Hermite Polynomial Expansion Research Articles

Complete third-order polynomial expansion-based gas kinetic flux solver for flows from continuum regime to rarefied regime

The Grad's 13 distribution function was derived through a third-order Hermite polynomial expansion in terms of peculiar velocity. Recently, it has been adopted to construct a gas kinetic flux solver called G13-GKFS for simulation of flows from the continuum regime to the rarefied regime. However, this Grad's distribution function only considers the contracted polynomials that strictly satisfy orthogonality. In other words, the third-order terms of CiC12, CiC22, and CiC32 share the same coefficients (γi). However, the results from the discrete velocity method reveal that those coefficients could be different, especially in the rarefied regime. This may affect the accuracy of numerical results in the rarefied region. In order to consider different coefficients of the third-order terms, we propose a complete third-order polynomial expansion to approximate the distribution function in this work. To show the capability of current distribution function, a new GKFS is developed for flows from the continuum regime to the rarefied regime. Some benchmark cases are solved to demonstrate that the new GKFS outperforms the G13-GKFS in the rarefied regime.

Constructing copulas using corrected Hermite polynomial expansion for estimating cross foreign exchange volatility

A finite order multivariate Hermite polynomial expansion, as an approximation of a joint density function, can handle complex correlation structures. However, it does not construct copulas, because the density function can take negative values. In this study, we propose a formulation of multivariate Hermite polynomial expansion suitable for the application of correction that recovers non-negativity and construct a copula. Several useful expressions for integrals with the copula are derived from this formulation. We also apply this copula to estimate the volatility smile of cross-currency pairs in the foreign exchange option market. In the numerical experiments, we compare the estimation results of the volatility smile of EUR–JPY, GBP–JPY, and AUD–JPY for the proposed and other copulas to confirm the validity of the proposed copula.

High-order modeling of multiphase flows: Based on discrete Boltzmann method

The high-order kinetic model for compressible multiphase flow is presented within the framework of the discrete Boltzmann method (DBM). Based on the Carnahan–Starling state equation, this model can describe the phase transition by introducing a source term of molecular interaction on the right-hand side of the Boltzmann equation. Meanwhile, the force term is incorporated to describe the external force. Through Hermite polynomial expansion, the equilibrium distribution function is expressed. Compared to the Navier–Stokes equations, the DBM provides more detailed and accurate information on both hydrodynamic and thermodynamic non-equilibrium effects. Finally, the model is verified through several typical benchmarks, including the liquid–vapor coexistence curve, free-falling process, shock wave, sound wave, thermal phase separation, and two-bubble oblique collision.

Chapman–Enskog derivation of multicomponent Navier–Stokes equations

There are several reasons to extend the presentation of Navier–Stokes equations to multicomponent systems. Many technological applications are based on physical phenomena that are present in neither pure elements nor in binary mixtures. Whereas Fourier's law must already be generalized in binaries, it is only with more than two components that Fick's law breaks down in its simple form. The emergence of dissipative phenomena also affects the inertial confinement fusion configurations, designed as prototypes for the future fusion nuclear plants hopefully replacing the fission ones. This important topic can be described in much simpler terms than it is in many textbooks since the publication of the formalism put forward recently by Snider [Phys. Rev. E 82, 051201 (2010)]. In a very natural way, it replaces the linearly dependent atomic fractions by the independent set of partial densities. Then, the Chapman–Enskog procedure is hardly more complicated for multicomponent mixtures than for pure elements. Moreover, the recent proposal of a convergent kinetic equation by Baalrud and Daligault [Phys. Plasmas 26, 082106 (2019)] demonstrates that the Boltzmann equation with the potential of mean force is a far better choice in situations close to equilibrium, as described by the Navier–Stokes equations, than Landau or Lenard–Balescu equations. In our comprehensive presentation, we emphasize the physical arguments behind Chapman–Enskog derivation and keep the mathematics as simple as possible. This excludes, as a technical non-essential aspect, the solution of the linearized Boltzmann equation through an expansion in Hermite polynomials. We discuss the link with the second principle of thermodynamics of entropy increase, and what can be learned from this exposition.

Minkowski Functionals of SDSS-III BOSS: Hints of Possible Anisotropy in the Density Field?

