Gauss Hermite Research Articles

Timing jitter’s influence on the symbol error rate performance of the L -ary pulse position modulation free-space optical link in atmospheric turbulent channels with pointing errors

An analytical approach is proposed to evaluate the impact of timing jitter on the error performance of the L-ary pulse position modulation (L-PPM) free-space optical (FSO) link, under the gamma–gamma (ΓΓ) turbulence with pointing errors. The expression of the conditional symbol error rate (SER) for a certain timing jitter is developed while the Gaussian–Hermite polynomial approximation is utilized to derive the closed-form expression of the average SER by the jitter’s variance. It is discovered that the timing jitter contributes to an error floor on the SER in both Monte–Carlo simulations and the theoretical results. Also, the jitter with small variance could be tolerable. What is more, the PPM system with lower order is more sensitive to the timing jitter than the higher ones.

Low-rank canonical-tensor decomposition of potential energy surfaces: application to grid-based diagrammatic vibrational Green's function theory

ABSTRACTA new method is proposed for a fast evaluation of high-dimensional integrals of potential energy surfaces (PES) that arise in many areas of quantum dynamics. It decomposes a PES into a canonical low-rank tensor format, reducing its integral into a relatively short sum of products of low-dimensional integrals. The decomposition is achieved by the alternating least squares (ALS) algorithm, requiring only a small number of single-point energy evaluations. Therefore, it eradicates a force-constant evaluation as the hotspot of many quantum dynamics simulations and also possibly lifts the curse of dimensionality. This general method is applied to the anharmonic vibrational zero-point and transition energy calculations of molecules using the second-order diagrammatic vibrational many-body Green's function (XVH2) theory with a harmonic-approximation reference. In this application, high dimensional PES and Green's functions are both subjected to a low-rank decomposition. Evaluating the molecular integrals over a low-rank PES and Green's functions as sums of low-dimensional integrals using the Gauss–Hermite quadrature, this canonical-tensor-decomposition-based XVH2 (CT-XVH2) achieves an accuracy of 0.1 cm−1 or higher and nearly an order of magnitude speedup as compared with the original algorithm using force constants for water and formaldehyde.

Euler–euler anisotropic gaussian mesoscale simulation of homogeneous cluster‐induced gas–particle turbulence

An Euler–Euler anisotropic Gaussian approach (EE‐AG) for simulating gas–particle flows, in which particle velocities are assumed to follow a multivariate anisotropic Gaussian distribution, is used to perform mesoscale simulations of homogeneous cluster‐induced turbulence (CIT). A three‐dimensional Gauss–Hermite quadrature formulation is used to calculate the kinetic flux for 10 velocity moments in a finite‐volume framework. The particle‐phase volume‐fraction and momentum equations are coupled with the Eulerian solver for the gas phase. This approach is implemented in an open‐source CFD package, OpenFOAM, and detailed simulation results are compared with previous Euler–Lagrange simulations in a domain size study of CIT. The results demonstrate that the proposed EE‐AG methodology is able to produce comparable results to EL simulations, and this moment‐based methodology can be used to perform accurate mesoscale simulations of dilute gas–particle flows. © 2017 American Institute of Chemical Engineers AIChE J, 63: 2630–2643, 2017

Performance Analysis of Multi-Hop Underwater Wireless Optical Communication Systems

In this letter, we analytically evaluate the end-to-end bit error rate (BER) of point-to-point multi-hop underwater wireless optical communication (UWOC) systems with respect to all degrading effects of the UWOC channel, namely absorption, scattering, and turbulence-induced fading. To do so, we first derive the BER expression of a single-hop UWOC link as the building block for end-to-end BER evaluation. We also apply Gauss–Hermite quadrature formula to obtain the closed-form solution for the system BER in the case of lognormal underwater fading channel. Numerical results demonstrate that multi-hop transmission, by alleviating channel impairments, can significantly improve the system performance and extend the viable end-to-end communication distance.

