A probability density function discretization and approximation method for the dynamic load identification of stochastic structures

Abstract. It is generally known that the tropical cyclone (TC) cloud systems (TCCSs) in the North Atlantic region frequently occur during boreal summer, while the Saharan dust outbreaks occur concurrently. The Sahara air layer (SAL), an elevated layer containing Saharan dry air and mineral dust, has crucial impacts on the generation and evolution of TCs. However, the effects of SAL on the physical (macro and micro) characteristics of the Atlantic TCCSs have not been well constrained, and the interaction mechanisms between them still need further investigation. In this study, our primary interest is to distinguish the various effects of SAL on different intensities of TCs and further find out the probable causes of the varied feedback mechanisms. Therefore, we attempt to identify whether and how the effects of the SAL play a positive or negative role on the TCCSs and to draw a qualitative conclusion on how SAL affects the various intensities of the TCs. This paper focuses on the 70 TC samples from July to September in the years of 2003–2019 to investigate the physical effects of SAL on three intensities of TCs, i.e., the tropical depression (TD), tropical storm (TS), and hurricane (HU). The results show that SAL has a positive impact on the macro properties of HU but significantly suppresses the TD. It appears that the SAL attributes little to the variation of the ice cloud effective radius (CERi) for TS, whereas CERi changes significantly and differentially for TD and HU. When affected by SAL, the probability density function (PDF) curve of CERi generally shifts to the smaller value for TD, whereas the PDF curve becomes flatter for HU. Our analysis indicates that the various responses of TCCSs to SAL are determined by the combined effects of dry air masses, the dust aerosols as ice nuclei, and the thermodynamic and moisture conditions. Based on the observation data analysis, a concept scheme description has been concluded to deepen our recognition of the effects of SAL on the TCCSs.

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

This paper presents a controller design for the shape control of output probability density functions for non-Gaussian dynamic stochastic systems represented by an ARMAX model. The coefficients in the ARMAX model are random and are represented by their known probability density functions and the system is also subjected to a random noise. All these random parameters and noise are assumed bounded and non-Gaussian. To formulate a simple relationship among the probability density function of the system output and those of random parameters and the noise term, the Laplace transform is applied to all the probability density functions. As a result, a simple mathematical relationship amongst all the transferred probability density functions of the system output and random parameters has been established. Since controlling the shape of the output probability density function is equivalent to controlling the shape of its Laplace transform, a new performance function is introduced, whose minimization is performed so as to design an optimal control input sequence that makes the shape of the output probability density function follow a target distribution. A gradient search technique is applied in the optimization.

The class of location-scale finite mixtures is of enduring interest both from applied and theoretical perspectives of probability and statistics. We establish and prove the following results: to an arbitrary degree of accuracy, (a) location-scale mixtures of a continuous probability density function (PDF) can approximate any continuous PDF, uniformly, on a compact set; and (b) for any finite location-scale mixtures of an essentially bounded PDF can approximate any PDF in in the norm.

Two issues in using mixtures of polynomials for inference in hybrid Bayesian networks

The determination of structural dynamic characteristics can be challenging, especially for complex cases. This can be a major impediment for dynamic load identification in many engineering applications. Hence, avoiding the need to find numerous solutions for structural dynamic characteristics can significantly simplify dynamic load identification. To achieve this, we rely on machine learning. The recent developments in machine learning have fundamentally changed the way we approach problems in numerous fields. Machine learning models can be more easily established to solve inverse problems compared to standard approaches. Here, we propose a novel method for dynamic load identification, exploiting deep learning. The proposed algorithm is a time-domain solution for beam structures based on the recurrent neural network theory and the long short-term memory. A deep learning model, which contains one bidirectional long short-term memory layer, one long short-term memory layer and two full connection layers, is constructed to identify the typical dynamic loads of a simply supported beam. The dynamic inverse model based on the proposed algorithm is then used to identify a sinusoidal, an impulsive and a random excitation. The accuracy, the robustness and the adaptability of the model are analyzed. Moreover, the effects of different architectures and hyperparameters on the identification results are evaluated. We show that the model can identify multi-points excitations well. Ultimately, the impact of the number and the position of the measuring points is discussed, and it is confirmed that the identification errors are not sensitive to the layout of the measuring points. All the presented results indicate the advantages of the proposed method, which can be beneficial for many applications.

