Uncertainty Analysis Problems Research Articles

Propagation algorithm for hybrid uncertainty parameters based on polynomial chaos expansion

AbstractThis article presents an uncertainty analysis method for systems with hybrid stochastic and fuzzy uncertainty parameters based on polynomial chaos expansion (PCE). Parameters in the system are described by probability boxes, interval numbers, and fuzzy sets, respectively, based on the differences in their limited stochastic knowledge. First, this method transforms the uncertain parameters into standard normal distribution and interval variables through equal probability transformation or ‐cut operations. Second, the Legendre and Hermite polynomials are used as the PCE model's primary functions, and the expansion coefficients are calculated by the Galerkin projection method based on sparse grid numerical integration. Then, the system response bounds under the pre‐defined confidence level can be obtained using a genetic algorithm to solve the optimization problem constructed based on PCE models. Finally, the feasibility and effectiveness of the method are illustrated by taking the tank bi‐directional stabilized system and the double‐pendulum‐controlled system as examples. The numerical results show that the system response bounds obtained by the PCE model optimization algorithm are consistent with the Monte Carlo simulation. Still, the computational efficiency is much higher. The proposed method effectively combines fuzzy sets and probability boxes and dramatically simplifies the analysis process of uncertain systems. The method exhibits fine precision even in high‐dimensional uncertainty analysis problems.

Meta-MgNet: Meta multigrid networks for solving parameterized partial differential equations

This paper studies numerical solutions for parameterized partial differential equations (PDEs) with deep learning. Parametrized PDEs arise in many important application areas, including design optimization, uncertainty analysis, optimal control, and inverse problems. The computational cost associated with these applications using traditional numerical schemes can be exorbitant, especially when the parameters fall into a particular range, and the underlying PDE model is required to be solved with high accuracy using a fine spatial-temporal mesh. Recently, solving PDEs with deep learning has become an emerging field in scientific computing. Existing works demonstrate great potentials of the deep learning-based approach in speeding up numerical solutions of various types of PDEs. However, there is still limited research on the deep learning approach for solving parameterized PDEs. If we directly apply existing deep supervised learning models to solving parameterized PDEs, the models need to be constantly fine-tuned or retrained when the parameters of the PDE change. This limits the applicability and utility of these models in practice. To resolve this issue, we propose a meta-learning based method that can efficiently solve parameterized PDEs with a wide range of parameters without retraining. Our key observation is to regard training a solver for the parameterized PDE with a given set of parameters as a learning task. Then, training a solver for the parameterized PDEs with varied parameters can be viewed as a multi-task learning problem, to which meta-learning is one of the most effective approaches. This new perspective can be applied to many existing PDE solvers to make them suitable for solving parameterized PDEs. As an example, we adopt the Multigrid Network (MgNet) [21] as the base solver. To achieve multi-task learning, we introduce a new hypernetwork, called Meta-NN, in MgNet and refer to the entire network as the Meta-MgNet. Meta-NN takes the differential operators and the right-hand-side of the underlying parameterized PDEs as inputs and generates appropriate smoothers for MgNet, which are essential ingredients for multigrid methods and can significantly affect the convergent speed. The proposed Meta-NN is carefully designed so that Meta-MgNet has guaranteed convergence for Poisson's equation. Finally, extensive numerical experiments demonstrate that Meta-MgNet is more efficient in solving parameterized PDEs than the MG methods and MgNet trained by supervised learning.

Методи та підходи до моделювання ефективності цілей операцій в транспортних системах