A comparison of analytic approaches for individual patient data meta-analyses with binary outcomes

BackgroundIndividual patient data meta-analyses (IPD-MA) are often performed using a one-stage approach-- a form of generalized linear mixed model (GLMM) for binary outcomes. We compare (i) one-stage to two-stage approaches (ii) the performance of two estimation procedures (Penalized Quasi-likelihood-PQL and Adaptive Gaussian Hermite Quadrature-AGHQ) for GLMMs with binary outcomes within the one-stage approach and (iii) using stratified study-effect or random study-effects.MethodsWe compare the different approaches via a simulation study, in terms of bias, mean-squared error (MSE), coverage and numerical convergence, of the pooled treatment effect (β1) and between-study heterogeneity of the treatment effect (τ12). We varied the prevalence of the outcome, sample size, number of studies and variances and correlation of the random effects.ResultsThe two-stage and one-stage methods produced approximately unbiased β1 estimates. PQL performed better than AGHQ for estimating τ12 with respect to MSE, but performed comparably with AGHQ in estimating the bias of β1 and of τ12. The random study-effects model outperformed the stratified study-effects model in small size MA.ConclusionThe one-stage approach is recommended over the two-stage method for small size MA. There was no meaningful difference between the PQL and AGHQ procedures. Though the random-intercept and stratified-intercept approaches can suffer from their underlining assumptions, fitting GLMM with a random-intercept are less prone to misfit and has good convergence rate.

On the approximation of electronic wavefunctions by anisotropic Gauss and Gauss–Hermite functions

The electronic Schr\"odinger equation describes the motion of N electrons under Coulomb interaction forces in a field of clamped nuclei. The solutions of this equation, the electronic wavefunctions, depend on 3N variables, three spatial dimensions for each electron. We study the approximability of these wavefunctions by linear combinations of anisotropic Gauss functions, or more precisely Gauss-Hermite functions, products of polynomials and anisotropic Gauss functions in the narrow sense. We show that the original, singular wavefunctions can up to given accuracy and a negligibly small residual error be approximated with only insignificantly more such terms than their convolution with a Gaussian kernel of sufficiently small width and that basically arbitrary orders of convergence can be reached. This is a fairly surprising result, since it essentially means that by this type of approximation, the intricate hierarchies of non-smooth cusps in electronic wavefunctions have almost no impact on the convergence, once the global structure is resolved.

Population abundance estimation with heterogeneous encounter probabilities using numerical integration

ABSTRACTEstimation of population abundance is a common problem in wildlife ecology and management. Capture‐mark‐reencounter (CMR) methods using marked animals are a standard approach, particularly in recent history with the development of innovative methods of marking using camera traps or DNA samples. However, estimates of abundance from multiple encounters of marked individuals are biased low when individual heterogeneity of encounter probabilities is not accounted for in the estimator. We evaluated the operating characteristics of the Huggins logit‐normal estimator through computer simulations, using Gaussian–Hermite quadrature to model individual encounter heterogeneity. We simulated individual encounter data following a factorial design with 2 levels of sampling occasions (t = 5, 10), 3 levels of abundance (N = 100, 500, 1,000), 4 levels of median detection probabilities (p = 0.1, 0.2, 0.4, 0.6) for each sampling occasion (on the probability scale), and 4 levels of individual heterogeneity (σp = 0, 0.5, 1, 2; on the logit normal scale), resulting in a design space consisting of 96 simulation scenarios (2 × 3 × 4 × 4). For each scenario, we performed 1,000 simulations using the Huggins estimators Mt, M0, MtRE, and M0RE, where the RE subscript corresponds to the random effects model. As expected, the Mt and M0 estimators were biased when individual heterogeneity was present but unbiased for σp = 0 data. The estimators for MtRE and M0RE were biased high for N = 100 and median p ≤ 0.2 but showed little bias elsewhere. The bias is attributed to the occasional sets of data that result in a low overall detection probability and a resulting highly skewed sampling distribution of . This result is confirmed in that the median of the sampling distributions was only slightly biased high. The random effects estimators performed poorly for σp = 0 data, mainly because a log link function forces the estimate of σp > 0. However, the Fletcher statistic provided useful evidence to evaluate σp > 0, as did likelihood ratio tests of the null hypothesis σp = 0. Generally, confidence interval coverage of N appears close to the nominal 95% expected when the estimator is not biased. © 2017 The Wildlife Society.