An approximation of small-time probability density functions in a general jump diffusion model

A function approximation based on an orthonormal wave function expansion in a complex space is derived. Although a probability density function (PDF) cannot always be expanded in an orthogonal series in a real space because a PDF is a positive real function, the function approximation can approximate an arbitrary PDF with high accuracy. It is applied to an actor-critic method of reinforcement learning to derive an optimal policy expressed by an arbitrary PDF in a continuous-action continuous-state environment. A chaos control problem and a PDF approximation problem are solved using the actor-critic method with the function approximation, and it is shown that the function approximation can approximate a PDF well and that the actor-critic method with the function approximation exhibits high performance.

In recent decades, analysis of structures considering variability of some parameters for more reliable design has attracted the attention of researchers. In this paper, the stochastic analysis of a cantilever deep beam made of large deformable neo-hookean material is carried out. For this purpose, the meshless local Petrov-Galerkin (MLPG) method is developed to obtain the geometrically non-linear equilibrium equations. The radial point interpolation method is used for generating the shape functions. The incremental iterative Newton-Raphson method with suitable load steps is used to solve the non-linear governing equations. The results of deterministic analysis obtained with proposed method are compared with the finite element results and good agreement is achieved. The initial elasticity modulus of neo-hookean material is considered to be uncertain variable. To generate random field of uncertain variable with normal, lognormal and uniform probability density functions (PDFs), the Monte Carlo Simulation (MCS) technique was employed. The sufficient number of simulations for convergence the results was determined experimentally. The effect of elasticity modulus, PDF and coefficients of variation (COV) on maximum vertical displacement, PDF and COV of results are studied in details. Comparing the stochastic and deterministic results shows that the uncertainty in mechanical properties has significant effect on results.

Uncertainty assessment of flow and contaminant transport in the vadose zone entails probability density functions (PDFs) of soil hydraulic parameters. An unconventional maximum likelihood (ML) approach was used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten α and m) for a situation common in field‐scale modeling where core samples are sparse and prior PDFs of the parameters are unknown. In this situation, the unconventional ML approach approximates the PDFs as multivariate Gaussian. This study developed a method of estimating the mean and covariance of the multivariate Gaussian PDF based on the results of least square methods that can be easily obtained in practice. The developed method was applied to and evaluated through numerical simulation of unsaturated flow and tracer transport at the proposed Yucca Mountain geologic repository. Another focus of this study was to investigate the effect of uncertainty in the water retention parameters on predictive uncertainty. By comparing the predictive uncertainty before and after incorporating random water retention parameters, it was found that the random water retention parameters had limited effects on the mean predictions of the state variables including percolation flux and tracer travel time from the potential repository to the water table. Incorporating the uncertainty in the water retention parameters, however, significantly increased the magnitude and spatial extent of predictive uncertainty of the state variables. In particular, incorporating the random water retention parameters significantly changed the 5th and 95th percentiles of the tracer travel time by tens of thousands of years.

