Low-order Statistical Moments Research Articles

Constructing efficient strategies for the process optimization by restart.

Optimization of the mean completion time of random processes by restart is a subject of active theoretical research in statistical physics and has long found practical application in computer science. Meanwhile, one of the key issues remains largely unsolved: how to construct a restart strategy for a process whose detailed statistics are unknown to ensure that the expected completion time will reduce? Addressing this query here we propose several constructive criteria for the effectiveness of various protocols of noninstantaneous restart in the mean completion time problem and in the success probability problem. Being expressed in terms of a small number of easily estimated statistical characteristics of the original process (MAD, median completion time, low-order statistical moments of completion time), these criteria allow informed restart decision based on partial information.

Data-driven stochastic particle scheme for collisional plasma simulations

We present a novel framework, enabled by machine learning (ML) trainable models, for collisional plasma flow simulations governed by the Rosenbluth-Fokker-Planck (RFP) equation. The maximum entropy objective is applied to close the particle velocity distribution. The closure offers a flexible tool to reconstruct a large variety of distributions, using a few low-order statistical moments. The methodology is extended by introducing an artificial neural network (ANN) to couple the moments and the transport coefficients. In order to ensure the model's training and input data quality, we generate diverse training data sets using the high-fidelity Direct Simulation Monte-Carlo (DSMC) method. Moreover, energy constraints are integrated in the model training, thereby improving prediction accuracy. As a final step, the ANN is combined with a stochastic particle system, where the accuracy of the whole machinery for non-equilibrium thermal relaxation scenarios is evaluated. Simulation results indicate that our approach reproduces the correct relaxation behavior at lower computational cost compared to standard methods.

Reconstruction of neutron multiplicity distributions from low-order statistical information

A point-kinetic methodology for rapid and accurate computation of neutron multiplicity distributions in fixed time intervals is demonstrated. The approach uses an optimal combination of low-order count probabilities and low-order statistical moments, both of which are inexpensively computed from a backward Master equation, to reconstruct distributions valid for all count numbers. For high count numbers a generalized Laguerre polynomial representation is shown to accurately reconstruct the multiplicity distribution from five or fewer statistical moments when the zero-count probability is negligible. For intermediate count numbers and when the zero-count probability is significant, maximum information entropy reconstruction constrained by low-order moments and low-order count probabilities reproduces the overall count distribution very accurately. Numerical results for count number distributions and reconstruction accuracy as a function of number of constraints, relative to a system state-updating stochastic simulation MC algorithm (SSA), are presented to demonstrate the efficacy of this approach.

Uncertainty propagation for deterministic models of biochemical networks using moment equations and the extended Kalman filter.

Differential equation models of biochemical networks are frequently associated with a large degree of uncertainty in parameters and/or initial conditions. However, estimating the impact of this uncertainty on model predictions via Monte Carlo simulation is computationally demanding. A more efficient approach could be to track a system of low-order statistical moments of the state. Unfortunately, when the underlying model is nonlinear, the system of moment equations is infinite-dimensional and cannot be solved without a moment closure approximation which may introduce bias in the moment dynamics. Here, we present a new method to study the time evolution of the desired moments for nonlinear systems with polynomial rate laws. Our approach is based on solving a system of low-order moment equations by substituting the higher-order moments with Monte Carlo-based estimates from a small number of simulations, and using an extended Kalman filter to counteract Monte Carlo noise. Our algorithm provides more accurate and robust results compared to traditional Monte Carlo and moment closure techniques, and we expect that it will be widely useful for the quantification of uncertainty in biochemical model predictions.

Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes

This paper deals with the applications of stochastic spectral methods for structural topology optimization in the presence of uncertainties. A non-intrusive polynomial chaos expansion is integrated into a topology optimization algorithm to calculate low-order statistical moments of the mechanical-mathematical model response. This procedure, known as robust topology optimization, can optimize the mean of the compliance while simultaneously minimizing its standard deviation. In order to address possible variabilities in the loads applied to the mechanical system of interest, magnitude and direction of the external forces are assumed to be uncertain. In this probabilistic framework, forces are described as a random field or a set of random variables. Representation of the random objects and propagation of load uncertainties through the model are efficiently done through Karhunen-Lo\`{e}ve and polynomial chaos expansions. We take advantage of using polygonal elements, which have been shown to be effective in suppressing checkerboard patterns and reducing mesh dependency in the solution of topology optimization problems. Accuracy and applicability of the proposed methodology are demonstrated by means of several topology optimization examples. The obtained results, which are in excellent agreement with reference solutions computed via Monte Carlo method, show that load uncertainties play an important role in optimal design of structural systems, so that they must be taken into account to ensure a reliable optimization process.

Intermittency and stochastic modeling of hydrodynamic pressure fluctuations in the near field of compressible jets

The intermittent statistics of the pressure fluctuations measured in the near field of a compressible jet are investigated under several flow conditions. An experiment is carried out in a semi-anechoic chamber on a single stream compressible jet at Mach numbers varying from 0.5 to 0.9 and measuring the fluctuating pressure in several positions in the near field. The main quantities analyzed are the intermittent time and the energy amplitude of events that are extracted from the experimental data through a wavelet-based tracking algorithm. As an extension of a previous paper (Camussi et al., 2017), low order statistical moments (mean and variance) and Probability Distribution Functions are parametrized in terms of three relevant quantities characterizing the jet flow physics: the Mach number, the radial distance from the jet axis and the axial position. It is observed that the non-dimensional statistical quantities are weakly dependent upon the flow conditions allowing for simple stochastic models to be introduced on the basis of suitable fittings of averaged statistical properties and of the Probability Distribution Functions.

A Spatial-Random-Process Based Multidisciplinary System Uncertainty Propagation Approach With Model Uncertainty

The performance of a multidisciplinary system is inevitably affected by various sources of uncertainties, usually categorized as aleatory (e.g., input variability) or epistemic (e.g., model uncertainty) uncertainty. In the framework of design under uncertainty, all sources of uncertainties should be aggregated to assess the uncertainty of system quantities of interest (QOIs). In a multidisciplinary design system, uncertainty propagation (UP) refers to the analysis that quantifies the overall uncertainty of system QOIs resulting from all sources of aleatory and epistemic uncertainty originating in the individual disciplines. However, due to the complexity of multidisciplinary simulation, especially the coupling relationships between individual disciplines, many UP approaches in the existing literature only consider aleatory uncertainty and ignore the impact of epistemic uncertainty. In this paper, we address the issue of efficient uncertainty quantification of system QOIs considering both aleatory and epistemic uncertainties. We propose a spatial-random-process (SRP) based multidisciplinary uncertainty analysis (MUA) method that, subsequent to SRP-based disciplinary model uncertainty quantification, fully utilizes the structure of SRP emulators and leads to compact analytical formulas for assessing statistical moments of uncertain QOIs. The proposed method is applied to a benchmark electronic packaging design problem. The estimated low-order statistical moments of the QOIs are compared to the results from Monte Carlo simulations (MCSs) to demonstrate the effectiveness of the method. The UP result is then used to facilitate the robust design optimization of the electronic packaging system.

A Method to Deconvolve Errors in GPS RO-Derived Water Vapor Histograms

AbstractWater vapor is an important constituent in the earth’s atmosphere that must be understood to predict weather and climate. Water vapor is challenging to measure, and observations of water vapor must be as independent of weather and climate models as possible in order to assess and improve those models. When combined with independent atmospheric temperature estimates, GPS radio occultation (RO) refractivity profiles yield unique, all-weather, high-vertical-resolution, globally distributed profiles of tropospheric water vapor whose vertical extent is maximum at low latitudes. A method for retrieving water vapor, known as the direct method, combines GPS RO-derived refractivity with temperatures from global weather analyses. Unlike variational methods, the direct method does not use the water vapor estimates from analyses because of their potential systematic errors. While utilization of the direct method has been limited because it produces negative values, these unphysical negative values reveal unique information about errors. A deconvolution technique is presented here that uses the negative values to estimate and remove random errors in histograms of water vapor derived via the direct method. Results indicate direct method specific humidity errors at low latitudes range from 0.4 g kg−1 at 725 hPa to 0.14 g kg−1 near 350 hPa, which can be removed via deconvolution. The deconvolved histograms extend to about the 240-K level in the troposphere, corresponding to 10 km at low latitudes. Unlike the limited information of low-order statistical moments like means and variances, histograms capture the full range of variability, which opens a new window into water vapor behavior and increases understanding of processes at work in the hydrological cycle.