Experimental technique for multi-qubit nuclear magnetic resonance system

With the development of quantum information and quantum computation science, quantum information processor has been widely used in different areas such as quantum simulation, quantum computation and quantum metrology and so on. To make quantum computer come true, we need to increase the number of controllable qubits of the system and improve the controllability to perform complex quantum manipulation. As a good experimental testbed for quantum information processing, nuclear magnetic resonance (NMR) spin system provides rich and sophisticated quantum control methods. In recent years a lot of multi-qubit experiments have been performed on the platform and a series of experimental technologies have been developed. In this paper, we firstly explain the difficulties of multi-qubit NMR experiments. Then by focusing on the experiment of 7-qubit labelled pseudo-pure state preparation and other relevant experiments, we review the technologies in multi-qubit experiments. Using the radio frequency selective method, the inhomogeneities of the radio frequency pulses are reduced and the spectral resolution is improved. After performing 1/2 spin selective sequence, we can regard the three methyl protons in the sample of crotonic acid as a single 1/2 spin nucleus and treat the whole molecule as a 7-qubit quantum information processor. We utilize Gauss pulses, Hermite pulses, composite pulses and gradient ascent pulse engineering (GRAPE) pulses to implement basic /2 and rotation operations. The GRAPE pulses are calculated by subspace GRAPE program to speed up the computation greatly. The errors of the basic pulses caused by chemical shift and J-coupling evolution can be estimated by the program of pulse compilation. It divides the errors of the pulses into a series of post-errors and pre-errors. A program of sequence compilation is used to eliminate the accumulated error of the whole pulse sequence, reduce the number of pulses and optimize the experimental duration. A variety of methods of quantum state tomography have been proposed to improve the efficiency of reading out information about quantum state. As an experimental example, we combine the above experimental technologies and perform the experiment of 7-qubit labelled pseudo-pure state preparation by using the method of cat state preparation. The sequence of cat state preparation consists of three steps:encoding procedure, phase cycling and decoding procedure. We use 14 experiments to realize the phase cycling and acquire the final 7-qubit labelled pseudo-pure state. The total duration of experimental sequence is about 132 ms. All the readout spectra have the similar shapes to the theoretical expectations. Finally we give an outlook for further research in this direction.

WavePacket: A Matlab package for numerical quantum dynamics. I: Closed quantum systems and discrete variable representations

WavePacket is an open-source program package for the numerical simulation of quantum-mechanical dynamics. It can be used to solve time-independent or time-dependent linear Schrödinger and Liouville–von Neumann-equations in one or more dimensions. Also coupled equations can be treated, which allows to simulate molecular quantum dynamics beyond the Born–Oppenheimer approximation. Optionally accounting for the interaction with external electric fields within the semiclassical dipole approximation, WavePacket can be used to simulate experiments involving tailored light pulses in photo-induced physics or chemistry. The graphical capabilities allow visualization of quantum dynamics ‘on the fly’, including Wigner phase space representations. Being easy to use and highly versatile, WavePacket is well suited for the teaching of quantum mechanics as well as for research projects in atomic, molecular and optical physics or in physical or theoretical chemistry. The present Part I deals with the description of closed quantum systems in terms of Schrödinger equations. The emphasis is on discrete variable representations for spatial discretization as well as various techniques for temporal discretization. The upcoming Part II will focus on open quantum systems and dimension reduction; it also describes the codes for optimal control of quantum dynamics. The present work introduces the MATLAB version of WavePacket 5.2.1 which is hosted at the Sourceforge platform, where extensive Wiki-documentation as well as worked-out demonstration examples can be found. Program summaryProgram Title: WavePacketProgram Files doi:http://dx.doi.org/10.17632/tc5mrhx9sb.1Licensing provisions: GPLv3Programming language:MatlabNature of problem:Schrödingerś equations are of fundamental importance in non-relativistic quantum mechanics of distinguishable particles. The solutions of the time-independent equation (TISE) are wavefunctions in coordinate space, the absolute squares of which are usually interpreted as probability density. The time-dependent equation (TDSE) describes the dynamics of a quantum system evolving in time. It plays a crucial role for the simulation, understanding, and prediction of modern experiments in atomic, molecular and optical physics where systems are driven by temporally shaped external fields.Solution method:All numerical methods compiled in WavePacket are based on discrete variable representations. Currently implemented are Gauss–Hermite, Gauss–Legendre and FFT-based schemes. The TISE is solved either by direct diagonalization or by propagation in imaginary time. For the TDSE there is a choice between second order differencing, operator splitting and Chebychev polynomial methods.Additional comments including Restrictions and Unusual features:The WavePacket program package is rather easy and intuitive to use, providing visualization of quantum dynamics ‘on the fly’. It is mainly intended for low-dimensional systems, typically not exceeding three to five degrees of freedom. Detailed user guides and reference manuals are available through numerous Wiki pages hosted at the SourceForge platform where also a large number of well documented demonstration examples can be found.