Methods and approaches to modeling the goals of operations in transport systems are given, which allows to obtain not only purposeful, but also flexibly oriented solutions when choosing a strategy for forecasting their work. Partial decision-making tasks for structuring the initial information using the mapping method are defined. The classification of factors by the nature of uncertainty is given, which is one of the results of solving the problem of uncertainty analysis. Tasks are formulated that allow to quickly receive, analyze and process information. A formalized formulation of the problem of forming the initial set of strategies for modeling the problem situation in the transport system is presented. It was found that on the one hand, the set of alternatives between the desired result and the conditions of its implementation should be as wide as possible, which will ensure freedom of choice of the decision-maker and minimize the possibility of missing the optimal decision. It is determined that the initial set of strategies should be comprehensive and, therefore, quite narrow, which will allow the decision-maker to verify the alternatives on the available models with resource constraints. It was found that many components of the task of forming the initial set of strategies are clearly not set and are heuristically formed by the decision maker. Taking into account the indicator of reliability of transport systems, the main directions of solving the problem of satisfying conflicting requirements for many initial strategies are identified. It is found that the need, in the general case, to consider not one model, but their multilevel hierarchy is determined by the peculiarity of the efficiency study scheme, in which the description of the higher level depends on generalized and factorized lower level variables. The mathematical formula of the criterion of adaptability in the choice of solutions is given. The set of features that can be used as a basis for the classification of decision-making tasks in the transport system on the basis of the degree of certainty of information about the problem situation is generalized.

An Uncertainty Analysis Method for Artillery Dynamics with Hybrid Stochastic and Interval Parameters

This paper proposes a non-intrusive uncertainty analysis method for artillery dynamics involving hybrid uncertainty using polynomial chaos expansion (PCE). The uncertainty parameters with sufficient information are regarded as stochastic variables, whereas the interval variables are used to treat the uncertainty parameters with limited stochastic knowledge. In this method, the PCE model is constructed through the Galerkin projection method, in which the sparse grid strategy is used to generate the integral points and the corresponding integral weights. Through the sampling in PCE, the original dynamic systems with hybrid stochastic and interval parameters can be transformed into deterministic dynamic systems, without changing their expressions. The yielded PCE model is utilized as a computationally efficient, surrogate model, and the supremum and infimum of the dynamic responses over all time iteration steps can be easily approximated through Monte Carlo simulation and percentile difference. A numerical example and an artillery exterior ballistic dynamics model are used to illustrate the feasibility and efficiency of this approach. The numerical results indicate that the dynamic response bounds obtained by the PCE approach almost match the results of the direct Monte Carlo simulation, but the computational efficiency of the PCE approach is much higher than direct Monte Carlo simulation. Moreover, the proposed method also exhibits fine precision even in high-dimensional uncertainty analysis problems.

Managing Wind Power Generation via Indexed Semi-Markov Model and Copula

Because of the stochastic nature of wind turbines, the output power management of wind power generation (WPG) is a fundamental challenge for the integration of wind energy systems into either power systems or microgrids (i.e., isolated systems consisting of local wind energy systems only) in operation and planning studies. In general, a wind energy system can refer to both one wind farm consisting of a number of wind turbines and a given number of wind farms sited at the area in question. In power systems (microgrid) planning, a WPG should be quantified for the determination of the expected power flows and the analysis of the adequacy of power generation. Concerning this operation, the WPG should be incorporated into an optimal operation decision process, as well as unit commitment and economic dispatch studies. In both cases, the probabilistic investigation of WPG leads to a multivariate uncertainty analysis problem involving correlated random variables (the output power of either wind turbines that constitute wind farm or wind farms sited at the area in question) that follow different distributions. This paper advances a multivariate model of WPG for a wind farm that relies on indexed semi-Markov chains (ISMC) to represent the output power of each wind energy system in question and a copula function to reproduce the spatial dependencies of the energy systems’ output power. The ISMC model can reproduce long-term memory effects in the temporal dependence of turbine power and thus understand, as distinct cases, the plethora of Markovian models. Using copula theory, we incorporate non-linear spatial dependencies into the model that go beyond linear correlations. Some copula functions that are frequently used in applications are taken into consideration in the paper; i.e., Gumbel copula, Gaussian copula, and the t-Student copula with different degrees of freedom. As a case study, we analyze a real dataset of the output powers of six wind turbines that constitute a wind farm situated in Poland. This dataset is compared with the synthetic data generated by the model thorough the calculation of three adequacy indices commonly used at the first hierarchical level of power system reliability studies; i.e., loss of load probability (LOLP), loss of load hours (LOLH) and loss of load expectation (LOLE). The results will be compared with those obtained using other models that are well known in the econometric field; i.e., vector autoregressive models (VAR).

