an increased population of ON bipolar cells, and decreased numbers of a subset of OFF bipolar cells. We also examined the effects of overexpression of Zhx2. Although Zhx2 acts as a tumor suppressor, its overexpression in developing retinas did not lead to any discernible difference in retinal thickness, suggesting that proliferation activity was not affected. After 14 days of explant culture, the total number of bipolar cells decreased, and subset composition was altered. Taken together, these results suggest that Zhx2 plays roles in the regulation of bipolar cell subset fate determination during retinal development.

60 APPLIED LIFE SCIENCES↗