Large Number Of Random Variables Research Articles

STOCHASTIC GALERKIN METHOD AND PORT-HAMILTONIAN FORM FOR LINEAR FIRST-ORDER ORDINARY DIFFERENTIAL EQUATIONS

We consider linear first-order systems of ordinary differential equations (ODEs) in port-Hamiltonian (pH) form. Physical parameters are remodeled as random variables to conduct an uncertainty quantification. A stochastic Galerkin projection yields a larger deterministic system of ODEs, which does not exhibit a pH form in general. We apply transformations of the original systems such that the stochastic Galerkin projection becomes structure-preserving. Furthermore, we investigate meaning and properties of the Hamiltonian function belonging to the stochastic Galerkin system. A large number of random variables implies a high-dimensional stochastic Galerkin system, which suggests itself to apply model order reduction (MOR) generating a low-dimensional system of ODEs. We discuss structure preservation in projection-based MOR, where the smaller systems of ODEs feature pH form again. Results of numerical computations are presented using two test examples.

Law of large numbers for weak Fubini-independent random variables under capacities without sub-additivity

This article attempts to establish the law of large numbers for random variables in Choquet expectation spaces where the Choquet integral is not sub-additive. As a basis, the Fubini-independence of random variables is generalized to a weaker version. Then the strong and weak law of large numbers are proved for weak-Fubini independent random variables under capacities which are not sub-additive.

A single-loop time-variant reliability evaluation via a decoupling strategy and probability distribution reconstruction

In this paper, a single-loop approach for time-variant reliability evaluation is proposed based on a decoupling strategy and probability distribution reconstruction. The most attractive feature of the proposed method is that the reliability at a specified time instant can be captured by performing time-invariant reliability analysis only once. In this method, the expansion optimal linear estimation is first employed to discretize the loading stochastic process. Then, a decoupling strategy that decouples the loading stochastic process and degradation processes is developed to formulate a single-loop method for time-variant reliability analysis, where an equivalent extreme value limit state function (EEV-LSF) is obtained. To improve the accuracy and robustness, the Box–Cox transformation is applied to get a transformed EEV-LSF. The maximum entropy method with fractional exponential moments is employed to robustly derive the probability distribution of transformed EEV-LSF. Once the probability distribution is captured, the time-variant failure probability can be readily computed. To handle a large number of random variables, a weighted sampling method is applied for moment assessment to ensure an efficient solution. Numerical examples including a complex real-world case are studied to validate the proposed method, where pertinent Monte Carlo simulations and PHI2 method are conducted for comparisons.

Laws of large numbers under model uncertainty with an application to m-dependent random variables

In this paper, we study weak laws of large numbers for random variables under model uncertainty. We introduce the concepts of conditional upper and lower expectations induced by a stable set of probability measures, and obtain weak laws of large numbers for arrays of random variables without the assumption of identical distribution. It shows that, under model uncertainty, the sample average of the observed random variables has its limit points within an identified set determined by the limits of upper and lower expectations. Further, we illustrate an application of our result to m-dependent random variables and obtain a general weak law of large numbers for sequences of m-dependent random variables.

A Nonintrusive Reduced-Order Model for Uncertainty Quantification in Numerical Solution of One-Dimensional Free-Surface Water Flows Over Stochastic Beds

Free-surface water flows over stochastic beds are complex due to the uncertainties in topography profiles being highly heterogeneous and imprecisely measured. In this study, the propagation and influence of several uncertainty parameters are quantified in a class of numerical methods for one-dimensional free-surface flows. The governing equations consist of both single-layer and two-layer shallow water equations on either flat or nonflat topography. For this purpose, the free-surface profiles are computed for different realizations of the random variables when the bed is excited with sources whose statistics are well defined. Many research studies have been dedicated to the development of numerical methods to achieve some order of accuracy in free-surface flows. However, little concern was given to examine the performance of these numerical methods in the presence of uncertainty. This work addresses this specific area in computational hydraulics with regards to the uncertainty generated from bathymetric forces. As numerical solvers for the one-dimensional shallow water equations, we implement four finite volume methods. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial chaos expansions for efficient uncertainty quantifications of complex hydraulic problems with large number of random variables. Numerical results are shown for several test examples including dam-break problems for single-layer and two-layer shallow water flows. The problem of flow exchange through the Strait of Gibraltar is also solved in this study. The obtained results demonstrate that in some hydraulic applications, a highly accurate numerical method yields an increase in its uncertainty and makes it very demanding to use in an operational manner with measured data from the field. On the other hand, when the complexity of physics increases, these highly accurate numerical methods display less uncertainty compared to the low accurate methods.

