Novel gradient-enhanced Bayesian neural networks for uncertainty propagation

Causality and Bayesian Network PDEs for multiscale representations of porous media

Complete Analytical Solution of the Asian Option Pricing and Asian Option Value-at-Risk Problems: A Probability Density Function Approach

Complete Analytical Solution of the American Style Option Pricing with Constant and Stochastic Volatilities: A Probability Density Function Approach

The three-condition version of the uniform conditional state combustion model makes use of the mixture fraction, progress variable, and normalized total enthalpy as conditioning variables to build a three-dimensional conditional manifold for chemistry. In order to map the solution in conditional space into the flow domain, the joint Probability Density Function (PDF) of the conditioning variables needs to be modeled. In simulations, presumed functions (i.e., β-PDF for the mixture fraction and progress variable and δ-PDF for total enthalpy) are often used for modeling the marginal PDFs. In this work, the measurements from the Cambridge/Sandia burner are employed to obtain the marginal PDFs for the conditioning variables at various points in the reacting domain. The measurements are then combined from all positions in space to form conditional PDFs of the normalized total enthalpy for various values of the other two variables. In the vicinity of the flame brush, the marginal PDF of the normalized total enthalpy resembles a bimodal Gaussian distribution; nonetheless, the conditional PDFs for this variable are nearly Gaussian distributions. The correlation coefficients between the conditioning variables are also investigated, and the assumption of their statistical independence is examined. To consider the association between the conditioning variables for modeling, the copula concept is introduced, and the performances of three different copulas are tested. Furthermore, the statistical moments of the conditioning variables are computed from the experimental data at different points and are utilized for modeling the joint PDF of the conditioning variables from two different approaches that are compared.

Environmental noise energy can be effectively captured by an energy harvester (EH) and then converted into usable electricity for low-powered monitoring sensors in aerospace vehicles. It is important to investigate the influence of potential well parameters on the energy harvesting performance under random excitation. Therefore, this paper theoretically derives the marginal probability density function (PDF), the joint PDF, and the averaging output power of the decoupled bistable energy harvester driven by the Gaussian white noise. First, the bistable energy harvester is linearized using the statistical linearization method, which is advantageous in obtaining coarse mean square values of outputs for further analysis. Second, the dimension of Fokker–Planck–Kolmogorov (FPK) equations is reduced by the state-space-split (SSS) method. Then, the approximate marginal and joint PDF can be obtained. Third, the influence of potential well width and height on PDFs and the averaging output power are, respectively, analyzed and discussed, respectively. Finally, the theoretical derivation is successfully validated by Monte Carlo simulations. Results show that the large potential well width can improve the averaging output power of the bistable EH under random excitation.

In recent years, the demand to provide best-estimate predictions with confident bounds is increasing for the nuclear reactor performance and safety analysis. The Organisation for Economic Co-operation and Development Nuclear Energy Agency has been developing an international benchmark of the light water reactor (LWR) uncertainty analysis in modeling (UAM) for the examination of uncertainty quantification and propagation methodologies with various modeling and simulation code systems. The objective of the present work is to develop an uncertainty propagation mechanism based on the stochastic sampling method by taking into account the uncertainties of both basic nuclear data and fuel modeling parameters in the simulation of pressurized water reactors (PWRs) that can be incorporated in the conventional LWR simulation approach. More specifically, the Three Mile Island Unit 1–related exercises from the LWR-UAM benchmark have been modeled using the coupled TRACE/PARCS code system in the three-dimensional core representation. The input uncertainties of the neutronics simulation include few-group cross sections and kinetics parameters generated using the Sampler/Polaris sequence of SCALE 6.2.1. Several heat transfer–related variables for the fuel modeling were considered as sources of input uncertainty of the thermal-hydraulics simulations, including the thermal conductivity of fuel and cladding, fuel heat capacity, and the gap conductance. Dakota was used to sample input parameters of the coupled code system and to perform the uncertainty analysis. Two types of simulations were conducted: steady-state calculation at hot full-power condition and transient scenario initiated by the spatially asymmetric rod ejection accident. Quantities of interest for the steady-state calculation, including core multiplication factor and power peaking factors, were calculated with associated uncertainties. For transient calculations, best-estimate plus uncertainty results of the time evolution of core reactivity, core power, and peak fuel temperature were generated and analyzed. The Wilks’ formula was used to determine the necessary sample size to achieve a 95% confidence of the 95% limit of output quantities of interest. Although the uncertainty propagation and quantification method presented in this paper was developed for PWRs, it could be in general applicable to the multiphysics uncertainty quantification of other types of LWR cores.