Uncertainty, Sensitivity and Geostatistical Studies of Flow and Contaminant Transport in Heterogeneous Unsaturated Zone by Feng Pan Dr. Zhongbo Yu, Examination Committee Chair Associate Professor of Hydrology University of Nevada Las Vegas Dr. Jianting Zhu, Examination Committee Co-Chair Associate Professor of Hydrologic Science Desert Research Institute The objectives of this study are: (1) to develop a methodology of estimating probability density functions (PDFs) of unsaturated hydraulic parameters when field samples are sparse, (2) to evaluate the predictive uncertainties in flow and contaminant transport due to parameter uncertainties in the layerand local-scale heterogeneities of hydraulic parameters in unsaturated zone (UZ), (3) to investigate the contributions of the parameter uncertainties to the flow and transport uncertainties, and (4) to estimate the spatial correlation structures of hydraulic parameters by incorporating prior information and site measurements. At layer scale, the uncertainty assessment of flow and contaminant transport in UZ entails PDFs of the hydraulic parameters. A non-conventional maximum likelihood (ML) approach is used in this study to estimate the PDFs of water retention parameters (e.g., van Genuchten a and ri) for situations common in field scale applications where core samples are sparse and prior PDFs of the parameters are unknown. This study also investigates the effects of the uncertainties in the water retention parameters on the predictive uncertainties in flow and transport in UZ. By comparing the predictive uncertainties with and without incorporating the random water retention parameters, it is found that the random water retention parameters have limited effects on the mean predictions of the state variables including percolation flux, normalized cumulative mass arrival, and contaminant travel time. However, incorporating the uncertainties in the water retention parameters significantly increases the magnitude and spatial extent of predictive uncertainties of the state variables. The layer-scale uncertainty is specific to hydrogeologic layers, while the localscale heterogeneity refers to the spatial variation of hydraulic properties within a layer. The local-scale heterogeneity is important in predicting flow path, velocity, and travel time of contaminants, but it is often neglected in modeling practices. This study incorporates the local-scale heterogeneity and examines its relative effects to the layerscale uncertainty on flow and transport uncertainties in UZ. Results illustrate that localscale heterogeneity significantly increases predictive uncertainties in the percolation fluxes and contaminant plumes, whereas the mean predictions are only slightly affected by the local-scale heterogeneity. Layer-scale uncertainty is more important than localscale heterogeneity for simulating overall contaminant travel time, suggesting that it would be more cost-effective to reduce the layer-scale parameter uncertainty in order to reduce predictive uncertainty in contaminant transport.

Levy flights are generalised random walk processes where the independent stationary increments are drawn from a long-tailed α-stable jump length distribution. We consider the formulation of Levy flights, for 0 < α < 1, in terms of a space-fractional diffusion equation which fundamental solutions are the probability density functions. First, we present how to obtain the governing equation of Levy motion from the Fourier transform of the jump distribution. Then, we derive a family of implicit numerical methods to determine the numerical solutions and we study their consistency and stability. Although numerical algorithms for the case 1 < α < 2 have been widely discussed, very few works paid attention to the case we discuss here. We present numerical experiments to show the performance of the numerical methods and to highlight the advantages and disadvantages of the different approaches. In the end we determine the numerical solutions of an initial value problem, that considers an approximation of the Dirac delta function as the initial condition, in order to obtain approximations of the probability density functions.

In this paper, piecewise linear approximation of the probability density function is done. On the basis of the obtained approximated probability density function, a companding quantizer is designed. The signal at the entrance of the companding quantizer is modeled by Laplacian probability density function. Performances of the proposed companding quantizer are estimated on the basis of comparison of calculated values of signal to quantization noise ratio and approximation error, with the calculated values that correspond to the model of piecewise uniform scalar quantizer, that is also proposed in this paper.

In this paper, we consider automatic modulation classification (AMC) from received signals over flat fading channels. AMC is a technique to identify the modulation type of the transmitted symbols by using received symbols. First, we develop a method to determine the modulation type based on the Neyman-Pearson (NP) detector. Then, to reduce the numerical complexity, a method by using probability density function (PDF) approximation is derived. To certify our detector can be scarcely effected by the uncertainty in noise variance, we compare the case that the noise variance is available with the case that the noise variance is unavailable. Furthermore numerical simulations are conducted to assess the performance of our proposed methods even when only finite samples are limited.

A generalized method for the stationary probabilistic response of nonlinear dynamical system