A comparative study of Bayesian inverse analyses of spatially varying soil parameters for slope reliability updating

ABSTRACT Bayesian estimation of spatially varying soil parameters is a challenging task in geotechnical engineering because a large number of random variables need to be learned. To address this challenge, three Bayesian methods are revisited, including Differential Evolution Adaptive Metropolis with sampling from past states [DREAM(zs)] method, Bayesian Updating with Structural reliability methods using Subset Simulation (BUS + SS), and modified BUS with Subset Simulation (mBUS + SS). The differences between the performances (i.e. convergences, computational accuracies, and efficiencies) of these three methods are not well understood. This study systematically investigates the differences of these three methods in the generation of random samples, convergence criterion, model evidence, and estimation of posterior probability of failure in slope reliability updating. Two slope examples are used for the comparative study. It is found that the BUS + SS method performs well not only in the low-dimensional Bayesian inverse problems but also in the high-dimensional Bayesian inverse problems of spatially varying soil parameters. The DREAM(zs) method is preferentially recommended to deal with the low-dimensional Bayesian inverse problems whereas the mBUS + SS method can well tackle the high-dimensional problems.

A New Methodology for Solving Multiobjective Chance-Constrained Problems: An Application on IoT Systems

This study presents a newly developed methodology to transform the chance-constrained problem into a deterministic problem and then solving this multiobjective deterministic problem with the proposed method. Chance-constrained problem contains independent gamma random variables that are denoted as a i j . Two methods are proposed to obtain the deterministic equivalent of chance-constrained problem. The first of the methods is directly based on using the distribution, and the second consists of normalizing probabilistic constraints using Lyapunov’s central limit theorem. An algorithm which uses the Global Criterion Method is developed to solve the multiobjective deterministic equivalent of chance-constrained problem. The methodology is applied to a real-life engineering problem that consists of an IoT device and its data sending process. Using Lyapunov’s central limit theorem for large numbers of random variables is found to be more appropriate.

A surrogate model for efficient quantification of uncertainties in multilayer shallow water flows

In this study, we investigate the implementation of a Proper Orthogonal Decomposition (POD) Polynomial Chaos Expansion (PCE) POD-PCE surrogate model for the propagation and quantification of the uncertainty in hydraulic modelling. The considered model consists of a system of multilayer shallow water equations with a mass exchange between the layers and over stochastic beds. As a numerical solver, we propose a finite volume characteristics method that does not require eigenstructure of the system in its implementation. The method is fast, accurate and can be used for both slowly and rapidly hydraulic simulations. The propagation and influence of several uncertainty parameters are quantified in the considered numerical methods for multilayer shallow water flows. To reduce the required number of samples for uncertainty quantification, we combine the proper orthogonal decomposition method with the polynomial Chaos expansions for efficient uncertainty quantification of complex hydraulic problems with a large number of random variables. Numerical results are shown for several test examples including a dam-break problem over a flat bed, and a wind-driven recirculation flow on flat and non-flat bottoms. Results are also presented for the case study of a recirculation flow problem in the Strait of Gibraltar. The results demonstrate the robustness of the uncertainty quantification method compared to the standard Monte-Carlo simulations. The results presented in this study suggest that the use of surrogate modelling may save a considerable amount of the necessary computational cost for all the considered cases.

