Number Of Random Variables Research Articles

Fast Variability Analysis of General Nonlinear Circuits Using Decoupled Polynomial Chaos

This paper presents a new approach aimed at limiting the growth of the computational cost of variability analysis, of nonlinear circuits, using the Hermite-based polynomial chaos (PC), with the increase in the number of random variables. The proposed technique is based on deriving a closed-form formula for the structure of the augmented Jacobian matrix generated by the PC approach, and then showing that this structure can be approximated with a different structure that can be decoupled into independent diagonal blocks.

Overlapping Decomposition for Gaussian Graphical Modeling

Correlation based graphical models are developed to detect the dependence relationships among random variables and provide intuitive explanations for these relationships in complex systems. Most of the existing works focus on learning a single correlation based graphical model for all the random variables. However, it is difficult to understand and interpret the massive dependencies of the variables learned from a single graphical model at a global level especially when the graph is large. In order to provide a clearer understanding for the dependence relationships among a large number of random variables, in this paper, we propose the problem of estimating an overlapping decomposition for the Gaussian graphical model of a large scale to generate overlapping sub-graphical models, where strong and meaningful correlations remain in each subgraph with a small scale. Specifically, we propose a greedy algorithm to achieve the overlapping decomposition for the Gaussian graphical model. A key technique of the algorithm is that the problem of solving a $(k+1)$ -node Gaussian graphical model can be approximately reduced to the problem of solving a one-step vector regularization problem based on a solved $k$ -node Gaussian graphical model with theoretical guarantee. Based on this technique, a greedy expansion algorithm is proposed to generate the overlapping subgraphs. Moreover, we extend the proposed method to deal with dynamic graphs where the dependence relationships among random variables vary with the time. We evaluate the proposed methods on synthetic dataset and a real-life traffic dataset, and the experimental results show the superiority of the proposed methods.

A hybrid method for stochastic response analysis of a vibrating structure

Response analysis of a linear structure with uncertainties in both structural parameters and external excitation is considered here. When such an analysis is carried out using the spectral stochastic finite element method (SSFEM), often the computational cost tends to be prohibitive due to the rapid growth of the number of spectral bases with the number of random variables and the order of expansion. For instance, if the excitation contains a random frequency, or if it is a general random process, then a good approximation of these excitations using polynomial chaos expansion (PCE) involves a large number of terms, which leads to very high cost. To address this issue of high computational cost, a hybrid method is proposed in this work. In this method, first the random eigenvalue problem is solved using the weak formulation of SSFEM, which involves solving a system of deterministic nonlinear algebraic equations to estimate the PCE coefficients of the random eigenvalues and eigenvectors. Then the response is estimated using a Monte Carlo (MC) simulation, where the modal bases are sampled from the PCE of the random eigenvectors estimated in the previous step, followed by a numerical time integration. It is observed through numerical studies that this proposed method successfully reduces the computational burden compared with either a pure SSFEM of a pure MC simulation and more accurate than a perturbation method. The computational gain improves as the problem size in terms of degrees of freedom grows. It also improves as the timespan of interest reduces.

Extreme negative dependence and risk aggregation

We introduce the concept of an extremely negatively dependent (END) sequence of random variables with a given common marginal distribution. An END sequence has a partial sum which, subtracted by its mean, does not diverge as the number of random variables goes to infinity. We show that an END sequence always exists for any given marginal distributions with a finite mean and we provide a probabilistic construction. Through such a construction, the partial sum of identically distributed but dependent random variables is controlled by a random variable that depends only on the marginal distribution of the sequence. We provide some properties and examples of our construction. The new concept and derived results are used to obtain asymptotic bounds for risk aggregation with dependence uncertainty.

Decoupled Polynomial Chaos and Its Applications to Statistical Analysis of High-Speed Interconnects

This paper presents a new Hermite-based approach to circuit variability analysis using the polynomial-chaos (PC) paradigm. The new approach is aimed at limiting the growth of the computational cost with the increase in the number of random variables and the number of Hermite coefficients used to represent the circuit response in each random variable. The proposed method is based on deriving a closed form for the structure of the augmented matrices generated by the PC approach. It then develops an algorithm to decouple the large augmented matrices into independent matrices that can be factorized in parallel. Several numerical examples are presented to demonstrate the computational efficiency of the proposed method and its ability to handle the increasing dimension of random space.

