Search NASA⌕ Search

SEARCH · Search NASA

Results for “Pade approximant”

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.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 19 records

Convergence of multipoint Pade approximants of piecewise analytic functions

The behaviour as n{yields}{infinity} of multipoint Pade approximants to a function which is (piecewise) holomorphic on a union of finitely many continua is investigated. The convergence of multipoint Pade approximants is proved for a function which extends holomorphically from these continua to a union of domains whose boundaries have a certain symmetry property. An analogue of Stahl's theorem is established for two-point Pade approximants to a pair of functions, either of which is a multivalued analytic function with finitely many branch points. Bibliography: 11 titles.

97 MATHEMATICS AND COMPUTING↗

Exponential time differencing scheme for mass transport and depletion in molten salt reactors

This work extends the capability previously shown for addressing the problem of computing depletion and mass transport calculations in molten salt reactors (MSRs) by calculating matrix exponentials. Additional algorithms are implemented to compute the matrix exponential and the action of the matrix exponential on a matrix. These algorithms include two methods based on the Pade approximation, a Taylor series method, and three methods based on Cauchy's integral formula. In addition to the added matrix exponential solvers, a variable-order total variation diminishing scheme is applied to the convective flux approximation to provide enhanced accuracy. Finally, a simplified MSR problem is shown for each of the exponential time differencing solvers along with classical backwards differencing integrators. The results show excellent convergence for exponential time differencing methods. Computation time is a key element for selecting the optimal solver in these problems, and this work shows that Pade and Cauchy-based solvers may provided the fastest and most accurate solutions. (authors)

21 SPECIFIC NUCLEAR REACTORS AND ASSOCIATED PLANTS↗

A continued fraction resummation form of bath relaxation effect in the spin-boson model

In the spin-boson model, a continued fraction form is proposed to systematically resum high-order quantum kinetic expansion (QKE) rate kernels, accounting for the bath relaxation effect beyond the second-order perturbation. In particular, the analytical expression of the sixth-order QKE rate kernel is derived for resummation. With higher-order correction terms systematically extracted from higher-order rate kernels, the resummed quantum kinetic expansion approach in the continued fraction form extends the Pade approximation and can fully recover the exact quantum dynamics as the expansion order increases.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Recommended nuclear data for medical radioisotope production: diagnostic gamma emitters

An extensive series of evaluations have been performed as part of an IAEA coordinated research project to study a set of nuclear reactions that produce the diagnostic gamma-ray emitting radionuclides Cr- 51 , Tc- 99m , In- 111 , I-123 and Tl- 201 . Recommended cross-section data in the form of excitation functions have been derived, along with quantifications of their uncertainties. These evaluations compiled the compilation of all previously published values and newly measured experimental data, followed by critical assessments and selection of those experimental datasets and accompanying uncertainties judged to be fully valid and statistically consistent for model-independent least-squares fitting by means of Pade approximations. Integral yields as a function of the energy were also calculated on the basis of the recommended cross sections deduced from these various fits. All evaluated numerical conclusions and their corresponding uncertainties are available online at www-nds.iaea.org/medical/gamma_emitters.html and also on the medical portal of the International Atomic Energy Agency/Nuclear Data Section (IAEA-NDS) www-nds.iaea.org/medportal

38 RADIATION CHEMISTRY, RADIOCHEMISTRY, AND NUCLEA↗

Tunable noninteracting free-energy density functionals for high-energy-density physics applications

In this work, we introduce the concept of a tunable noninteracting free-energy density functional and present two examples realized: (i) via a simple one-parameter convex combination of two existing functionals and (ii) via the construction of a generalized gradient approximation (GGA) enhancement factor that contains one free parameter and is designed to satisfy a set of incorporated constraints. Functional (i), constructed as a combination of the local Thomas–Fermi and a pseudopotential-adapted GGA for the noninteracting free-energy, has already demonstrated its practical usability for establishing the high temperature end of the equation of state of deuterium [Phys. Rev. B 104, 144104 (2021)] and CHON resin [Phys. Rev. E 106, 045207 (2022)] for inertial confinement fusion applications. Hugoniot calculations for liquid deuterium are given as another example of how the application of computationally efficient orbital-free density functional theory (OF-DFT) can be utilized with the employment of the developed functionals. Once the functionals have been tuned such that the OF-DFT Hugoniot calculation matches the Kohn–Sham solution at some low-temperature point, agreement with the reference Kohn–Sham results for the rest of the high temperature Hugoniot path is very good with relative errors for compression and pressure on the order of 2% or less.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