Random matrix eigenvalue problems in structural dynamics: An iterative approach

Uncertainties need to be taken into account in the dynamic analysis of complex structures. This is because in some cases uncertainties can have a significant impact on the dynamic response and ignoring it can lead to unsafe design. For complex systems with uncertainties, the dynamic response is characterised by the eigenvalues and eigenvectors of the underlying generalised matrix eigenvalue problem. This paper aims at developing computationally efficient methods for random eigenvalue problems arising in the dynamics of multi-degree-of-freedom systems. There are efficient methods available in the literature for obtaining eigenvalues of random dynamical systems. However, the computation of eigenvectors remains challenging due to the presence of a large number of random variables within a single eigenvector. To address this problem, we project the random eigenvectors on the basis spanned by the underlying deterministic eigenvectors and apply a Galerkin formulation to obtain the unknown coefficients. The overall approach is simplified using an iterative technique. Two numerical examples are provided to illustrate the proposed method. Full-scale Monte Carlo simulations are used to validate the new results.

Failure probability estimation of a class of series systems by multidomain Line Sampling

This contribution proposes an approach for the assessment of the failure probability associated with a particular class of series systems. The type of systems considered involves components whose response is linear with respect to a number of Gaussian random variables. Component failure occurs whenever this response exceeds prescribed deterministic thresholds. We propose multidomain Line Sampling as an extension of the classical Line Sampling to work with a large number of components at once. By taking advantage of the linearity of the performance functions involved, multidomain Line Sampling explores the interactions that occur between failure domains associated with individual components in order to produce an estimate of the failure probability. The performance and effectiveness of multidomain Line Sampling is illustrated by means of two test problems and an application example, indicating that this technique is amenable for treating problems comprising both a large number of random variables and a large number of components.

Integration of a Hybrid Method into an Efficient Reliability-Based Design Optimization Technique for Large Structures

The computational burden imposed by the repetitive and lengthy nature of the structural analyses of large structures has lead researches to continuously devise efficient methods for conducting their reliability analysis in addition to powerful methods for solving the design optimization problem. The presence of a considerably large number of random variables in the reliability analyses of large structure is another issue that tends to slow down the design optimization problem. In this paper, a hybrid method that was recently presented by the first author will be integrated into a structural reliability-based design optimization technique for large structures. In this technique, modified concepts of the weighted average simulation method are used to determine the most probable point of failure in a computationally efficient manner. The most probable point of failure is then transferred into the standard normal space, where the reliability index is calculated in closed-form. The firefly algorithm will be used for solving the optimization problem. It will be shown that the integrated approach significantly reduces the computational resources required for solving the design problem. The technique will be tested on a truss bridge. The results will shed light on the efficiency of the proposed technique in solving reliability-based design optimization problems.

Hybrid System that Integrates the Lost Energy Recovery on the Water-Water Heat Pump Exhaust Circuit

Exceeding intermitence and the stochastic character associated with most renewable energy sources in order to maintain the balance between generation and consumption require the most efficient energy storage methods. This balance is important to maintain the quality of the power supply by adjusting the frequency and voltage of the grid. Storage is the capture of the energy produced at a given time for later use. It is known that electricity cannot be stored directly unless it is stored in condenser batteries or in superconducting coils. Other methods of electricity storage presuppose the conversion of these into other forms of energy: mechanical storage, electrochemical storage. The optimization of RES operation and, implicitly, of the energy system implies the choice of a suitable conversion / storage system that must take into account a number of factors, such as: capacity and minimum storage time, loading and unloading conditions, minimum number of cycles download loading, power supply required by the storage system, system lifetime. It is not to be neglected the space required for the installation of the storage systems and their impact on the environment during their exploitation as well as during the post-use and recycling period. There are RES that contain natural storage systems (hydro-electric power plants and biomass and biogas generators). The article proposes an analysis of how to store electricity by approach a Pelton water-water-turbine hybrid system. The heat pump consumes electrical energy for action, producing a thermal effect. By integrating a system with Pelton water storage tank and turbine on the heat pump outlet circuit, the lost hydraulic energy can be recovered. The experimental hybrid system was made to develop HRES optimization models. Hybrid systems may exceed the limits of individual generators in terms of efficiency, economy, reliability and flexibility. An energy storage system can alleviate the problems associated with uncertainties and fluctuations from renewable sources. The large number of random variables and parameters in a hybrid energy system requires optimization to increase the efficiency of hybrid system components to achieve economic and technical benefits.