Analytical uncertainty quantification for modal frequencies with structural parameter uncertainty using a Gaussian process metamodel

Quantifying the uncertainty in the dynamic properties of large-scale complex engineering structures presents significant computational challenges. Monte Carlo simulation (MCS) method is extensively employed to perform uncertainty quantification (UQ) because of its generality, stability, and easy implementation. However, a brute-force MCS approach may be unaffordable and impractical when the target model contains a large number of uncertain parameters. In this circumstance, MCS requires a potentially burdensome (if not computationally intractable) number of model evaluations to obtain a credible estimate of the global statistics. In this study, a general framework for analytical UQ of model outputs using a Gaussian process (GP) metamodel is presented, where case inputs are characterized as normal and/or uniform random variables. A detailed derivation of important low-order statistical moments (mean and variance) is given analytically. This analytical method is adopted to characterize the uncertainty of modal frequencies of two bridges with assumed normally- and uniformly-distributed parameters. Meanwhile, the brute-force MCS approach is used for comparison of GP metamodel-derived statistics. Results show that the GP method outperforms the MCS methodology in terms of computational cost, with consistency in the ”true” values obtained by MCS. It demonstrates that this GP method is feasible and reliable for modal frequency UQ of complex structures.

Nonparametric segmentation of nonstationary time series

The nonstationary evolution of observable quantities in complex systems can frequently be described as a juxtaposition of quasistationary spells. Given that standard theoretical and data analysis approaches usually rely on the assumption of stationarity, it is important to detect in real time series intervals holding that property. With that aim, we introduce a segmentation algorithm based on a fully nonparametric approach. We illustrate its applicability through the analysis of real time series presenting diverse degrees of nonstationarity, thus showing that this segmentation procedure generalizes and allows one to uncover features unresolved by previous proposals based on the discrepancy of low order statistical moments only.

On the Probability Structure of Output Noise From a Digital Phase Demodulation System Subject to Biased Intensity-Based Input Noise

Interferometry is a common technique used in fiber sensing that requires a demodulation algorithm to extract the signal of interest. Of great interest for sensor characterization is the performance of the demodulation scheme under the influence of input noise. Here we consider correlated, biased intensity noise corrupting an interferometer. We analytically compute a probability density function of this output noise only and use this to compute low-order statistical moments of the output noise. We compare the analytical formulations with simulated data from a representative demodulation scheme used in a currently existing fiber Bragg grating sensor system and find excellent agreement within the example of Gaussian input noise.

Output-Noise Statistical Characterization for Digital-Phase-Demodulation Systems With Intensity-Based Input Noise

A large fraction of fiber-optic-sensor systems make use of interferometry in the measurement process, implying the need for demodulation to obtain the encoded signal of interest. Of great interest for sensor characterization is the performance of the demodulation scheme under the influence of noise. In this paper, we construct a very general transfer function that relates intensity-induced input noise to its phase noise realization in the output for a wide class of digital-demodulation techniques. We proceed to compute analytically a probability density function of this output noise and use this to compute low-order statistical moments of the output noise. We compare the analytical formulations with simulated data from a representative demodulation scheme used in a currently existing fiber Bragg grating (FBG) sensor system and find excellent agreement

Assessment of spectral representation and Karhunen–Loève expansion methods for the simulation of Gaussian stochastic fields