Uncertainty Analysis for Return Trajectory of Vertical Takeoff and Vertical Landing Reusable Launch Vehicle

The uncertainties during the return trajectory of vertical takeoff and vertical landing reusable launch vehicle weaken the ability of precision landing and make the return process more challenging. This paper is devoted to quantifying the probability uncertainty of return trajectory with uncertain parameters. The uncertainty model of return multi-flight-phase under the uncertainties of initial flight path angle, axial aerodynamic coefficient, and atmospheric density is established using the generalized polynomial chaos expansion method. By parameterizing random uncertainties and introducing random parameters into the uncertainty model, the uncertainty analysis problem of return trajectory is transformed into stochastic trajectory approximation problem. The coefficients of the polynomial basis function are solved by the stochastic collocation method. Then state solutions, statistical properties, and global sensitivity with Sobol index are established based on coefficients. The simulation results show the efficiency and accuracy of this method compared with the Monte Carlo method, the evolution process of main output parameters under random parameters, and relative importance for random parameters. Through the uncertainty analysis of the return trajectory, the robustness of return trajectory can be quantified, which is contributed to improving the safety, reliability, and robustness of recovery and landing mission.

A Vine Copula-Based Hierarchical Framework for Multiscale Uncertainty Analysis

Abstract Uncertainty analysis is an effective methodology to acquire the variability of composite material properties. However, it is hard to apply hierarchical multiscale uncertainty analysis to the complex composite materials due to both quantification and propagation difficulties. In this paper, a novel hierarchical framework combined R-vine copula with sparse polynomial chaos expansions is proposed to handle multiscale uncertainty analysis problems. According to the strength of correlations, two different strategies are proposed to complete the uncertainty quantification and propagation. If the variables are weakly correlated or mutually independent, Rosenblatt transformation is used directly to transform non-normal distributions into the standard normal distributions. If the variables are strongly correlated, the multidimensional joint distribution is obtained by constructing R-vine copula, and Rosenblatt transformation is adopted to generalize independent standard variables. Then, the sparse polynomial chaos expansion is used to acquire the uncertainties of outputs with relatively few samples. The statistical moments of those variables that act as the inputs of next upscaling model can be gained analytically and easily by the polynomials. The analysis process of the proposed hierarchical framework is verified by the application of a 3D woven composite material system. Results show that the multidimensional correlations are modeled accurately by the R-vine copula functions, and thus uncertainty propagations with the transformed variables can be done to obtain the computational results with consideration of uncertainties accurately and efficiently.

Uncertainty analysis of composite laminated plate with data-driven polynomial chaos expansion method under insufficient input data of uncertain parameters

The uncertainty information related to uncertain structural, material and geometric parameters is included in the available input uncertainty data, and there are multiple uncertainty types when only insufficient input data is acquired from experimental or numerical analyses. In this study, an uncertainty analysis method for composite laminated plate is proposed using a data-driven polynomial chaos expansion (PCE) method under insufficient input data related to uncertain design parameters. An identification criterion for uncertain types of design parameters is constructed based on the Akaike information criterion and observation significance level method, and the uncertain design parameters are subsequently divided into strong statistical variables, sparse variables, and interval variables. A data-driven PCE model of composite laminated plate is constructed by simultaneously considering the three uncertainty types, in which the polynomial coefficients are calculated based on the statistical moment of both strong statistical variables and sparse variables, and the PCE coefficients are quadratic polynomial functions of interval variables. The uncertainty analysis problems of the natural frequency of 5-layer and 10-layer composite laminated plates are presented to verify the effectiveness of the proposed algorithm.

Uncertainty and Reliability Analysis of Open Pit Rock Slopes: A Critical Review of Methods of Analysis

The stability analyses of slope excavations in rock mass require reliable geomechanical input parameters such as rock mass strength, friction angle and cohesion of sliding surface. These parameters are naturally uncertain and their exact values cannot be known, therefore, their variability must be properly accounted for in the stability analyses. Deterministic approaches such as the limit equilibrium methods, numerical methods and kinematic analysis methods do not account for the variability in any of the input parameter. This paper therefore provides a review of uncertainty and uncertainty analysis methods, problems and developments in geotechnical modelling of rock slope stability. The review is motivated by the availability of qualitative and number of methods for uncertainty analysis. The paper examines the various definitions and description of uncertainty and the different vocabularies that are used, and also summarises and categorises the different sources of uncertainty as well as integrating uncertainty for rock slope assessment problems. The paper discussed a simple survey of probability-based reliability methods that have been used for rock slope stability analysis in the past 3 decades.