Uncertainty Exchange Through Multiple Quadrature Kalman Filtering

One of the major challenges in Bayesian filtering is the curse of dimensionality. The quadrature Kalman filter (QKF) is the method of choice in many real-life Gaussian problems, but its computational complexity increases exponentially with the dimension of the state. As a promising solution to overcome the filter limitations in such scenarios, we further explore the multiple state-partitioning approach, which considers the partition of the original space into several subspaces, with the goal to apply a low-dimensional filter at each partition. In this contribution, the key idea is to take advantage of the estimation uncertainty provided by the QKF to improve the interaction among filters and avoid the point estimate approximation performed in the original Multiple QKF (MQKF). The new filter formulation, named Improved MQKF, considers Gauss–Hermite quadrature rules to propagate the subspaces of interest, together with cubature rules for marginalization purposes. The nested quadrature–cubature approximation provides robustness and improves the filter performance. Simulation results for a multiple target tracking scenario are provided to support the discussion.

Improving the full spectrum fitting method: accurate convolution with Gauss–Hermite functions

I start by providing an updated summary of the penalized pixel-fitting (pPXF) method, which is used to extract the stellar and gas kinematics, as well as the stellar population of galaxies, via full spectrum fitting. I then focus on the problem of extracting the kinematic when the velocity dispersion $\sigma$ is smaller than the velocity sampling $\Delta V$, which is generally, by design, close to the instrumental dispersion $\sigma_{\rm inst}$. The standard approach consists of convolving templates with a discretized kernel, while fitting for its parameters. This is obviously very inaccurate when $\sigma<\Delta V/2$, due to undersampling. Oversampling can prevent this, but it has drawbacks. Here I present a more accurate and efficient alternative. It avoids the evaluation of the under-sampled kernel, and instead directly computes its well-sampled analytic Fourier transform, for use with the convolution theorem. A simple analytic transform exists when the kernel is described by the popular Gauss-Hermite parametrization (which includes the Gaussian as special case) for the line-of-sight velocity distribution. I describe how this idea was implemented in a significant upgrade to the publicly available pPXF software. The key advantage of the new approach is that it provides accurate velocities regardless of $\sigma$. This is important e.g. for spectroscopic surveys targeting galaxies with $\sigma\ll\sigma_{\rm inst}$, for galaxy redshift determinations, or for measuring line-of-sight velocities of individual stars. The proposed method could also be used to fix Gaussian convolution algorithms used in today's popular software packages.