A multi-fidelity boundary element method for structural reliability analysis with higher-order sensitivities

A novel multi-fidelity modelling methodology for structural reliability analysis using the Boundary Element Method (BEM) with an Implicit Differentiation Method (IDM) is presented. Reliability analyses are conducted with methods such as Monte Carlo Simulation (MCS) and the First-Order Reliability Method (FORM). The higher-order sensitivities of the elastostatic Boundary Element Method equations with respect to changes in several geometric variables have been derived for the first time for use with the IDM for the purpose of conducting reliability analyses with the Second-Order Reliability Method (SORM), a more accurate alternative to FORM for problems with non-linear limit state functions. Multi-fidelity formulations involving the IDM have also been derived for the first time, making use of the metamodelling technique Kriging. The use of multi-fidelity modelling enables the creation of a model that has similar accuracy to a high-fidelity model, but with a computational cost similar to that of a low-fidelity model. By combining the accuracy of the IDM with the efficiency of multi-fidelity modelling the proposed methodology has the capability to be very effective when used for structural reliability analysis. The IDM is validated through a numerical example for which the analytical solution is known. A further two numerical examples featuring an I-beam section and a triangular support bracket with a large number of variables are also investigated. Results show that the employed multi-fidelity models were up to 6000 times faster in terms of CPU-time than the high-fidelity model, while also providing probabilities of failure that were up to 2225 times more accurate than the low-fidelity model. Overall, it has been shown that the use of the proposed IDM/multi-fidelity modelling methodology significantly improved the efficiency and accuracy of the above reliability analysis techniques when applied to complex problems involving a large number of random variables under high levels of uncertainty.

Structural reliability analysis based on polynomial chaos, Voronoi cells and dimension reduction technique

Polynomial chaos expansions (PCEs) has been widely used to construct meta-models for structural reliability analysis. The computational effort of classical PCEs is unaffordable as the required number of deterministic model analyses grows exponentially with the dimension. Alternatively, the sparse PCEs are always built to alleviate this problem. This paper proposes an efficient method, which combines the sparse PCE with a novel unequal-weighted sampling strategy, i.e. Voronoi cells and the dimension reduction technique for structural reliability analysis. The unequal-weighted sampling strategy could converge fast to the ultimate goal of sequentially building a sparse PCE. Besides, when the dimension is high, the sliced inverse regression technique is employed to convert the original high-dimensional problem to a low-dimensional one. Then, a stepwise weighted regression method is involved to automatically determine the significant terms of the PCE and discard the insignificant ones for the reduced model. In this regard, the sparsity of the basis, the dimension reduction technique and the fast convergence of unequal-weighted sampling strategy lead to a considerably reduced computational cost. Four numerical examples with a large number of random variables are presented to validate the proposed method. The computational results show that the proposed method can establish fairly accurate meta-models for structural reliability assessment with low computational effort.

Projective Latent Dependency Forest Models

Latent dependence forest models (LDFM) are a new type of probabilistic models with the advantage of not requiring the difficult procedure of structure learning in model learning. However, normalized joint probability computation and marginal inference are intractable for LDFM. In this paper, we proposed projective LDFMs (PLDFMs), a variant of LDFM, for which joint and marginal probabilities become tractable (cubic time with respect to the number of random variables) to compute while learning remains easy. We show that PLDFMs can be seen as a special case of sum-product networks (SPNs). We then propose sum-product projective dependence networks, a combination of PLDFMs and SPNs that scales up to a large number of random variables. Our extensive experiments on 29 datasets show that our models achieve competitive results with other probabilistic models.

Efficient uncertainty quantification of stochastic heat transfer problems by combination of proper orthogonal decomposition and sparse polynomial chaos expansion

To increase the contribution and reliability of computational fluid dynamics efforts in design process of industrial equipments, it is necessary to quantify the effects of uncertainties on the system performance. Due to exponentially increment of the computational cost with number of uncertain variables for uncertainty quantification using classical polynomial chaos expansion methodology, reducing the required number of samples for uncertainty quantification is a real engineering challenge. In this paper, the proper orthogonal decomposition method based on the multifidelity approach is combined with the full and sparse polynomial chaos expansions for efficient uncertainty quantification of complex heat transfer problems with large number of random variables. The conjugate conduction heat transfer in NASA C3X cooled gas turbine blade with geometrical uncertainties and the convective heat transfer in ribbed passage with the stochastic wall heat flux boundary condition are considered as the test cases. Results of uncertainty quantification analysis in both test cases showed that proposed multi-fidelity approaches are able to produce the statistical quantities with much lower computational cost compare to the classical regression-based polynomial chaos method. It is shown that the combination of the proper orthogonal decomposition with the sparse polynomial chaos gives a computational gain at least 2 times greater than combination of the proper orthogonal decomposition with the full polynomial chaos expansion.