PADÉ APPROXIMANTS FOR THE EQUATION OF STATE FOR RELATIVISTIC HYDRODYNAMICS BY KINETIC THEORY

A two-point Padé approximant (TPPA) algorithm is developed for the equation of state (EOS) for relativistic hydrodynamic systems, which are described by the classical Maxwell–Boltzmann statistics and the semiclassical Fermi–Dirac statistics with complete degeneracy. The underlying rational function is determined by the ratios of the macroscopic state variables with various orders of accuracy taken at the extreme relativistic limits. The nonunique TPPAs are validated by Taub's inequality for the consistency of the kinetic theory and the special theory of relativity. The proposed TPPA is utilized in deriving the EOS of the dilute gas and in calculating the specific heat capacity, the adiabatic index function, and the isentropic sound speed of the ideal gas. Some general guidelines are provided for the application of an arbitrary accuracy requirement. The superiority of the proposed TPPA is manifested in manipulating the constituent polynomials of the approximants, which avoids the arithmetic complexity of struggling with the modified Bessel functions and the hyperbolic trigonometric functions arising from the relativistic kinetic theory.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Structures of the fractional spaces generated by the difference neutron transport operator

The initial boundary value problem for the neutron transport equation is considered. The first, second and third order of accuracy difference schemes for the approximate solution of this problem are presented. Highly accurate difference schemes for neutron transport equation based on Padé approximation are constructed. In applications, stability estimates for solutions of difference schemes for the approximate solution of the neutron transport equation are obtained.The positivity of the neutron transport operator in Slobodeckij spaces is proved. Numerical techniques are developed and algorithms are tested on an example in MATLAB.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

Noise effects on Padé approximants and conformal maps

Here, we analyze the properties of Padé and conformal map approximants for functions with branch points, in the situation where the expansion coefficients are only known with finite precision or are subject to noise. We prove that there is a universal scaling relation between the strength of the noise and the expansion order at which Padé or the conformal map breaks down. We illustrate this behavior with some physically relevant model test functions and with two non-trivial physical examples where the relevant Riemann surface has complicated structure.

97 MATHEMATICS AND COMPUTING↗

The Hubble flow of plateau inflation

In the absence of CMB precision measurements, a Taylor expansion has often been invoked to parametrize the Hubble flow function during inflation. The standard ''horizon flow'' procedure implicitly relies on this assumption. However, the recent Planck results indicate a strong preference for plateau inflation, which suggests the use of Padé approximants instead. We propose a novel method that provides analytic solutions of the flow equations for a given parametrization of the Hubble function. This method is illustrated in the Taylor and Padé cases, for low order expansions. We then present the results of a full numerical treatment scanning larger order expansions, and compare these parametrizations in terms of convergence, prior dependence, predictivity and compatibility with the data. Finally, we highlight the implications for potential reconstruction methods.

79 ASTRONOMY AND ASTROPHYSICS↗

Comparison between explicit and implicit discretization strategies for a dissipative thermal environment

We investigate strategies for simulating open quantum systems coupled to dissipative baths by comparing explicit wave function-based discretization [via multi-layer multi-configuration time-dependent Hartree (ML-MCTDH)] and the implicit density matrix-based master equation method [via tree tensor network hierarchical equations of motion (TTN-HEOM)]. For dissipative baths characterized by exponentially decaying bath correlation functions, the implicit discretization approach of HEOM—rooted in bath correlation function decompositions—proves significantly more efficient than explicit discretization of the bath into discrete harmonic modes. Explicit methods, like ML-MCTDH, require extensive mode discretization to approximate continuum baths, leading to computational bottlenecks. Case studies for two-level systems and a Fenna–Matthews–Olson complex model highlight TTN-HEOM’s superiority in capturing dissipative dynamics with relaxations with a minimal number of auxiliary modes, while the explicit methods are as exact as the HEOM in pure dephasing regimes. This comparison is enabled by the TENSO package, which has both ML-MCTDH and TTN-HEOM implemented using the same computational structure and propagation strategy.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

Metaplectic geometrical optics for ray-based modeling of caustics: Theory and algorithms

The optimization of radio frequency-wave (RF) systems for fusion experiments is often performed using ray-tracing codes, which rely on the geometrical-optics (GO) approximation. However, GO fails at caustics such as cutoffs and focal points, erroneously predicting the wave intensity to be infinite. This is a critical shortcoming of GO, since the caustic wave intensity is often the quantity of interest, e.g., RF heating. Full-wave modeling can be used instead, but the computational cost limits the speed at which such optimizations can be performed. Here, we have developed a less expensive alternative called metaplectic geometrical optics (MGO). Instead of evolving waves in the usual x (coordinate) or k (spectral) representation, MGO uses a mixed X$\equiv$Ax+Bk representation. By continuously adjusting the matrix coefficients A and B along the rays, one can ensure that GO remains valid in the X coordinates without caustic singularities. The caustic-free result is then mapped back onto the original x space using metaplectic transforms. Here, we overview the MGO theory and review algorithms that will aid the development of an MGO-based ray-tracing code. We show how using orthosymplectic transformations leads to considerable simplifications compared to previously published MGO formulas. We also prove explicitly that MGO exactly reproduces standard GO when evaluated far from caustics (an important property that until now has only been inferred from numerical simulations), and we relate MGO to other semiclassical caustic-removal schemes published in the literature. Finally this discussion is then augmented by an explicit comparison of the computed spectrum for a wave bounded between two cutoffs.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Data driven analytic continuation for one particle spectral functions