Inference of reaction rate parameters based on summary statistics from experiments

We present the results of an application of Bayesian inference and maximum entropy methods for the estimation of the joint probability density for the Arrhenius rate parameters of the rate coefficient of the H2/O2-mechanism chain branching reaction H+O2→OH+O. Available published data is summary statistics in terms of nominal values and error bars of the rate coefficient of this reaction at a number of temperature values obtained from shock-tube experiments. Our approach relies on generating data, in this case OH concentration profiles, consistent with the given summary statistics, using Approximate Bayesian Computation methods and a Markov chain Monte Carlo procedure. The approach permits the forward propagation of parametric uncertainty through the computational model in a manner that is consistent with the published statistics. A consensus joint posterior on the parameters is obtained by pooling the posterior parameter densities given each consistent data set. To expedite this process, we construct efficient surrogates for the OH concentration using a combination of Padé and polynomial approximants. These surrogate models adequately represent forward model observables and their dependence on input parameters and are computationally efficient to allow their use in the Bayesian inference procedure. We also utilize Gauss–Hermite quadrature with Gaussian proposal probability density functions for moment computation resulting in orders of magnitude speedup in data likelihood evaluation. Despite the strong non-linearity in the model, the consistent data sets all result in nearly Gaussian conditional parameter probability density functions. The technique also accounts for nuisance parameters in the form of Arrhenius parameters of other rate coefficients with prescribed uncertainty. The resulting pooled parameter probability density function is propagated through stoichiometric hydrogen–air auto-ignition computations to illustrate the need to account for correlation among the Arrhenius rate parameters of one reaction and across rate parameters of different reactions.

Scale invariants from Gaussian–Hermite moments

Invariants to image scaling composed of Gaussian–Hermite moments are introduced in this paper for the first time. To achieve the invariance, we propose to modulate the Gaussian–Hermite polynomial basis using variable parameter σ, the value of which depends on the input image. The scaling invariance property can be coupled with the rotation invariance presented earlier. This approach is applicable in 2D as well as in 3D and provides very good numerical stability, as demonstrated by several experiments on real data.

Robust visual tracking framework in the presence of blurring by arbitrating appearance- and feature-based detection

Abstract This paper proposes a new visual tracking framework and demonstrates its merits via mobile robot experiments. An image sequence from the vision system of a mobile robot is not static when a mobile robot is moving, since slipping and vibration occur. These problems cause image blurring. Therefore, in this paper, we address the problem of robust object tracking under blurring and introduce a novel robust visual tracking framework based on the arbitration of the AdaBoost-based detection method and the appearance-based detection method to overcome the blurring problem. The proposed framework consists of three parts: (1) distortion error compensation and feature extraction using the Modified Discrete Gaussian–Hermite Moment (MDGHM) and fuzzy-based distortion error compensation, (2) object detection using arbitration of appearance- and feature-based object detection, and (3) object tracking using a Finite Impulse Response (FIR) filter. To demonstrate the performance of the proposed framework, mobile robot visual tracking experiments are carried out. The results show that the proposed framework is more robust against blurring than the conventional feature- and appearance-based methods.

Hard-wall and non-uniform lattice Monte Carlo approaches to one-dimensional Fermi gases in a harmonic trap

We present in detail two variants of the lattice Monte Carlo method aimed at tackling systems in external trapping potentials: a uniform-lattice approach with hard-wall boundary conditions, and a non-uniform Gauss–Hermite lattice approach. Using those two methods, we compute the ground-state energy and spatial density profile for systems of N=4–8 harmonically trapped fermions in one dimension. From the favorable comparison of both energies and density profiles (particularly in regions of low density), we conclude that the trapping potential is properly resolved by the hard-wall basis. Our work paves the way to higher dimensions and finite temperature analyses, as calculations with the hard-wall basis can be accelerated via fast Fourier transforms; the cost of unaccelerated methods is otherwise prohibitive due to the unfavorable scaling with system size. To illustrate this point, we show a brief performance comparison of accelerated versus unaccelerated methods across spatial dimensions.