Abstract We present measurements of the Minkowski functionals extracted from the SDSS-III BOSS catalogs. After defining the Minkowski functionals, we describe how an unbiased reconstruction of these statistics can be obtained from a field with masked regions and survey boundaries, validating our methodology with Gaussian random fields and mock galaxy snapshot data. From the BOSS galaxy data, we generate a set of four density fields in three dimensions corresponding to the northern and southern skies of LOWZ and CMASS catalogs, smoothing over large scales (Gaussian smoothing scale of 35 Mpc) such that the field is perturbatively non-Gaussian. We extract the Minkowski functionals from each data set separately, and measure their shapes and amplitudes by fitting a Hermite polynomial expansion. For the shape parameter of the Minkowski functional curves a 0, that is related to the bispectrum of the field, we find that the LOWZ-South data presents a systematically lower value of a 0 = −0.080 ± 0.040 than its northern sky counterpart a 0 = 0.032 ± 0.024. Although the significance of this discrepancy is not very high, it potentially indicates some systematics in the data or that the matter density field exhibits anisotropy at low redshift. By assuming a standard isotropic flat ΛCDM cosmology, the amplitudes of the Minkowski functionals from the combination of northern and southern sky data give the constraints Ωc h 2 n s = 0.110 ± 0.006 and 0.111 ± 0.008 for CMASS and LOWZ, respectively, which is in agreement with the Planck ΛCDM best-fit Ωc h 2 n s = 0.116 ± 0.001.

Разработка явных и консервативных схем для решеточных уравнений Больцмана с адаптивным переносом

The Lattice Boltzmann Method (LBM) has several limitations for velocity and temperature. One can consider distribution function in moving frame to overcome these limitations as in PonD. In PonD, values of distribution functions are streamed from off-lattice points, so value estimation is needed. It leads to the implicit and non-conservative numerical scheme. Earlier, for the one-dimensional case, the approach of moments prediction was found, which leads to an explicit and conservative numerical scheme. We apply this approach to the two-dimensional and three-dimensional cases in this work. Requirements to interpolation stencil, quadrature, and Hermite polynomial expansion which guarantee moment matching, conservation, and exact calculation, were studied. The resulting schemes were implemented and tested on several tasks.

A Comparative Study of Likelihood Approximations for Univariate Diffusions

Abstract Maximum likelihood estimation of the parameters of stochastic differential equations commonly used in finance requires numerical approximation of their transitional probability density functions. This article undertakes a comparative study of the accuracy of Hermite polynomial expansion approximations for univariate diffusions and checks how the accuracy of the existing methods responds to increasing the order of the approximation. It is found that one class of expansion dealing with irreducible diffusions is particularly problematic due to the need to evaluate a number of troublesome integrals. A Gaussian quadrature is introduced which resolves the problem and improves the reliability of the expansion. A simulation study demonstrates all the methods in action and provides insight into the practical aspects of using these expansions. An empirical application using data on the VIX indicates that the proposed method based on the Gaussian quadrature performs very well when applied to financial data.

Onsager-regularized lattice Boltzmann method: A nonequilibrium thermodynamics-based regularized lattice Boltzmann method.

The regularized class of lattice Boltzmann methods (LBMs) leverage the potency of the standard lattice-Bhatnagar-Gross-Krook method by filtering out spurious nonhydrodynamic moments from the moment space; this is achieved through evaluating regularized populations via a multiscale or a Hermite polynomial expansion approach. In this paper, we propose an alternative approach for evaluating the lattice populations. This approach is based on a kinetic theory that is consistent with nonequilibrium thermodynamics and obeys the Onsager-reciprocity principle. The proposed method is verified and validated for a number of canonical problems such as the athermal shock tube, the double periodic shear layer, the lid driven cavity, flow past square cylinder, and Poiseuille flow at nonvanishing Knudsen numbers. Additionally, the proposed method is compared to existing regularized LBM schemes and is shown to yield significant improvement in the stability and accuracy of the simulations.