From the wide variety of methods developed for the simulation of Gaussian stochastic processes and fields, two are most often used in applications: the spectral representation method and the Karhunen–Loève (K–L) expansion. In this paper, an in-depth assessment on the capabilities of the two methods is presented. The spectral representation method expands the stochastic field as a sum of trigonometric functions with random phase angles and/or amplitudes. The version having only random phase angles is used in this work. A wavelet-Galerkin scheme is adopted for the efficient numerical solution of the Fredholm integral equation appearing in the K–L expansion. A one-dimensional homogeneous Gaussian random field with two types of autocovariance function, exponential and square exponential, is used as the benchmark test. The accuracy achieved and the computational effort required by the K–L expansion and the spectral representation for the simulation of the stochastic field are investigated. The accuracy obtained by the two approaches is examined by comparing their ability to produce sample functions that match the target correlation structure and the Gaussian probability distribution or, alternatively, its low order statistical moments (mean, variance and skewness).

Multifractal modelling and spectrum analysis: Methods and applications to gamma ray spectrometer data from southwestern Nova Scotia, Canada

Multifractal spectrum, autocorrelation/semivariogram and power spectrum are three different functions characterizing a field or measure from different aspects. These functions are interre- lated in such that the autocorrelation/semivariogram and power spectrum are related to the low order statistical moments (0 to 2 nd) which may determine the local multifractality ( τ ²(1)) of a multifractal measure. A better understanding of the interrelationships among these three functions is important because, on one hand, the multifractal modelling characterizes a multifractal measure in a more de- tailed manner since it involves moments of all orders. On the other hand, the commonly used semivariogram and power spectrum functions can be used as alternatives to study the dominant multifractal properties around the mean measure. Moreover, semivariogram and power-spectrum functions provide spatial and spectral information, which is highly valued in geological applications. A new fractal relation found between area and power-spectrum will be useful for investigation of new meth- ods of spatial-spectral analysis for pattern recognition, anomaly separation, classification, etc. These results have been demonstrated with the case study of modelling gamma ray spectrometer data from the mineral district, southwestern Nova Scotia, Canada. The results have shown that the values of uranium (U), thorium (Th) and potassium (K) may possess monofractal properties whereas the ratios of these values show high multifractality. The values of the ratios U/K and U/Th show relatively large variances and may provide more information for distinguishing the distinct phases of the granites, country rocks as well as possible gold mineralization-associated regional hydrothermal alteration zones. In addition, the power spectra for U, Th, K, U/Th and U/K consistently show two distinct power-law relationships for two ranges of wave number 12≤ ω ≤160 km and 0≤ ω ≤12 km. These properties might provide useful thresholds for separating the power-spectrum values into two types of patterns to reflect different influences of possible geological processes such as hydrothermal alteration in the study area.

The use of wind probability distributions derived from the maximum entropy principle in the analysis of wind energy. A case study

This paper analyses the use of a general probability distribution obtained through application of the maximum entropy principle (MEP), constrained by the low-order statistical moments of a given set of wind speed data, in the estimation of wind energy. For this purpose, a comparison is made between the two parameter Weibull distribution and the distributions obtained through the MEP. This comparison is based on an analysis of the level of fit to the cumulative frequencies of the hourly mean wind speeds recorded at weather stations located in the Canarian Archipelago. A comparison is also made of the ability to describe the experimental mean wind power density. The application of the probability plot correlation coefficient R2, adjusted for degrees of freedom, shows that the Weibull distribution, whose parameters are estimated using the maximum likelihood principle, provide worse fits in all the cases analysed than those obtained through the maximum entropy distributions constrained by the low-order statistical moments. It is, thus, shown that maximum entropy distributions constrained by the three low-order statistical moments, in addition to representing the probabilities of observed periods of null wind speeds, offer less relative errors in determining the mean wind power density than the Weibull distribution. However, among other advantages of the Weibull distribution, is the greater simplicity of the calculations involved.

Flow Induced Orientation Of Fibers In CouetteFlow Between Eccentric Cylinders

The microstructure and rheology of suspensions can be characterized by using low-order statistical moments of the orientation distribution function. This is illustrated by calculating the orientation dyad and the fiber induced stress for Couette flow between two eccentrically situated cylinders. Under the conditions studied, the eigenvalues of the orientation dyad remain nonnegative throughout the flow domain. The results are compared with other theories for the microstructure that show unrealizable behavior under the same conditions.