On the Uncertainty Control in the Complex Multiphysics Systems in the Task of Multi-Scale Stochastic GHG and Carbon Balance Modeling

The problem of uncertainty analysis in complex multi-component systems is considered. The problem of decision making with uncertainty in tasks modeling carbon balance and the analysis of greenhouse gas (GHG) emissions using satellite tools was considered. Approaches to decision making under uncertainty are described in terms of interval, fuzzy, and stochastic assessments. Different approaches and algorithms to calculate carbon and GHG emissions are described. For every algorithm, errors and uncertainties are analyzed and estimated. Algorithms for uncertainty analysis based on integrated interlinked models of the system are presented. Algorithms for the analysis of components of vegetation productivity assessment using satellite data are proposed. Uncertainty component analysis allows the understanding of important properties of the system studied, and its feedback as to its anthropogenic load and impact on the climate. It was demonstrated that the comprehensive analysis of uncertainties allows not only the reduction of errors, but new knowledge about the studied systems.

Stochastic Modeling of Groundwater Extractions over a Data-Sparse Region of Australia.

Setting limit on groundwater extractions is important to ensure sustainable groundwater management. Lack of extraction data can affect interpretations of historical pressure changes, predictions of future impacts, accuracy of groundwater model calibration, and identification of sustainable management options. Yet, many groundwater extractions are unmetered. Therefore, there is a need for models that estimate extraction rates and quantify model outputs uncertainties arising due to a lack of data. This paper develops such a model within the Generalized Linear Modeling (GLM) framework, using a case study of stock and domestic (SD) extractions in the Surat Cumulative Management Area, a predominantly cattle farming region in eastern Australia. Various types of extraction observations were used, ranging from metering to analytically-derived estimates. GLMs were developed and applied to estimate the property-level extraction amounts, where observation types were weighted by perceived relative accuracy, and well usage status. The primary variables found to affect property-level extraction rates were: yearly average temperature and rainfall, pasture, property area, and number of active wells; while variables most affecting well usage were well water electrical conductivity, spatial coordinates, and well age. Results were compared with analytical estimates of property-level extraction, illustrating uncertainties and potential biases across 20 hydrogeological units. Spatial patterns of mean extraction rates (and standard deviations) are presented. It is concluded that GLMs are well suited to the problem of extraction rate estimation and uncertainty analysis, and are ideal when model verification is supported by measurement of a random sample of properties.

Uncertainty analysis of reachable set for planetary entry using polynomial chaos

Reachable set (RS) is a useful tool in assessing the flight capability of the entry vehicle with the given initial entry state and the required control and path constraints. The evolution of the RS under the effects of uncertainties of some important entry parameters is also beneficial to the mission design and analysis. The objective of this paper is to investigate the RS evolution under parameter uncertainties. It would be computationally intensive to compute all the elements of the RS, and therefore only the RS boundary (RSB) extreme points, which well characterize the RS, are considered. The method of tracking drag profile is employed to generate each point of the RSB. By parameterizing the random uncertainties and introducing random parameters in uncertainties, the uncertainty analysis problem of the RSB is converted to a stochastic trajectory approximation problem, which can be solved by the polynomial chaos expansion-based method. The evolutions of the RSB under uncertainties of initial flight path angle, lift-to-drag ratio and atmospheric density are illustrated by a specific Mars entry mission. With uncertainty analysis of the RSB, the robustness of deployment site selection and the satisfaction of path constraints under uncertainties can be quantified in the entry mission design, thus enhancing the safety, reliability and accuracy of entry missions.