Total variation estimates in the Breuer–Major theorem

This paper provides estimates for the convergence rate of the total variation distance in the framework of the Breuer–Major theorem, assuming some smoothness properties of the underlying function. The results are proved by applying new bounds for the total variation distance between a random variable expressed as a divergence and a standard Gaussian random variable, which are derived by a combination of techniques of Malliavin calculus and Stein’s method. The representation of a functional of a Gaussian sequence as a divergence is established by introducing a shift operator on the expansion in Hermite polynomials. Some applications to the asymptotic behavior of power variations of the fractional Brownian motions and to the estimation of the Hurst parameter using power variations are presented.

Sparse-grid polynomial interpolation approximation and integration for parametric and stochastic elliptic PDEs with lognormal inputs

By combining a certain approximation property in the spatial domain, and weighted 𝓁2-summability of the Hermite polynomial expansion coefficients in the parametric domain obtained in Bachmayr et al. [ESAIM: M2AN 51 (2017) 341–363] and Bachmayr et al. [SIAM J. Numer. Anal. 55 (2017) 2151–2186], we investigate linear non-adaptive methods of fully discrete polynomial interpolation approximation as well as fully discrete weighted quadrature methods of integration for parametric and stochastic elliptic PDEs with lognormal inputs. We construct such methods and prove convergence rates of the approximations by them. The linear non-adaptive methods of fully discrete polynomial interpolation approximation are sparse-grid collocation methods which are certain sums taken over finite nested Smolyak-type indices sets of mixed tensor products of dyadic scale successive differences of spatial approximations of particular solvers, and of successive differences of their parametric Lagrange interpolating polynomials. The Smolyak-type sparse interpolation grids in the parametric domain are constructed from the roots of Hermite polynomials or their improved modifications. Moreover, they generate in a natural way fully discrete weighted quadrature formulas for integration of the solution to parametric and stochastic elliptic PDEs and its linear functionals, and the error of the corresponding integration can be estimated via the error in the Bochner space L1 (ℝ∞, V, γ) norm of the generating methods where γ is the Gaussian probability measure on ℝ∞ and V is the energy space. We also briefly consider similar problems for parametric and stochastic elliptic PDEs with affine inputs, and problems of non-fully discrete polynomial interpolation approximation and integration. In particular, the convergence rates of non-fully discrete polynomial interpolation approximation and integration obtained in this paper significantly improve the known ones.

Generating Accurate Pseudo-labels in Semi-Supervised Learning and Avoiding Overconfident Predictions via Hermite Polynomial Activations.

Rectified Linear Units (ReLUs) are among the most widely used activation function in a broad variety of tasks in vision. Recent theoretical results suggest that despite their excellent practical performance, in various cases, a substitution with basis expansions (e.g., polynomials) can yield significant benefits from both the optimization and generalization perspective. Unfortunately, the existing results remain limited to networks with a couple of layers, and the practical viability of these results is not yet known. Motivated by some of these results, we explore the use of Hermite polynomial expansions as a substitute for ReLUs in deep networks. While our experiments with supervised learning do not provide a clear verdict, we find that this strategy offers considerable benefits in semi-supervised learning (SSL) / transductive learning settings. We carefully develop this idea and show how the use of Hermite polynomials based activations can yield improvements in pseudo-label accuracies and sizable financial savings (due to concurrent runtime benefits). Further, we show via theoretical analysis, that the networks (with Hermite activations) offer robustness to noise and other attractive mathematical properties. Code is available on //GitHub.

Lévy walk with parameter dependent velocity: Hermite polynomial approach and numerical simulation

To analyze stochastic processes, one often uses integral transform (Fourier and Laplace) methods. However, for the time-space coupled cases, e.g. the Lévy walk, sometimes the integral transform method may fail. Here we provide a Hermite polynomial expansion approach, being complementary to the integral transform method, to the Lévy walk. Two approaches are compared for some already known results. We also consider the generalized Lévy walk with parameter dependent velocity. Namely, we consider the Lévy walk with velocity which depends on the walking length or on the duration of each step. Some interesting features of the generalized Lévy walk are observed, including the special shapes of the probability density function, the first passage time distributions, and various diffusive behaviors of the mean squared displacement.

Sieve extremum estimation of a semiparametric transformation model

This paper considers the estimation of a semiparametric transformation model, Λ(yt,β0)=g(xt)+ut, where Λ(⋅,β0) is a strictly increasing function known up to an ℓ-dimensional parameter β0, g is an unknown link function. Hermite polynomial expansion is used to approximate the link function g, which leads to an extreme estimator for β0 and a plug-in estimator for g. Asymptotic properties of the estimators are established. Simulation results demonstrate that the estimators perform well in finite samples. An example on Canadian occupation prestige is provided to illustrate the practical value of the proposed model.