Bayesian Updating with Structural Reliability Methods

Bayesian updating is a powerful method to learn and calibrate models with data and observations. Because of the difficulties involved in computing the high-dimensional integrals necessary for Bayesian updating, Markov chain Monte Carlo (MCMC) sampling methods have been developed and successfully applied for this task. The disadvantage of MCMC methods is the difficulty of ensuring the stationarity of the Markov chain. We present an alternative to MCMC that is particularly effective for updating mechanical and other computational models, termed Bayesian updating with structural reliability methods (BUS). With BUS, structural reliability methods are applied to compute the posterior distribution of uncertain model parameters and model outputs in general. An algorithm for the implementation of BUS is proposed, which can be interpreted as an enhancement of the classic rejection sampling algorithm for Bayesian updating. This algorithm is based on the subset simulation, and its efficiency is not dependent on the number of random variables in the model. The method is demonstrated by application to parameter identification in a dynamic system, Bayesian updating of the material parameters of a structural system, and Bayesian updating of a random field–based finite-element model of a geotechnical site.

Reliability Analysis of Automobile Engine Connecting Rod Using Fourier Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical procedure, such as finite element method (FEM). To reduce the computational effort required for reliability analysis, response surface method could be used. However the conventional response surface method is still time consuming especially when the number of random variables is large. In this paper, a Fourier orthogonal neural network (FONN)-based response surface method is adopted to solve the reliability analysis of the automobile engine. The working process of the connecting rod is simulated with UG software, the dynamics analysis on crank-connecting rod-piston mechanism is performed with ANSYS and ADAMS software, with FEM analysis results, the stress information of the critical point of the structure can be obtained, so the performance function of the structure can be established. The FONN method is used to fit the performance function as well as its derivatives, so as to calculate the reliability of the structure.

Sparse Grid Collocation Method for an Optimal Control Problem Involving a Stochastic Partial Differential Equation with Random Inputs

AbstractIn this article, we propose and analyse a sparse grid collocation method to solve an optimal control problem involving an elliptic partial differential equation with random coefficients and forcing terms. The input data are assumed to be dependent on a finite number of random variables. We prove that an optimal solution exists, and derive an optimality system. A Galerkin approximation in physical space and a sparse grid collocation in the probability space is used. Error estimates for a fully discrete solution using an appropriate norm are provided, and we analyse the computational efficiency. Computational evidence complements the present theory, to show the effectiveness of our stochastic collocation method.

Structural Reliability Analysis Using Legendre Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be estimated by a numerical analysis such as finite element method. The response surface method could be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However the conventional response surface method is time-consuming or cumbersome if the number of random variables is large. This paper presents a Legendre orthogonal neural network (LONN)-based response surface method to predict the reliability of a structure. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability method, i.e. first-order reliability methods (FORM) to predict the failure probability of the structure. Numerical example has shown that the proposed approach is applicable to structural reliability analysis involving implicit performance functions.

Monte Carlo Finite Volume Element Methods for the Convection-Diffusion Equation with a Random Diffusion Coefficient

The paper presents a framework for the construction of Monte Carlo finite volume element method (MCFVEM) for the convection-diffusion equation with a random diffusion coefficient, which is described as a random field. We first approximate the continuous stochastic field by a finite number of random variables via the Karhunen-Loève expansion and transform the initial stochastic problem into a deterministic one with a parameter in high dimensions. Then we generate independent identically distributed approximations of the solution by sampling the coefficient of the equation and employing finite volume element variational formulation. Finally the Monte Carlo (MC) method is used to compute corresponding sample averages. Statistic error is estimated analytically and experimentally. A quasi-Monte Carlo (QMC) technique with Sobol sequences is also used to accelerate convergence, and experiments indicate that it can improve the efficiency of the Monte Carlo method.

