An explicit conservative finite volume scheme based on multi-scale characteristic lines is proposed for the simulation of advection and convection in fluid flows. By representation of the Navier–Stokes equation in a kinetic manner, the evolution of particle distribution functions on discrete velocity space is governed by the kinetic advection equation. The control equation is discretized and integrated along particle velocity characteristic lines via a temporal/spatial related reconstruction to obtain accurate macroscopic fluxes. With a D2Q37 lattice velocity model, the collision processes are modeled by a relaxation term expressed with Hermite expansion to reduce complexity and computational cost. Several cases, including the Sod shock tube case, lid-driven cavity flows, and natural convection flows on Rayleigh numbers of 103–106, Rayleigh–Bénard convection flows on Rayleigh numbers of 104–106 and Rayleigh–Taylor instability on the Reynolds number of Re = 62 990 are simulated and analyzed. Simulation results show satisfactory accuracy with a relative L2-norm error less than 1% as compared with benchmark data, and reveal several steady and unsteady mechanisms, including advection in the Riemann problem, buoyancy-driven motion, temperature-driven mass transfer, interface dynamical behaviors, and vortex evolution in convective flows.

Higher Order Riesz Transforms and Almost Diagonality for Hermite Expansions

Context. Dynamical models enable the recovery of galaxy mass profiles, intrinsic shapes, and, in the case of Schwarzschild orbit-superposition methods, orbital structure. An accurate dynamical inference relies on the precise recovery of the line-of-sight velocity distribution from an integrated spectrum. Aims. The main goal of this work is to evaluate the effect of different methods for recovering stellar kinematics from an integrated spectrum on the resulting orbit distribution and subsequent implications on the recovered mass and velocity anisotropy profiles. Methods. We applied the commonly used penalised pixel fitting method pPXF on archival SAURON datacubes of NGC 4550 and NGC 2768, a well-studied counter-rotating and a regular-rotating elliptical galaxy, for a parametrised description of the stellar line-of-sight velocity distribution based on Gauss Hermite expansion. We also used Bayes-LOSVD, a method based on Bayesian framework, to recover a non-parametric representation of the stellar kinematics. Both datasets are used as input for the orbit-superposition code DYNAMITE. Results. We find that the inferred dynamical properties of NGC 4550 are strongly affected by the difference in the method used to derive the stellar kinematics. Using non-parametric kinematics, we recover two distinct, dynamically cold counter-rotating disks; the orbit solution resulting from the Gauss Hermite stellar kinematics indicates dynamically warmer disks. In addition, the parametrised approach results in notably higher predicted total mass of the galaxy and more radial velocity anisotropy. For NGC 2768, these differences are less significant, but still noticeable, especially in the orbit distribution. We also find that the non-parametric kinematics results are more robust against changes to spatial binning and in the uncertainty computation. Conclusions. We argue for the advantages of using a non-parametric description of the stellar kinematics for the dynamical modelling of galaxies.

Kinetic simulations are computationally intensive due to six-dimensional phase space discretization. Many kinetic spectral solvers use the asymmetrically weighted Hermite expansion due to its conservation and fluid-kinetic coupling properties, i.e., the lower-order Hermite moments capture and describe the macroscopic fluid dynamics, and higher-order Hermite moments describe the microscopic kinetic dynamics. We leverage this structure by developing a parametric data-driven reduced-order model based on the proper orthogonal decomposition, which projects the higher-order kinetic moments while retaining the fluid moments intact. We demonstrate analytically and numerically that the method ensures local and global mass, momentum, and energy conservation. The numerical results show that the proposed method effectively replicates the high-dimensional spectral simulations at a fraction of the computational cost and memory, as validated on the weak Landau damping and two-stream instability benchmark problems.

We introduce Littlewood Paley functions defined in terms of a reparameterization of the Ornstein-Uhlenbeck semigroup obtaining that these operators are bounded in \(L^p\), \(1<p<\infty\), with respect to the unidimensional gaussian measure, by means of singular integrals theory. In addition, we study the Abel summability of the Fourier Hermite expansions considering their pointwise convergence and their convergence in the \(L^p\) sense, obtaining a version of Tauber's theorem.