Kinetic multiscale scheme based on the discrete-velocity and lattice-Boltzmann methods

A novel hybrid computational method based on the discrete-velocity (DV) approximation, including the lattice-Boltzmann (LB) technique, is proposed. Numerical schemes for the kinetic equations are used in regions of rarefied flows, and LB schemes are employed in continuum flow zones. The schemes are written under the finite-volume (FV) formulation to achieve the flexibility of local mesh refinement. The truncated Hermite polynomial expansion is used for matching of DV and LB solutions. Special attention is paid to preserving conservation properties in the coupling algorithm. The test results obtained for the Couette flow of a rarefied gas are in excellent agreement with the benchmark solutions, mostly thanks to mesh refinement (both in the physical and velocity spaces) in the Knudsen layer.

Universal formulation of central-moments-based lattice Boltzmann method with external forcing for the simulation of multiphysics phenomena

The cascaded or central-moments-based lattice Boltzmann method (CM-LBM) is a robust alternative to the more conventional Bhatnagar-Gross-Krook-LBM for the simulation of high-Reynolds number flows. Unfortunately, its original formulation makes its extension to a broader range of physics quite difficult. In addition, it relies on CMs that are derived in an ad hoc manner, i.e., by mimicking those of the Maxwell-Boltzmann distribution to ensure their Galilean invariance a posteriori. This work aims at tackling both issues by deriving Galilean invariant CMs in a systematic and a priori manner, thanks to the Hermite polynomial expansion framework. More specifically, the proposed formalism fully takes advantage of the D3Q27 discretization by relying on the corresponding set of 27 Hermite polynomials (up to the sixth-order) for the derivation of both the discrete equilibrium state and the forcing term in an a priori manner. Furthermore, while keeping the numerical properties of the original CM-LBM, this work leads to a compact and simple algorithm, representing a universal methodology based on CMs and external forcing within the lattice Boltzmann framework. To support these statements, mathematical derivations and a comparative study with four other forcing schemes are provided. The universal nature of the proposed methodology is eventually proved through the simulation of single phase, multiphase (using both pseudopotential and color-gradient formulations), and magnetohydrodynamic flows.

Time-stepping wave-equation solution for seismic modeling using a multiple-angle formula and the Taylor expansion

We have developed an analytical solution for wave equations using a multiple-angle formula. The new solution based on the multiple-angle expansion allows us to generate a family of solutions for the acoustic-wave equation, which may be combined with Taylor-series, Chebyshev, Hermite, and Legendre polynomial expansions or any other expansion for the cosine function and used for seismic modeling, reverse time migration, and inverse problems. Extension of this method to the solution of elastic and anisotropic wave equations is also straightforward. We also derive a criterion using the stability and dispersion relations to determine the order of the solution for a given time step and, thus, obtaining stable wavefields free of numerical dispersion. Afterward, numerical tests are performed using complex 2D velocity models to evaluate the effectiveness and robustness of our method, combined with second- or fourth-order Taylor approximations. Our multiple-angle approach is stable and provides reliable seismic modeling results for larger times steps than those usually used by conventional finite-difference methods. Moreover, multiple-angle schemes using a second-order Taylor approximation for each cosine term have a lower computational cost than the mixed wavenumber-space rapid expansion method.

On-node lattices construction using partial Gauss–Hermite quadrature for the lattice Boltzmann method**Project supported by the National Science and Technology Major Project, China (Grant No. 2017ZX06002002).

A concise theoretical framework, the partial Gauss–Hermite quadrature (pGHQ), is established to construct on-node lattices of the lattice Boltzmann (LB) method under a Cartesian coordinate system. Compared with the existing approaches, the pGHQ scheme has the following advantages: extremely concise algorithm, unifies the constructing procedure for symmetric and asymmetric on-node lattices, and covers a full-range quadrature degree of a given discrete velocity set. We employ the pGHQ scheme to search the local optimal and asymmetric lattices for moment degree equilibrium distribution discretization on the range . The search reveals a surprising abundance of available lattices. Through a brief analysis, the discrete velocity set shows a significant influence on the positivity of equilibrium distributions, which is considered as one of the major impacts of the numerical stability of the LB method. Hence, the results of the pGHQ scheme lay a foundation for further investigations to improve the numerical stability of the LB method by modifying the discrete velocity set. It is also worth noting that pGHQ can be extended into the entropic LB model, even though it was proposed for the Hermite polynomial expansion LB theory.