FINDING THE BEST-FIT POLYNOMIAL APPROXIMATION IN EVALUATING DRILL DATA: THE APPLICATION OF A GENERALIZED INVERSE MATRIX / POSZUKIWANIE NAJLEPSZEJ ZGODNOŚCI W PRZYBLIŻENIU WIELOMIANOWYM WYKORZYSTANEJ DO OCENY DANYCH Z ODWIERTÓW – ZASTOSOWANIE UOGÓLNIONEJ MACIERZY ODWROTNEJ

Abstract In mining, various estimation models are used to accurately assess the size and the grade distribution of an ore body. The estimation of the positional properties of unknown regions using random samples with known positional properties was first performed using polynomial approximations. Although the emergence of computer technologies and statistical evaluation of random variables after the 1950s rendered the polynomial approximations less important, theoretically the best surface passing through the random variables can be expressed as a polynomial approximation. In geoscience studies, in which the number of random variables is high, reliable solutions can be obtained only with high-order polynomials. Finding the coefficients of these types of high-order polynomials can be computationally intensive. In this study, the solution coefficients of high-order polynomials were calculated using a generalized inverse matrix method. A computer algorithm was developed to calculate the polynomial degree giving the best regression between the values obtained for solutions of different polynomial degrees and random observational data with known values, and this solution was tested with data derived from a practical application. In this application, the calorie values for data from 83 drilling points in a coal site located in southwestern Turkey were used, and the results are discussed in the context of this study.

Structural Reliability Analysis Using Fourier Orthogonal Neural Network Method

In order to predict the failure probability of a complicated structure, the structural responses usually need to be predicted by a numerical procedure, such as FEM method. The response surface method could be used to reduce the computational effort required for reliability analysis. However the conventional response surface method is still time consuming when the number of random variables is large. In this paper, a Fourier orthogonal neural network (FONN)-based response surface method is proposed. In this method, the relationship between the random variables and structural responses is established using FONN models. Then the FONN model is connected to the first order and second moment method (FORM) to predict the failure probability. Numerical example result shows that the proposed approach is efficient and accurate, and is applicable to structural reliability analysis.

Simulation of the Stochastic Wave Loads Using a Physical Modeling Approach

In analyzing stochastic dynamic systems, analysis of the system uncertainty due to randomness in the loads plays a crucial role. Typically time series of the stochastic loads are simulated using traditional random phase method. This approach combined with fast Fourier transform algorithm makes an efficient way of simulating realizations of the stochastic load processes. However it requires many random variables, i.e. in the order of magnitude of 1000, to be included in the load model. Unfortunately having too many random variables in the problem makes considerable difficulties in analyzing system reliability or its uncertainty. Moreover applicability of the probability density evolution method on engineering problems faces critical difficulties when the system embeds too many random variables. Hence it is useful to devise a method which can make realization of the stochastic load processes with low, say less than 20, number of random variables. In this article we introduce an approach, so-called "physical modeling of stochastic processes", and show its applicability for simulation of the wave surface elevation.

Risk assessment of water quality using Monte Carlo simulation and artificial neural network method

There is always uncertainty in any water quality risk assessment. A Monte Carlo simulation (MCS) is regarded as a flexible, efficient method for characterizing such uncertainties. However, the required computational effort for MCS-based risk assessment is great, particularly when the number of random variables is large and the complicated water quality models have to be calculated by a computationally expensive numerical method, such as the finite element method (FEM). To address this issue, this paper presents an improved method that incorporates an artificial neural network (ANN) into the MCS to enhance the computational efficiency of conventional risk assessment. The conventional risk assessment uses the FEM to create multiple water quality models, which can be time consuming or cumbersome. In this paper, an ANN model was used as a substitute for the iterative FEM runs, and thus, the number of water quality models that must be calculated can be dramatically reduced. A case study on the chemical oxygen demand (COD) pollution risks in the Lanzhou section of the Yellow River in China was taken as a reference. Compared with the conventional risk assessment method, the ANN-MCS-based method can save much computational effort without a loss of accuracy. The results show that the proposed method in this paper is more applicable to assess water quality risks. Because the characteristics of this ANN-MCS-based technique are quite general, it is hoped that the technique can also be applied to other MCS-based uncertainty analysis in the environmental field.