In this proceeding, an idea is outlined suggesting a generic treatment on any type of input data for a numerical analytic continuation problem, which is needed when dynamical information is to be extracted from a calculationally convenient one particle imaginary time Green function. The quality of the resulting spectral function will rely only on the data to be treated, viz, data-driven. This is different from the Maximum Entropy or the Stochastic method which relies on an entropy term to guide convergence of the resulting spectral function.

36 MATERIALS SCIENCE↗

Jets from shocked metal surfaces with grooves: Missing experiments

Many studies have investigated the mass outflows generated when a planar shock transits an imperfect (“defected”) metal surface, where the defects are symmetric triangular or sinusoidal grooves. Yet a fundamental question remains unanswered: how does the quantity of outflow mass and its maximum velocity vary as a function of the groove cross-sectional aspect ratio? We identify two sets of missing experiments that must be addressed to answer the question. The aspect ratio (groove depth over width) is equivalently represented by θ, the cross-sectional half angle, or by η 0 k, the amplitude multiplied by an effective wavenumber. Low θ (high η 0 k) grooves comprise the first set of missing experiments, which are necessary to determine the validity of theoretical predictions of the nonlinear regime (η 0 k ≥ 1, θ < 57.5°). The second set of missing experiments are those in which the volume of the groove (or equivalently, the axial cross-sectional area) has been held constant as θ or η 0 k are varied. Such experiments are necessary to independently measure the effects of variations in groove volume and groove aspect ratio on the resulting jets.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗

The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the S-property in a harmonic external field. The existence problem for S-compact sets is solved in Theorem 2. The main idea of the proof of Theorem 1 consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the '1/9'-conjecture. The relation of the result obtained here to some results and conjectures due to Nuttall is discussed. Bibliography: 51 titles.

97 MATHEMATICS AND COMPUTING↗

The leading term of the Plancherel-Rotach asymptotic formula for solutions of recurrence relations

Recurrence relations generating Padé and Hermite-Padé polynomials are considered. Their coefficients increase with the index of the relation, but after dividing by an appropriate power of the scaling function they tend to a finite limit. As a result, after scaling the polynomials 'stabilize' for large indices; this type of asymptotic behaviour is called Plancherel-Rotach asymptotics. An explicit expression for the leading term of the asymptotic formula, which is valid outside sets containing the zeros of the polynomials, is obtained for wide classes of three- and four-term relations. For three-term recurrence relations this result generalizes a theorem Van Assche obtained for recurrence relations with 'regularly' growing coefficients. Bibliography: 19 titles.

97 MATHEMATICS AND COMPUTING↗

Scale-fixed predictions for γ + η c production in electron-positron collisions at NNLO in perturbative QCD

In the paper, we present QCD predictions for γ + η c production at an electron-positron collider up to next-to-next-to-leading order (NNLO) accuracy without renormalization scale ambiguities. The NNLO total cross-section for e + + e – → γ + η c using the conventional scale-setting approach has large renormalization scale ambiguities, usually estimated by choosing the renormalization scale to be the e + e – center-of-mass collision energy $\sqrt{s}$. The Principle of Maximum Conformality (PMC) provides a systematic way to eliminate such renormalization scale ambiguities by summing the nonconformal β contributions into the QCD coupling α s ( Q 2 ). The renormalization group equation then sets the value of α s for the process. The PMC renormalization scale reflects the virtuality of the underlying process, and the resulting predictions satisfy all of the requirements of renormalization group invariance, including renormalization scheme invariance. After applying the PMC, we obtain a renormalization scale-and-scheme independent prediction, σ | NNLO,PMC ≃ 41.18 fb for $\sqrt{s}$=10.6 GeV. The resulting pQCD series matches the series for conformal theory and thus has no divergent renormalon contributions. The large K factor which contributes to this process reinforces the importance of uncalculated NNNLO and higher-order terms. Using the PMC scale-and-scheme independent conformal series and the Padé approximation approach, we predict σ | NNNLO,PMC+Pade ≃ 18.99 fb, which is consistent with the recent BELLE measurement ${\sigma}^{\mathrm{obs}}={16.58}_{-9.93}^{+10.51}$ fb at $\sqrt{s}$ ≃ 10.6 GeV. This procedure also provides a first estimate of the NNNLO contribution.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