In the d-dimensional Euclidean space, any set of n + 1 independent random points, uniformly distributed in the interior of a unit ball of center O, determines almost surely a circumsphere of center C and radius Ω (1 ≤ n ≤ d) and an n-flat (1 ≤ n ≤ d − 1). The orthogonal projection of O onto this flat is called O′ while Δ designates the distance O′C. The classical problem of the distance between two random points in a unit ball corresponds to n = 1. The focus is set on the family of circumspheres which are contained in this unit ball. For any d≥2 and 1≤n≤d−1, the joint probability density function of the distance Δ≡O′C and circumradius Ω has a simple closed-form expression. The marginal probability density functions of Δ and Ω are both products of powers and a Gauss hypergeometric function. Stochastic representations of the latter random variables are described in terms of geometric means of two independent beta random variables. For n=d≥1, Δ and Ω have a joint Dirichlet distribution with parameters (d, d2, 1) while Δ and Ω are beta distributed. Results of Monte Carlo simulations are in very good agreement with their calculated counterparts. The tail behavior of the circumradius probability density function has been studied by Monte Carlo simulations for 2≤n=d≤9, where all circumspheres are this time considered, regardless of whether or not they are entirely contained in the unit ball.

In this paper, we evaluate the root mean square error (RMSE) performance of MUSIC (Multiple Signal Classification) and COMET (COvariance Matching EsTimator) algorithms in the presence of dispersion in azimuth and elevation. We use a 3D hemishperoid scattering model to describe angle of arrival in azimuth and elevation. We derive a closed- form expression for the joint and marginal probability density function (PDF) of the uplink angle of arrival (AOA) in azimuth and elevation. The base station is elevated to support the macrocellular model where the scatterers are distributed about the mobile (MS). We also compare the PDF of AOA and the simulated histogram.

This paper is concerned with the development and analysis of a nonlinear predictor for a non-Gaussian process using a zero-memory nonlinearity (ZMNL) followed by a second order Volterra filter (SVF). The processor exploits partial statistical information such as marginal probability density function (PDF) and the covariance structure. The ZMNL transforms the process into a Gaussian process. The SVF can be implemented as a parallel combination of linear and quadratic filters. Another equivalent structure is to carry out a linear prediction in the transformed Gaussian domain and pass the linear predicted samples through a second order polynomial.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Random field theory is an effective tool for modelling spatial or temporal variability and uncertainty in natural phenomena, and it has been widely applied in many areas such as structural dynamics, geology, hydrology, and meteorology. Conventional methods of random field simulations are often parametric, in which explicit function forms for trend function, auto-correlation function, and marginal probability density function (PDF) should be selected, together with their corresponding parameters estimated from measurements. Complete random field measurements with many data points are indispensable for selection of the appropriate function forms and accurate estimates of their corresponding parameters. However, the measurements in practice are often incomplete. Without sufficient measurement data, the random field samples (RFSs) generated by conventional methods might contain unexpected uncertainty and might be misleading. To tackle this problem, a purely non-parametric random field simulation method is developed in this study that generates RFSs directly from incomplete measurement data using generative adversarial networks (GAN). Statistical analysis is performed to estimate statistical properties from the generated RFSs, including mean, standard deviation (SD), autocovariance function (AF), and cumulative density function (CDF). The results show that the proposed method properly simulates RFSs from incomplete measurement data in a purely data-driven manner.