Dimension adaptive finite difference decomposition using multiple sparse grids for stochastic computation

The present study aims to investigate uncertainty quantification followed by reliability analysis of structure with homogeneous non-normal random fields. In stochastic finite element formulation, these continuous fields are discretized by different methods (e.g. Karhunen-Loève Expansion) which transformed it into a set of random variables. However, this discretization often leads to large number of random variables, especially for multiple random fields. With this in view, two different meta-model based approaches are presented in this study using high dimensional model representation (HDMR) for efficient stochastic computation. First, an adaptive multiple finite difference HDMR (AMFD-HDMR) is proposed that decomposes the original performance function into summands of smaller dimensions. These subfunctions are modeled by polynomial chaos expansion (PCE) using moving least square technique which utilizes the benefits of orthogonality of the basis functions and provides adaptive interpolation between the support points. These support points are generated in a sparse grid framework based on the hierarchial tensor products of the sub-grids and the appropriate statistical properties. An iterative scheme is developed with the aim to create new support points in the desired locations such as most probable failure point and/or maxima/minima. Later, a dimension adaptive multiple finite difference HDMR (dAMFD-HDMR) is proposed utilizing sensitivity analysis to further improve the efficiency and accuracy. In the second proposal, an intermittent HDMR formulation is suggested based on the individual and mutual contributions of the significant dimensions. Once the meta-model is built, Monte Carlo simulation is performed over it, thus bypassing the time exhaustive computation of the original performance function. Numerical studies are carried out using composite plate to prove the merits of the proposed algorithms compared to other methods available in the literature.

Probabilistic failure analysis of infinite slopes under random rainfall processes and spatially variable soil

ABSTRACTRainfall-induced landslides occur during or immediately after rainfall events in which the pore water pressure builds up, leading to shallow slope failure. Thereby, low permeability layers result in high gradients in pore water pressure. The spatial variability of the soil permeability influences the probability such low permeability layers, and hence the probability of slope failure. In this paper, we investigate the influence of the vertical variability of soil permeability on the slope reliability, accounting for the randomness of rainfall processes. We model the saturated hydraulic conductivity of the soil with a one-dimensional random field. The random rainfall events are characterised by their duration and intensity and are modelled through self-similar random processes. The transient infiltration process is represented by Richards equation, which is evaluated numerically. The reliability analysis of the infinite slope is based on the factor of safety concept for evaluating slope stability. To cope with the large number of random variables arising from the discretization of the random field and the rainfall process, we evaluate the slope reliability through Subset Simulation, which is an adaptive Monte Carlo method known to be especially efficient for reliability analysis of such high-dimensional problems. Numerical investigations show higher probability of slope failure with increased spatial variability of the saturated hydraulic conductivity and with more uniform rainfall patterns.

Strong law of large numbers for random variables with multidimensional indices

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A BIRTH AND DEATH PROCESS FOR BAYESIAN NETWORK STRUCTURE INFERENCE

Bayesian networks are convenient graphical expressions for high-dimensional probability distributions which represent complex relationships between a large number of random variables. They have been employed extensively in areas such as bioinformatics, artificial intelligence, diagnosis, and risk management. The recovery of the structure of a network from data is of prime importance for the purposes of modeling, analysis, and prediction. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a Bayesian network from observed data. Typically, one assigns a score to various structures and the search becomes an optimization problem that can be approached with either deterministic or stochastic methods. In this paper, we introduce a new search strategy where one walks through the space of graphs by modeling the appearance and disappearance of edges as a birth and death process. We compare our novel approach with the popular Metropolis–Hastings search strategy and give empirical evidence that the birth and death process has superior mixing properties.