The occipital cortex responds to visual stimuli regardless of a patient's level of consciousness or attention, offering a noninvasive diagnostic tool for both ophthalmologists and neurologists. This response signal manifests as a unique waveform referred to as the visually evoked potential (VEP), which can be extracted from the electroencephalogram (EEG) activity of a human being. We propose a trainable VEP representation to disentangle the underlying explanatory factors of the data. To enhance the learning process with domain knowledge, we present an innovative parameterization of classical Hermite functions that effectively captures VEP pattern variations arising from patient-specific factors, disorders, and measurement setup influences. Then, we introduce a differentiable variable projection (VP) layer to fuse Hermite basis function expansions (BFEs) of VEP signals with machine learning (ML) approaches. We prove the existence of an optimal set of parameters in the least-squares sense, assess the representation power of such layers, and calculate their analytical derivatives, which allows us to utilize backpropagation for training. Finally, we evaluate the effectiveness of the proposed learning framework in VEP-based color classification. To achieve this, we have designed a novel measurement system dedicated to intraoperative clinical use cases, which presents new ways for patient monitoring during neurosurgical procedures.

ABSTRACTBased on the Hermite expansion approach, a type of discrete Boltzmann model is devised to simulate the open channel flows governed by the Saint‐Venant equations. In this model, a four‐discrete‐velocity set is adopted, and a local equilibrium distribution with the fourth‐order polynomials is kept to simulate the supercritical flows. To numerically solve the kinetic equation, the finite difference method is employed. The model is numerically validated by using four benchmark problems, i.e., dam‐break flows, hydraulic jump, steady flow over a bump, and the flume dam‐break flows with rectangular and triangular cross‐sections. Then, the thin film method is incorporated into the proposed model to deal with the wet‐dry boundaries, and this ability has been validated by two cases, i.e., the dam‐break flows in a converging–diverging channel and over a triangular obstacle. It is found that the present discrete Boltzmann model can accurately predict the subcritical, transcritical, and supercritical flows with source terms and wet‐dry boundaries.

This work presents an extensive analysis of the lattice Boltzmann method for solving rotating fluid flows inside channel-like geometries, a topic relevant to many scientific and engineering fields. The present research investigates the role of the collision operator, equilibrium and source term formulations, the number of discrete velocities in three-dimensional cubic lattices, and boundary schemes applied to this problem class. Here, it is considered the two-relaxation-time (TRT) collision operator with both equilibrium and source terms represented on the Hermite expansion formalism. Denoting by H(n) the n-order Hermite orthonormal basis, the TRT modeling of the isothermal Navier–Stokes equations expands the symmetric and anti-symmetric components of equilibrium up to H(2) and H(1), respectively, and at H(1) for the external source term, featuring the Coriolis force. This study proposes higher-order expansions of equilibrium and source terms to improve the accuracy of diffusion and advection in rotating fluids. Diffusion modeling is improved by including an H(3) correction to the source term expansion to remedy artifacts from the Coriolis force discretization. Advection modeling is improved by including an H(4) correction to the H(2) expansion of the symmetric equilibrium in the D3Q19 lattice to retrieve an isotropy comparable to the D3Q27. These improvements are derived based on the exact analytical solution of the TRT equation at the discrete level and the steady Chapman–Enskog fourth-order expansion of the TRT solution, respectively, applied to two well-known benchmarks in this problem class: the Poiseuille–Ekman channel flow and the rotating square duct flow.

In this paper, Hermite polynomials (HPs) are introduced to solve the 2nd kind Volterra-Fredholm integro-differential equations (VFIDEs) of the first and second order. This technique is based on replacing the unknown function “infinite series” by truncated series of that is well know by Hermite expansion of functions. The presented method converts the equation into matrix form or a system of algebraic equations with Hermite coefficients which they must be determined. The existence and uniqueness of the solution are proved.The convergence analysis of the method are studied.Some examples for the first, and second orders of 2nd kind VFIDEs are given to demonstrate the effectiveness and the precision of the proposed method.