Likelihood analysis for a class of spatial geostatistical compositional models

We propose a model-based geostatistical approach to deal with regionalized compositions. We combine the additive-log-ratio transformation with multivariate geostatistical models whose covariance matrix is adapted to take into account the correlation induced by the compositional structure. Such specification allows the usage of standard likelihood methods for parameters estimation. For spatial prediction we combined a back-transformation with the Gauss–Hermite method to approximate the conditional expectation of the compositions. We analyse particle size fractions of the top layer of a soil for agronomic purposes which are typically expressed as proportions of sand, clay and silt. Additionally a simulation study assesses the small sample properties of the maximum likelihood estimator.

Probabilistic load flow considering correlations of input variables following arbitrary distributions

For the purpose of accurately calculating probabilistic load flow (PLF) with correlated input variables following arbitrary distributions, this paper proposes a Latin hypercube sampling (LHS) based PLF method, which combines kernel density estimation with Nataf transformation to improve the calculation performance. First, enhanced kernel density estimation with adaptive-bandwidth is established to precisely depict arbitrary distributions, including the atypical probability density functions (PDF) of power injections from renewable energies such as wind and photovoltaic. Then, considering the correlations of renewable energies and the difficulties of kernel density estimation dealing with correlativity, Gauss–Hermite integral based Nataf transformation is proposed to handle the problem. The proposed method is demonstrated to be feasible and practicable in modified IEEE 14 and IEEE 118 systems with additional wind and photovoltaic power. The results suggest that proposed method is prominent in efficiency and accuracy, and applicable to arbitrary distributions of power injections in PLF calculation.

Modeling Longitudinal Obesity Data with Intermittent Missingness Using a New Latent Variable Model

We propose a latent variable model for informative missingness in longitudinal studies which is an extension of latent dropout class model. In our model, the value of the latent variable is affected by the missingness pattern and it is also used as a covariate in modeling the longitudinal response. So the latent variable links the longitudinal response and the missingness process. In our model, the latent variable is continuous instead of categorical and we assume that it is from a normal distribution. The EM algorithm is used to obtain the estimates of the parameter we are interested in and Gauss–Hermite quadrature is used to approximate the integration of the latent variable. The standard errors of the parameter estimates can be obtained from the bootstrap method or from the inverse of the Fisher information matrix of the final marginal likelihood. Comparisons are made to the mixed model and complete-case analysis in terms of a clinical trial dataset, which is Weight Gain Prevention among Women (WGPW) study. We use the generalized Pearson residuals to assess the fit of the proposed latent variable model.

Quadrature filters for one-step randomly delayed measurements

In this paper, two existing quadrature filters, viz., the Gauss–Hermite filter (GHF) and the sparse-grid Gauss–Hermite filter (SGHF) are extended to solve nonlinear filtering problems with one step randomly delayed measurements. The developed filters are applied to solve a maneuvering target tracking problem with one step randomly delayed measurements. Simulation results demonstrate the enhanced accuracy of the proposed delayed filters compared to the delayed cubature Kalman filter and delayed unscented Kalman filter.

Lattice Boltzmann models based on half-range Gauss–Hermite quadratures

We discuss general features of thermal lattice Boltzmann models based on half-range Gauss quadratures, specialising to the half-range Gauss-Hermite and Gauss-Laguerre cases. The main focus of the paper is on the construction of high order half-range Hermite lattice Boltzmann (HHLB) models. The performance of the HHLB models is compared with that of Laguerre lattice Boltzmann (LLB) and full-range Hermite lattice Boltzmann (HLB) models by conducting convergence tests with respect to the quadrature order on stationary profiles of the particle number density, macroscopic velocity, temperature and heat fluxes in the two-dimensional Couette flow. The Bhatnagar-Gross-Krook (BGK) collision term is used throughout the paper. To reduce the computational costs of the numerical simulations, we use mixed lattice Boltzmann models, constructed using different quadrature methods on each Cartesian axis. For Kn ? 0.01 , the HLB models require the least number of velocities to satisfy our convergence test. When Kn ? 0.05 , the HLB models are outperformed in terms of number of velocities employed by both the LLB and the HHLB models. Moreover, we find that the HHLB models require less quadrature points than the LLB models at all tested values of Kn, which we attribute to the Maxwellian form of the weight function for the half-range Hermite polynomials.