Simple measure of similarity for the market graph construction

A simple measure of similarity for the construction of the market graph is proposed. The measure is based on the probability of the coincidence of the signs of the stock returns. This measure is robust, has a simple interpretation, is easy to calculate and can be used as measure of similarity between any number of random variables. For the case of pairwise similarity the connection of this measure with the sign correlation of Fechner is noted. The properties of the proposed measure of pairwise similarity in comparison with the classic Pearson correlation are studied. The simple measure of pairwise similarity is applied (in parallel with the classic correlation) for the study of Russian and Swedish market graphs. The new measure of similarity for more than two random variables is introduced and applied to the additional deeper analysis of Russian and Swedish markets. Some interesting phenomena for the cliques and independent sets of the obtained market graphs are observed.

Solving Continuous Models with Dependent Uncertainty: A Computational Approach

This paper presents a computational study on a quasi-Galerkin projection-based method to deal with a class of systems of random ordinary differential equations (r.o.d.e.’s) which is assumed to depend on a finite number of random variables (r.v.’s). This class of systems of r.o.d.e.’s appears in different areas, particularly in epidemiology modelling. In contrast with the other available Galerkin-based techniques, such as the generalized Polynomial Chaos, the proposed method expands the solution directly in terms of the random inputs rather than auxiliary r.v.’s. Theoretically, Galerkin projection-based methods take advantage of orthogonality with the aim of simplifying the involved computations when solving r.o.d.e.’s, which means to compute both the solution and its main statistical functions such as the expectation and the standard deviation. This approach requires the previous determination of an orthonormal basis which, in practice, could become computationally burden and, as a consequence, could ruin the method. Motivated by this fact, we present a technique to deal with r.o.d.e.’s that avoids constructing an orthogonal basis and keeps computationally competitive even assuming statistical dependence among the random input parameters. Through a wide range of examples, including a classical epidemiologic model, we show the ability of the method to solve r.o.d.e.’s.

Approximation of the stochastic Galerkin matrix in the low‐rank canonical tensor format

AbstractIn this article we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log‐normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient low-rank approximation of the stochastic Galerkin matrix in tensor formats

In this article, we describe an efficient approximation of the stochastic Galerkin matrix which stems from a stationary diffusion equation. The uncertain permeability coefficient is assumed to be a log-normal random field with given covariance and mean functions. The approximation is done in the canonical tensor format and then compared numerically with the tensor train and hierarchical tensor formats. It will be shown that under additional assumptions the approximation error depends only on the smoothness of the covariance function and does not depend either on the number of random variables nor the degree of the multivariate Hermite polynomials.

Buckling Loads of Two-Layer Composite Columns with Interlayer Slip and Stochastic Material Properties

This paper presents an efficient stochastic buckling model for studying the structural reliability of layered composite columns with interlayer slip between the layers and random material and loading parameters. The model is based on the exact buckling model, response surface method, and Monte Carlo simulations. The probability of failure is investigated for a different number of random variables, sample points, and various degrees of response surfaces. The results show that the probability of failure is considerably affected by the type (deterministic or probabilistic) of the loading and its distribution.

Reliability Analysis of Flexural Concrete Member Based on the Randomness of Pitting

In order to analyze the influence of stochastic pitting corrosion to the reliability of service structure, According to the theory of extreme values of random variables and the basic theory of reliability analysis, the reliability of flexural concrete member based on the randomness of pitting, the change of statistical parameters for both corroded steel bar and resistance is also analyzed based on the randomness of pitting. Based on the checking point method, the development law of reliability of concrete bending members with the number of random variables and the service life are studied. The results indicated that the influence of the randomness of pitting to the reliability of flexural concrete members is of the most significance, the reliability index was found to decline with the increase of number of random variables and the service life. The analysis show the research presented in the paper can provide theoretical basis of life prediction.