ABSTRACTThis article combines methods from existing techniques to identify multiple changepoints in non‐Gaussian autocorrelated time series. A transformation is used to convert a Gaussian series into a non‐Gaussian series, enabling penalized likelihood methods to handle non‐Gaussian scenarios. When the marginal distribution of the data is continuous, the methods essentially reduce to the change of variables formula for probability densities. When the marginal distribution is count‐oriented, Hermite expansions and particle filtering techniques are used to quantify the scenario. Simulations demonstrating the efficacy of the methods are given and two data sets are analyzed: 1) the proportion of home runs hit by Major League Baseball batters from 1920 to 2023 and 2) a six‐dimensional series of tropical cyclone counts from the Earth's basins of generation from 1980 to 2023. In the first series, beta marginal distributions are used to describe the proportions; in the second, Poisson marginal distributions seem appropriate.

We prove a conjecture by Vemuri [Hermite expansions and Hardy’s theorem, arXiv:0801.2234] by proving sharp bounds on ℓ κ \ell ^{\kappa } sums of Hermite functions multiplied by an exponentially decaying factor. More explicitly, we prove that, for each y > 0 y>0 , we have \[ ∑ n ≥ 1 | h n ( x ) | κ e − κ n y n β ≪ y x 1 − κ 2 − 2 β e − κ x 2 tanh ⁡ ( y ) / 2 , ∀ x ∈ R sufficiently large. \sum _{n \ge 1} |h_n(x)|^{\kappa } \frac {e^{-\kappa n y}}{n^{\beta }} \ll _y x^{1-\frac {\kappa }{2} - 2\beta } e^{-\kappa x^2 \tanh (y)/2}, \, \forall \, x \in {\mathbb {R}}\text { sufficiently large.} \] Our proof involves the classical Plancherel-Rotach asymptotic formula for Hermite polynomials and a careful local analysis near the maximum point of such a bound.

Hermite Expansions of $C$-regularized cosine functions

Hermite expansions for spaces of functions with nearly optimal time-frequency decay

Two-relaxation-time regularized lattice Boltzmann model for convection-diffusion equation with spatially dependent coefficients

Maximum correntropy polynomial chaos Kalman filter for underwater navigation

Many macroscopic non-Fourier heat conduction models have been developed in the past decades based on Chapman-Enskog, Hermite, or other small perturbation expansion methods. These macroscopic models have achieved great success in capturing non-Fourier thermal behaviors in solid materials, but most of them are limited by small Knudsen numbers and incapable of capturing highly nonequilibrium or ballistic thermal transport. In this paper, we provide a different strategy for constructing macroscopic non-Fourier heat conduction modeling, that is, using data-driven deep-learning methods combined with nonequilibrium thermodynamics instead of small perturbation expansion. We present the mechanism-data fusion method, an approach that seamlessly integrates the rigorous framework of conservation-dissipation formalism (CDF) with the flexibility of machine learning to model non-Fourier heat conduction. Leveraging the conservation-dissipation principle with dual-dissipative variables, we derive an interpretable series of partial differential equations, fine tuned through a training strategy informed by data from the phonon Boltzmann transport equation. Moreover, we also present the inner-step operation to narrow the gap from the discrete form to the continuous system. Through numerical tests, our model demonstrates excellent predictive capabilities across various heat conduction regimes, including diffusive, hydrodynamic, and ballistic regimes, and displays its robustness and precision even with discontinuous initial conditions.

Lifting relations for a generalized total-energy double-distribution-function kinetic model and their impact on compressible turbulence simulation

Noncommutative analysis of Hermite expansions

Hermite Expansion Technique for Model Reduction of Circuit Systems with Delay Components

We introduce a class of rough symbols for pseudo-multipliers for Hermite expansions and obtain Lp and weighted Lp estimates. These symbols generalise the class of rough symbols introduced by Kenig–Staubach.