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Abstract Recently, the concept of complex fractional moments has been introduced as an extension to ordinary moments. It has been shown that complex fractional moments never diverge and they are able to represent both probability density function (PDF) and characteristic function (CF). In this paper, we develop an improved complex fractional moments-based approach to recover the PDF and the CF, in which the PDF is re-expressed in two parts such that the complex fractional moment coincide with Mellin transform in each part. The proposed PDF decomposition scheme is a most natural way to connect complex fractional moments and Mellin transform while it represent the PDF (or CF) in a more straightforward and more distinct way. By virtue of inverse Mellin transform theorem, the PDF (or CF) can be exactly recovered in terms of complex fractional moments. In addition, the developed approach is further extended to estimate the PDF (or CF) from a finite number of samples of variables. Three illustrative distributions are used to demonstrate the proposed complex fractional moment-based approach. [ABSTRACT FROM AUTHOR]

• A unified framework for simulating cross-correlated random fields with arbitrary correlation structure is established. • The developed method is conceptually simple and computationally efficient. • The method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. • Five examples illustrate the effectiveness of the developed method. Cross-correlated random fields are widely used to model multiple uncertain parameters and/or phenomena with inherent spatial/temporal variability in numerous engineering systems. The effective representation of such fields is therefore the key element in the stochastic simulation, reliability analysis and safety assessment of engineering problems with mutual correlations. However, the simulation of such fields is generally not straightforward given the complexity of correlation structure. In this paper, we develop a unified framework for simulating non-Gaussian and non-stationary cross-correlated random fields that have been specified by their correlation structure and marginal cumulative distribution functions. Our method firstly represents the cross-correlated random fields by means of a new general stochastic expansion, in which the fields are expanded in terms of a set of deterministic functions with corresponding random variables. A finite element discretization scheme is then developed to further approximate the fields, so that the sets of deterministic functions reflecting the cross-covariance structure can be straightforwardly determined from the spectral decomposition of the resulting discretized fields. For non-Gaussian random fields, an iterative mapping procedure is developed to generate random variables to fit non-Gaussian marginal distribution of the fields. By virtue of the remarkable property of the presented stochastic expansion, i.e., various random fields share an identical set of random variables, the framework we develop is conceptually simple for simulating non-Gaussian cross-correlated fields with arbitrary covariance functions, which need not be stationary. In particular, the developed method is further generalized to a consistent framework for the simulation of multi-dimensional random fields. Five illustrative examples, including a spatially varying non-Gaussian and nonstationary seismic ground motions, are used to demonstrate the application of the developed method. [ABSTRACT FROM AUTHOR]

• A unified non-Gaussian stochastic model from limited observations is developed. • The developed model is based on polynomial chaos (PC) and fractional moments. • The developed model can be embedded into PC framework for stochastic propagation. • Two numerical examples demonstrate the developed model. The reasonable representation of input random fields is the key element in the reliability analysis of practical engineering systems. In most engineering applications, the characterization of a random field often relies on limited measurements. Although the simulation of random fields with complete probabilistic information has been quite well-established, reconstructing a random field from limited observations is still a challenging task. In this paper, we develop a methodology for constructing non-Gaussian random model from limited observations based on polynomial chaos (PC) and fractional moments for real-life problems. Our method begins with the reduce-order representation of measurements by Karhunen-Loève (KL) expansion, followed by the PC representation of KL coefficients. The PC coefficients are further modeled as random variables, whose distributions are determined by a modified maximum entropy principle with fractional moments (ME-FM) procedure and a ME-FM-based bootstrapping. In this way, the developed non-Gaussian model enables to quantify the inherent randomness and the statistical uncertainty of the observed non-Gaussian field simultaneously. Since the developed non-Gaussian model is embedded into the well-established PC framework, our method facilitates the implementation of PC-based stochastic analysis in practical engineering applications, in which only limited probabilistic measures are available. Two numerical examples demonstrate the application of the developed method. [ABSTRACT FROM AUTHOR]

Abstract: In practice, reliability analysis of structures is often performed on the basis of limited data. Under this circumstance, there are practical difficulties in identifying unique distributions as input for a probabilistic analysis. But the selection of realistic probabilistic input is critical for the quality of the results of the reliability analysis. This problem can be addressed using an entire set of plausible distribution functions rather than one single distribution for random variables based on limited data. The uncertain nature of the available information is then reflected in the probabilistic input. An imprecise probability distribution can be modeled by a probability box, i.e., the bounds of the cumulative distribution function for the random variable. Sampling-based methods have been proposed to perform reliability analysis with probability boxes. However, direct sampling of probability boxes requires a large number of samples. The computational cost can be very high as each simulation involves an interval analysis (a range-finding problem). This study proposes an interval quasi-Monte Carlo simulation methodology to efficiently compute the bounds of structure failure probabilities. The methodology is based on deterministic low-discrepancy sequences, which are distributed more regularly than the (pseudo) random points in direct Monte Carlo simulation. The efficiency and accuracy of the present method is illustrated using two examples. The reliability implications of different approaches for construction of probability boxes are also investigated through the example. [Copyright &y& Elsevier]

This paper presents a new stochastic finite element method for computing structural stochastic responses of linear problems. The method provides a new expansion of stochastic response and decouples the stochastic response into a combination of a series of deterministic responses with random variable coefficients. A dedicated iterative algorithm is proposed to compute the deterministic responses and corresponding random variable coefficients one by one. The algorithm computes the deterministic responses and corresponding random variable coefficients in their individual spaces and is insensitive to stochastic dimensions, thus it can be applied to high-dimensional stochastic problems without extra difficulties. More importantly, the deterministic responses can be computed efficiently by use of existing finite element method (FEM) solvers, thus the proposed method can be easy to embed into existing FEM structural analysis software. Three practical examples, including low-dimensional and high-dimensional stochastic problems, are given to demonstrate the accuracy and effectiveness of the proposed method. • This paper develops a new method for solving stochastic finite element equations. • Computational costs are low for large-scale and high-dimensional stochastic problems. • The Curse of Dimensionality of high dimensions can be avoided with great success. [ABSTRACT FROM AUTHOR]

This paper presents a weak-intrusive stochastic finite element method for solving stochastic structural dynamics equations. In this method, the stochastic solution is decomposed into the summation of a series of products of random variables, spatial vectors and temporal functions. An iterative algorithm is proposed to compute each triplet of the random variable, spatial vector and temporal function one by one. The original stochastic dynamics problem is firstly transformed into spatial–temporal coupled problems (i.e. deterministic structural dynamics equations), which can be solved efficiently by existing FEM solvers. Based on the solution of the spatial–temporal coupled problem, the original problem is then transformed into stochastic-temporal coupled problems (i.e. one-dimensional second-order stochastic ordinary differential equations), which are solved by a proposed sampling method. All random sources are embedded into the stochastic-temporal coupled problems. The proposed sampling method can solve the stochastic-temporal problems of hundreds of dimensions with low computational costs. Thus the curse of dimensionality in high-dimensional stochastic spaces is avoided with great success. Three numerical examples, including low- and high-dimensional stochastic problems, are used to demonstrate the good accuracy and the high efficiency of the proposed method. • A universal decomposition is proposed to approximate the stochastic solutions of stochastic dynamics equations. • An iterative algorithm is developed to solve stochastic dynamics equations. • The proposed method combines the advantages of intrusive and nonintrusive methods. • The curse of dimensionality of high-dimensional stochastic problems is avoided with great success. [ABSTRACT FROM AUTHOR]