Likelihood-based representation of epistemic uncertainty and its application in robustness-based design optimization

In this paper, we propose a new likelihood-based methodology to represent epistemic uncertainty described by sparse point and/or interval data for input variables in uncertainty analysis and design optimization problems. A worst-case maximum likelihood-based approach is developed for the representation of epistemic uncertainty, which is able to estimate the distribution parameters of a random variable described by sparse point and/or interval data. This likelihood-based approach is general and is able to estimate the parameters of any known probability distributions. The likelihood-based representation of epistemic uncertainty is then used in the existing framework for robustness-based design optimization to achieve computational efficiency. The proposed uncertainty representation and design optimization methodologies are illustrated with two numerical examples including a mathematical problem and a real engineering problem.

Adjoint-Based Sensitivity and Uncertainty Analysis for Density and Composition: A User’s Guide

The evaluation of uncertainties is essential for criticality safety. This paper deals with material density and composition uncertainties and provides guidance on how traditional first-order sensitivity methods can be used to predict their effects. Unlike problems that deal with traditional cross-section uncertainty analysis, material density and composition-related problems are often characterized by constraints that do not allow arbitrary and independent variations of the input parameters. Their proper handling requires constrained sensitivities that take into account the interdependence of the inputs. This paper discusses how traditional unconstrained isotopic density sensitivities can be calculated using the adjoint sensitivity capabilities of the popular Monte Carlo codes MCNP6 and SCALE 6.2, and we also present the equations to be used when forward and adjoint flux distributions are available. Subsequently, we show how the constrained sensitivities can be computed using the unconstrained (adjoint-based) sensitivities as well as by applying central differences directly. Three distinct procedures are presented for enforcing the constraint on the input variables, each leading to different constrained sensitivities. As a guide, the sensitivity and uncertainty formulas for several frequently encountered specific cases involving densities and compositions are given. An analytic k∞ example highlights the relationship between constrained sensitivity formulas and central differences, and a more realistic numerical problem reveals similarities among the computer codes used and differences among the three methods of enforcing the constraint.

Methods for analyzing the uncertainty of a reconstructed result in a traffic accident with interval and probabilistic traces

In order to make the reconstructed result more reliable, a method named improved probabilistic-interval method was proposed to analyze the uncertainty of a reconstructed result in a traffic accident with probabilistic and interval traces. In the method, probabilistic traces are replaced by probabilistic sub-intervals firstly; secondly, these probabilistic sub-intervals and those interval traces will be combined to form many new uncertainty analysis problems with only interval traces; thirdly, the upper and lower bound of the reconstructed result and their probability were calculated in each new uncertainty analysis problem, and an algorithm was proposed to shorten the time taken for this step; finally, distribution functions of the upper and lower bound of the reconstructed result were obtained by doing statistic analysis. Through 2 numerical cases, results obtained from the proposed method were almost the same as results obtained from the Monte Carlo method, but the time taken for the proposed method was far less than the time taken for the Monte Carlo method and results obtained from the proposed method were more stable. Through applying the proposed method to a true vehicle–pedestrian accident, not only the upper and lower bound of the impact velocity (v) can be obtained; but also the probability that the upper bound and the lower bound of v falls in an arbitrary interval can be obtained; furthermore, the probability that the interval of v is less than an arbitrary interval can be obtained also. It is concluded that the proposed improved probabilistic-interval method is practical.

A simple algorithm for analyzing uncertainty of accident reconstruction results

In order to analyzing the uncertainty in accident reconstruction, based on the theory of extreme value and the convex model theory, the uncertainty analysis problem is turn to an extreme value problem. In order to calculate the range of the dependent variable, the extreme value in the definition domain and on the boundary of the definition domain are calculated independently, and then the upper and lower bound of the dependent variable can be given by these obtained extreme values. Based on such idea and through analyzing five numerical cases, a simple algorithm for calculating the range of an accident reconstruction result was given; appropriate results can be obtained through the proposed algorithm in these cases. Finally, a real world vehicle-motorcycle accident was given, the range of the reconstructed velocity of the vehicle was calculated by employing the Pc-Crash, the response surface methodology and the new proposed algorithm, the range was [66.1–67.3] km/h. This research will provide another choice for uncertainty analysis in accident reconstruction.