Chaos and quantum-classical correspondence via phase-space distribution functions

Quantum-classical correspondence in conservative chaotic Hamiltonian systems is examined using a uniform structure measure for quantal and classical phase space distribution functions. The similarities and differences between quantum and classical time-evolving distribution functions are exposed by both analytical and numerical means. The quantum-classical correspondence of low-order statistical moments is also studied. The results shed considerable light on quantum-classical correspondence.

A faster simulation method for the stochastic response of hysteretic structures subject to earthquakes

A semi-analytical forward-difference Monte Carlo simulation procedure is proposed for the determination of the lower order statistical moments and the joint probability density function of the stochastic response of hysteretic non-linear multi-degree-of-freedom structural systems subject to nonstationary gaussian white noise excitation, as an alternative to conventional direct simulation methods. The method generalizes the so-called Ermak-Allen algorithm developed for simulation applications in molecular dynamics to structural hysteretic systems. The proposed simulation procedure rely on an assumption of local gaussianity during each time step. This assumption is tantamount to various linearizations of the equations of motion. The procedure then applies an analytical convolution of the excitation process, hereby reducing the generation of stochastic processes and numerical integration to the generation of random vectors only. Such a treatment offers higher rates of convergence, faster speed and higher accuracy. The procedure has been compared to the direct Monte Carlo simulation procedure, which uses a fourth-order Runge-Kutta scheme with the white noise process approximated by a broad band Ruiz-Penzien broken line process. The considered system was a multi-dimenensional hysteretic shear frame, where the constitutive equation of the hysteretic shear forces are described by a bilinear hysteretic model. The comparisons show that significant savings in computer time and accuracy can be achieved.

A two-phase flow model of the Rayleigh–Taylor mixing zone

The Rayleigh–Taylor instability of an interface separating fluids of distinct density is driven by an acceleration across the interface. Low order statistical moments of fluctuating fluid quantities characterize the hydrodynamics of the mixing zone. A new model is proposed for the momentum coupling between the two phases. This model is validated against computational data for compressible flows, including flows near the incompressible limit. Our main result is a zero parameter first order closure for ensemble averaged two phase flow equations. We do not, however, fully solve the closure problem, as the equations we derive are missing an (internal) boundary condition along any surface for which either phase goes to zero volume fraction. In this sense, the closure problem is reduced from a volume to a surface condition, rather than being solved completely. We compare two formulations of the statistical moments, one based on two phase flow and the other on turbulence models. These formulations describe different aspects of the mixing process. For the problem considered, the two phase flow moments appear to be preferable, in that they subsume the turbulence moments but not conversely.

Statistical properties of chaotic dynamical systems which exhibit strange attractors

A path integral method is developed for the calculation of teh statistical properties of turbulent dynamical systems. The method is applicable to conservative systems which exhibit a transition to stochasticity as well as dissipative systems which exhibit strange attractors. A specific dissipative mapping is considered in detail which models the dynamics of a Brownian particle in a wave field with a broad frequency spectrum. Results are presented for the low order statistical moments for three turbulent regimes which exhibit strange attractors corresponding to strong, intermediate, and weak collisional dumping. In the dissipationless limit this map is equivalent to the conservative Chrikov- Taylor mapping. The turbulent behavior of the Chirikov-Taylor mapping is shown to be diffusive due to the intrinsic stochasticity; and the stochastic diffusion coefficient derived by Rechester and White is recovered. The statistical dynamics are significantly altered by the inclusion of damping which restricts the chaotic motion to a strange attractor which is bounded in phase space. Specifically, for weak and intermediate damping the long-time dynamics are no longer diffusive; and the corrections to the random phase results for the low order moments decay away for long times. In addition the “accelerator modes”, which are exhibited by the Chirikov-Taylor mapping, are destroyed by the drug. These results provide a new description of the effects of collisional damping on the stochastic heating of plasmas by electrostatic waves.