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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 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2016

Hamilton-Jacobi formalism for inflation with non-minimal derivative coupling

In inflation with nonminimal derivative coupling there is not a conformal transformation to the Einstein frame where calculations are straightforward, and thus in order to extract inflationary observables one needs to perform a detailed and lengthy perturbation investigation. In this work we bypass this problem by performing a Hamilton-Jacobi analysis, namely rewriting the cosmological equations considering the scalar field to be the time variable. We apply the method to two specific models, namely the power-law and the exponential cases, and for each model we calculate various observables such as the tensor-to-scalar ratio, and the spectral index and its running. We compare them with 2013 and 2015 Planck data, and we show that they are in a very good agreement with observations.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2013

Effect of the refractive index on the hawking temperature: an application of the Hamilton-Jacobi method

Hawking radiation of a non-asymptotically flat 4-dimensional spherically symmetric and static dilatonic black hole (BH) via the Hamilton-Jacobi (HJ) method is studied. In addition to the naive coordinates, we use four more different coordinate systems that are well-behaved at the horizon. Except for the isotropic coordinates, direct computation by the HJ method leads to the standard Hawking temperature for all coordinate systems. The isotropic coordinates allow extracting the index of refraction from the Fermat metric. It is explicitly shown that the index of refraction determines the value of the tunneling rate and its natural consequence, the Hawking temperature. The isotropic coordinates in the conventional HJ method produce a wrong result for the temperature of the linear dilaton. Here, we explain how this discrepancy can be resolved by regularizing the integral possessing a pole at the horizon.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Leveraging Multitime Hamilton–Jacobi PDEs for Certain Scientific Machine Learning Problems

Hamilton-Jacobi partial differential equations (HJ PDEs) have deep connections with a wide range of fields, including optimal control, differential games, and imaging sciences. By considering the time variable to be a higher dimensional quantity, HJ PDEs can be extended to the multi-time case. In this paper, we establish a novel theoretical connection between specific optimization problems arising in machine learning and the multi-time Hopf formula, which corresponds to a representation of the solution to certain multi-time HJ PDEs. Through this connection, we increase the interpretability of the training process of certain machine learning applications by showing that when we solve these learning problems, we also solve a multi-time HJ PDE and, by extension, its corresponding optimal control problem. As a first exploration of this connection, we develop the relation between the regularized linear regression problem and the Linear Quadratic Regulator (LQR). We then leverage our theoretical connection to adapt standard LQR solvers (namely, those based on the Riccati ordinary differential equations) to design new training approaches for machine learning. Lastly, we provide some numerical examples that demonstrate the versatility and possible computational advantages of our Riccati-based approach in the context of continual learning, post-training calibration, transfer learning, and sparse dynamics identification.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Modifying the Asynchronous Jacobi Method for Data Corruption Resilience

Moving scientific computation from high-performance computing (HPC) and cloud computing (CC) environments to devices on the edge, i.e., physically near instruments of interest, has received tremendous interest in recent years. Such edge computing environments can operate on data in situ, offering enticing benefits over data aggregation to HPC and CC facilities that include avoiding costs of transmission, increased data privacy, and real-time data analysis. Because of the inherent unreliability of edge computing environments, new fault-tolerant approaches must be developed before the benefits of edge computing can be realized. Motivated by algorithm-based fault tolerance, a variant of the asynchronous Jacobi (ASJ) method is developed that achieves resilience to data corruption by rejecting solution approximations from neighbor devices according to a bound derived from convergence theory. Numerical results on a two-dimensional Poisson problem show that the new rejection criterion, along with a novel approximation to the shortest path length on which the criterion depends, restores convergence for the ASJ variant in the presence of certain types data corruption. Numerical results are obtained for when the singular values in the analytic bound are approximated. Additional linear systems are also explored, one with a more dense sparsity pattern and one that includes advection. All results indicate that successful resilience to data corruption depends on whether the bound tightens fast enough to reject corrupted data before the iteration evolution deviates significantly from that predicted by the convergence theory defining the bound. This observation generalizes to future work on algorithm-based fault tolerance for other asynchronous algorithms, including upcoming approaches that leverage Krylov subspaces.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Supersymmetry and the celestial Jacobi identity

In this paper we study the simplifying effects of supersymmetry on celestial OPEs at both tree and loop level. We find at tree level that theories with unbroken supersymmetry around a stable vacuum have celestial soft current algebras satisfying the Jacobi identity, and we show at one loop that celestial OPEs in these theories have no double poles.

Supersymmetric Gauge Theory↗
DOE OSTI2015

A Semi-linear Backward Parabolic Cauchy Problem with Unbounded Coefficients of Hamilton–Jacobi–Bellman Type and Applications to Optimal Control

We obtain weighted uniform estimates for the gradient of the solutions to a class of linear parabolic Cauchy problems with unbounded coefficients. Such estimates are then used to prove existence and uniqueness of the mild solution to a semi-linear backward parabolic Cauchy problem, where the differential equation is the Hamilton–Jacobi–Bellman equation of a suitable optimal control problem. Via backward stochastic differential equations, we show that the mild solution is indeed the value function of the controlled equation and that the feedback law is verified.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

On Nonuniqueness of Solutions of Hamilton–Jacobi–Bellman Equations

An example of a nonunique solution of the Cauchy problem of Hamilton–Jacobi–Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t, x, p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Frankowska (SIAM J. Control Optim. 31:257–272, 1993) and Barron and Jensen (Commun. Partial Differ. Equ. 15(12):1713–1742, 1990). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2018

Generalized Jacobi method for linear bounded operators system

In this paper, we construct a generalization of the Jacobi iterative method adapted to a system of linear bounded operators. This new method is used with product integration method to solve a linear Fredholm integral equation of the second kind with a weakly singular kernel. The convergence analysis is proved. The numerical tests developed show its effectiveness.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2018

Jacobi collocation scheme for variable-order fractional reaction-subdiffusion equation

We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

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

Generalized Jacobi–Galerkin method for nonlinear fractional differential algebraic equations

In this paper, we provide an approximate approach based on the Galerkin method to solve a class of nonlinear fractional differential algebraic equations. The fractional derivative operator in the Caputo sense is utilized and the generalized Jacobi functions are employed as trial functions. The existence and uniqueness theorem as well as the asymptotic behavior of the exact solution are provided. It is shown that some derivatives of the solutions typically have singularity at origin dependence on the order of the fractional derivative. The influence of the perturbed data on the exact solutions along with the convergence analysis of the proposed scheme is also established. Some illustrative examples provided to demonstrate that this novel scheme is computationally efficient and accurate.

97 MATHEMATICS AND COMPUTING↗