Discrete Boltzmann method for non-equilibrium flows: Based on Shakhov model

A general framework for constructing discrete Boltzmann model for non-equilibrium flows based on the Shakhov model is presented. The Hermite polynomial expansion and a set of discrete velocity with high spatial isotropy are adopted to solve the kinetic moments of discrete equilibrium distribution function. Such a model possesses both an adjustable specific heat ratio and Prandtl number, and can be applied to a wide range of flow regimes including continuous, slip, and transition flows. To recover results for actual situations, the nondimensionalization process is demonstrated. To verify and validate the new model, several typical non-equilibrium flows including the Couette flow, Fourier flow, unsteady boundary heating problem, cavity flow, and Kelvin–Helmholtz instability are simulated. Comparisons are made between the results of discrete Boltzmann model and those of previous models including analytic solution in slip flow, Lattice ES-BGK, and DSMC based on both BGK and hard-sphere models. The results show that the new model can accurately capture the velocity slip and temperature jump near the wall, and show excellent performance in predicting the non-equilibrium flow even in transition flow regime. In addition, the measurement of non-equilibrium effects is further developed and the non-equilibrium strength Dn∗ in the nth order moment space is defined. The non-equilibrium characteristics and the advantage of using Dn∗ in Kelvin–Helmholtz instability are discussed. It concludes that the non-equilibrium strength Dn∗ is more appropriate to describe the interfaces than the individual components of Δn∗. Besides, the D3∗ and D3,1∗ can provide higher resolution interfaces in the simulation of Kelvin–Helmholtz instability.

Consistent second-order boundary implementations for convection-diffusion lattice Boltzmann method.

In this study, an alternative second-order boundary scheme is proposed under the framework of the convection-diffusion lattice Boltzmann (LB) method for both straight and curved geometries. With the proposed scheme, boundary implementations are developed for the Dirichlet, Neumann and linear Robin conditions in a consistent way. The Chapman-Enskog analysis and the Hermite polynomial expansion technique are first applied to derive the explicit expression for the general distribution function with second-order accuracy. Then, the macroscopic variables involved in the expression for the distribution function is determined by the prescribed macroscopic constraints and the known distribution functions after streaming [see the paragraph after Eq.(29) for the discussions of the "streaming step" in LB method]. After that, the unknown distribution functions are obtained from the derived macroscopic information at the boundary nodes. For straight boundaries, boundary nodes are directly placed at the physical boundary surface, and the present scheme is applied directly. When extending the present scheme to curved geometries, a local curvilinear coordinate system and first-order Taylor expansion are introduced to relate the macroscopic variables at the boundary nodes to the physical constraints at the curved boundary surface. In essence, the unknown distribution functions at the boundary node are derived from the known distribution functions at the same node in accordance with the macroscopic boundary conditions at the surface. Therefore, the advantages of the present boundary implementations are (i) the locality, i.e., no information from neighboring fluid nodes is required; (ii) the consistency, i.e., the physical boundary constraints are directly applied when determining the macroscopic variables at the boundary nodes, thus the three kinds of conditions are realized in a consistent way. It should be noted that the present focus is on two-dimensional cases, and theoretical derivations as well as the numerical validations are performed in the framework of the two-dimensional five-velocity lattice model.

Orthogonal Polynomials and Sharp Estimates for the Schrödinger Equation

In this article, we study sharp estimates for the Schrödinger operator via the framework of orthogonal polynomials. We use Hermite and Laguerre polynomial expansions to produce sharp Strichartz estimates for even exponents. In particular, for radial initial data in dimension |$2$|⁠, we establish an interesting connection of the Strichartz norm with a combinatorial problem about words with four letters. We use spherical harmonics and Gegenbauer polynomials to prove a sharpened weighted inequality for the Schrödinger equation that is maximized by radial functions.