Probabilistic estimation of plug-in electric vehicles charging load profile

Plug-in electric vehicles (PEVs) are widely considered as a sustainable mode of transport by countries worldwide due to high efficiency and low or zero carbon emissions. However, PEVs will add significant additional load to the existing power distribution system and it will be a challenge to meet the new demand. In this study, probabilistic modelling has been presented to estimate the system-wide PEV charging load within domestic grids. U.S. national household travel survey data set has been utilized to quantitatively determine the mobility behaviour of PEVs.Uncertain nature of the problem in modelling and data preparation should be taken into account. Due to the existence of complex interdependencies between the system inputs, the problem definition leads to a multivariate uncertainty analysis problem. The modelling procedure is decomposed into two basic components: the modelling of the marginal distributions; and that of the stochastic dependence structure. In addition, Copula theory is presented for the multivariate modelling of dependent random variable. The results indicate that the PEVs can contribute to increase the load demand at certain hours, although the charging demand is very limited most of the time. Moreover, the probabilistic distribution of aggregated PEV charging demand is compared with that obtained by the Monte Carlo simulation. The numerical results have shown the effectiveness of the proposed methodology.

An Optimization-Based Approach to Calculate Confidence Interval on Mean Value with Interval Data

In this paper, we propose a methodology for construction of confidence interval on mean values with interval data for input variable in uncertainty analysis and design optimization problems. The construction of confidence interval with interval data is known as a combinatorial optimization problem. Finding confidence bounds on the mean with interval data has been generally considered an NP hard problem, because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the confidence interval on mean values with interval data. With numerical experimentation, we show that the proposed confidence bound algorithms are scalable in polynomial time with respect to increasing number of intervals. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the current practice for the design optimization with interval data that typically implements the constraints on interval variables through the computation of bounds on mean values from the sampled data, the proposed approach of construction of confidence interval enables more complete implementation of design optimization under interval uncertainty.

Estimation of a quantity of interest in uncertainty analysis: Some help from Bayesian decision theory

In the context of risk analysis under uncertainty, we focus here on the problem of estimating a so-called quantity of interest of an uncertainty analysis problem, i.e. a given feature of the probability distribution function (pdf) of the output of a deterministic model with uncertain inputs. We will stay here in a fully probabilistic setting. A common problem is how to account for epistemic uncertainty tainting the parameter of the probability distribution of the inputs. In the standard practice, this uncertainty is often neglected (plug-in approach). When a specific uncertainty assessment is made, under the basis of the available information (expertise and/or data), a common solution consists in marginalizing the joint distribution of both observable inputs and parameters of the probabilistic model (i.e. computing the predictive pdf of the inputs), then propagating it through the deterministic model. We will reinterpret this approach in the light of Bayesian decision theory, and will put into evidence that this practice leads the analyst to adopt implicitly a specific loss function which may be inappropriate for the problem under investigation, and suboptimal from a decisional perspective. These concepts are illustrated on a simple numerical example, concerning a case of flood risk assessment.

Copula-based modeling of stochastic wind power in Europe and implications for the Swiss power grid

Large scale integration of wind energy poses new challenges to the European power system due to its stochastic nature and often remote location. In this paper a multivariate uncertainty analysis problem is formulated for the integration of stochastic wind energy in the European grid. By applying copula theory a synthetic set of data is generated from scarce wind speed reanalysis data in order to achieve the increased sample size for the subsequent Monte Carlo simulation.In the presented case study, European wind power samples are generated from the modeled stochastic process. Under the precondition of a modeled perfect market environment, wind power impacts dispatch decisions and therefore leads to alterations in power balances. Stochastic power balances are implemented in a detailed model of the European electricity network, based on the generated samples. Finally, a Monte Carlo method is used to determine power flows and contingencies in the system. An indicator is elaborated in order to analyze risk of overloading and to prioritize necessary grid reinforcements. Implications for the Swiss power grid are investigated in detail, revealing that the current system is significantly put at risk in certain areas by the further integration of wind power in Europe.It is the first time that the results of a probabilistic model for wind energy are further deployed within a power system analysis of the interconnected European grid. The method presented in this paper allows to account for stochastic wind energy in a load flow analysis and to evaluate deterministic indicators, such as the N−1 criterion, on a probabilistic basis. Thus, it constitutes an important extension of deterministic models.