Angle and time of arrival characteristics of 3D air-to-ground radio propagation environments

In this paper, a propagation model is presented for the land mobile radio cellular systems with 3-D Gaussian distributed scatterers around Mobile Station (MS). The Base Station (BS) equipped with a directional antenna is located outside of the scattering region. The effect of directional antenna is thoroughly observed on spatial and temporal characteristics of the proposed model. General expressions for joint and marginal probability density functions (PDFs) of Angle-of-Arrival (AoA) observed at MS and BS in correspondence with azimuth and elevation angles are derived. Novel expressions for the trivariate and the bivariate density functions for Time-of-Arrival (ToA) / AoA in correspondence with azimuth and elevation angles are derived. Exploiting the proposed model, the theoretical results along with observations are presented to illustrate the effect of directional antenna on the spatio-temporal characteristics of channel. Obtained theoretical results for spatial statistics at BS are compared with an empirical set of measured data, which also demonstrates the validity of proposed model.

Summary A sparse grid approach to orbit uncertainty propagation is presented. Efficient and accurate uncertainty propagation methods for nonlinear dynamic systems have been of enormous interest to space object tracking. Recent methods include those based on the time evolution of the probability density function, the statistical moments, the random samples, or a sum of Gaussian components. The idea of the sparse grid method for orbit uncertainty propagation is to represent the initial uncertainty by a sparse grid, propagate the sparse grid points individually through the nonlinear orbit dynamics, and compute the statistical moments from the propagated sparse grid points. The Smolyak rule is used to generate the sparse grid through linear combinations of low-level tensor products; the number of the resultant sparse grid points is polynomial in the dimension of the state. The sparse grid based uncertainty propagation may be considered a generalization of the Unscented Transformation (UT) of the well-known Unscented Kalman Filter but can achieve much higher accuracy levels than UT. Two examples of orbit propagation in different dynamic environments are used to show the efficacy of the sparse grid approach.

Uncertainty propagation (UP) of the frequency response is essential for the robust design of viscoelastic damping structures. One challenge in solving this problem is enormous computation cost from the UP analysis. This paper aims to develop a novel computational effective method based on the combination of the adaptive sparse grid collocation (ASGC) method and the high dimensional model representation (HDMR) technique to evaluate the variability of the frequency response of viscoelastic damping structures. First, a well-validated layer-wise finite element method is employed to model the viscoelastic damping structures. The direct frequency response (DFR) method is utilized to calculate the response. Then, a modified adaptive strategy using expectation increments as the sampling indicator is proposed for cost-effective sparse grid construction. Lastly, high dimensional model representation (HDMR) technique is introduced to address the difficulty in moderate and high random dimensional situations. Two numerical examples are provided to assess the performances of the proposed method. Variations in constitutive parameters of viscoelastic material and thicknesses of the viscoelastic layer are considered. Numerical results show that, compared to the original high-dimensional ASGC method and Monte Carlo Simulation (MCS), the proposed method accurately predicts the variability of frequency responses and significantly improves the computational efficiency.

Presented are novel theoretical probability density functions (PDF) for the magnitude and phase of electromagnetic fields inside reverberation chambers (RC) operating in a ‘good-but-imperfect regime’. The derivation is based on considering two Gaussian random variables with mean values, variances and correlation between them that depart from the ideal assumptions. A multivariate approach using a complete joint Gaussian distribution of these variables is defined. Marginal distributions obtained by integration of this two-dimensional joint PDF are compared with theoretical PDFs for ideal situations, and significantly lower rejection rates are experienced for field data measured in real RCs. Additionally, these novel marginal PDFs are highly general since they are able to describe both ideal and non-ideal stirred fields.