校准条件下的数值模拟不确定度量化方法

当存在众多不确定输入因素时,不确定度的传递分析往往导致对数值模拟不确定度的过高估计.利用校准行为能够消减系统级数值模拟中认知不确定度的客观机制,提出一个综合利用已有系统级试验对比信息和新增建模与模拟传递信息的不确定度量化方法,结合一个虚拟试验的例子对该方法进行展示和验证.     

A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos – capturing all scales of random modes on independent grids

In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection–diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.     

Probabilistic inference of reaction rate parameters from summary statistics

This investigation tackles the probabilistic parameter estimation problem involving the Arrhenius parameters for the rate coefficient of the chain branching reaction H + O₂ → OH + O. This is achieved in a Bayesian inference framework that uses indirect data from the literature in the form of summary statistics by approximating the maximum entropy solution with the aid of approximate bayesian computation. The summary statistics include nominal values and uncertainty factors of the rate coefficient, obtained from shock-tube experiments performed at various initial temperatures. The Bayesian framework allows for the incorporation of uncertainty in the rate coefficient of a secondary reaction, namely OH + H₂ → H₂O + H, resulting in a consistent joint probability density on Arrhenius parameters for the two rate coefficients. It also allows for uncertainty quantification in numerical ignition predictions while conforming with the published summary statistics. The method relies on probabilistic reconstruction of the unreported data, OH concentration profiles from shock-tube experiments, along with the unknown Arrhenius parameters. The data inference is performed using a Markov chain Monte Carlo sampling procedure that relies on an efficient adaptive quadrature in estimating relevant integrals needed for data likelihood evaluations. For further efficiency gains, local Padé–Legendre approximants are used as surrogates for the time histories of OH concentration, alleviating the need for 0-D auto-ignition simulations. The reconstructed realisations of the missing data are used to provide a consensus joint posterior probability density on the unknown Arrhenius parameters via probabilistic pooling. Uncertainty quantification analysis is performed for stoichiometric hydrogen–air auto-ignition computations to explore the impact of uncertain parameter correlations on a range of quantities of interest.     

A New Hybrid Possibilistic-Probabilistic Decision-Making Scheme for Classification

Uncertainty is at the heart of decision-making processes in most real-world applications. Uncertainty can be broadly categorized into two types: aleatory and epistemic. Aleatory uncertainty describes the variability in the physical system where sensors provide information (hard) of a probabilistic type. Epistemic uncertainty appears when the information is incomplete or vague such as judgments or human expert appreciations in linguistic form. Linguistic information (soft) typically introduces a possibilistic type of uncertainty. This paper is concerned with the problem of classification where the available information, concerning the observed features, may be of a probabilistic nature for some features, and of a possibilistic nature for some others. In this configuration, most encountered studies transform one of the two information types into the other form, and then apply either classical Bayesian-based or possibilistic-based decision-making criteria. In this paper, a new hybrid decision-making scheme is proposed for classification when hard and soft information sources are present. A new Possibilistic Maximum Likelihood (PML) criterion is introduced to improve classification rates compared to a classical approach using only information from hard sources. The proposed PML allows to jointly exploit both probabilistic and possibilistic sources within the same probabilistic decision-making framework, without imposing to convert the possibilistic sources into probabilistic ones, and vice versa.     

爆炸波中的混合不确定度量化方法

将认知不确定作为外层,偶然不确定作为内层,利用混合不确定理论处理爆炸波中的不确定度问题.分别使用非嵌入多项式混沌方法（NIPC）和Dempster-Shafer理论处理偶然不确定和认知不确定,用迎风格式求解确定系统.结果表明：NIPC和Dempster-Shafer结构为模型输入参数和初边值不确定性对输出响应量的影响提供一种有效方法,对各种建模与模拟不确定性评估和确认建模与模拟具有很好的参考价值.     

基于D-S理论的靶压幅度不确定性量化方法

针对关键参数测试样本数有限的情况下,概率理论、区间分析等方法在对输出靶压幅度进行不确定性定量评价时存在局限性和不合理性,将D-S理论引入到靶压幅度的不确定性量化中,根据小子样测试信息得出不确定性参数的基本信任分配,以信任函数和似然函数构造靶压幅度的上下界概率分布,并以Monte Carlo方法求解。实验和仿真得出了靶压幅度的近似概率分布、置信区间及期望值分布区间等信息,并表明：与传统的概率方法相比,该方法避免了根据小样本测试信息构造概率分布的难题;与区间分析方法相比,该方法可得到更丰富的信息。     

基于D-S理论的靶压幅度不确定性量化方法

 针对关键参数测试样本数有限的情况下,概率理论、区间分析等方法在对输出靶压幅度进行不确定性定量评价时存在局限性和不合理性,将D-S理论引入到靶压幅度的不确定性量化中,根据小子样测试信息得出不确定性参数的基本信任分配,以信任函数和似然函数构造靶压幅度的上下界概率分布,并以Monte Carlo方法求解。实验和仿真得出了靶压幅度的近似概率分布、置信区间及期望值分布区间等信息,并表明：与传统的概率方法相比,该方法避免了根据小样本测试信息构造概率分布的难题;与区间分析方法相比,该方法可得到更丰富的信息。     

A least-squares approximation of partial differential equations with high-dimensional random inputs

Uncertainty quantification schemes based on stochastic Galerkin projections, with global or local basis functions, and also stochastic collocation methods in their conventional form, suffer from the so called curse of dimensionality: the associated computational cost grows exponentially as a function of the number of random variables defining the underlying probability space of the problem. In this paper, to overcome the curse of dimensionality, a low-rank separated approximation of the solution of a stochastic partial differential (SPDE) with high-dimensional random input data is obtained using an alternating least-squares (ALS) scheme. It will be shown that, in theory, the computational cost of the proposed algorithm grows linearly with respect to the dimension of the underlying probability space of the system. For the case of an elliptic SPDE, an a priori error analysis of the algorithm is derived. Finally, different aspects of the proposed methodology are explored through its application to some numerical experiments.     

Measuring Uncertainty in the Negation Evidence for Multi-Source Information Fusion

Dempster–Shafer evidence theory is widely used in modeling and reasoning uncertain information in real applications. Recently, a new perspective of modeling uncertain information with the negation of evidence was proposed and has attracted a lot of attention. Both the basic probability assignment (BPA) and the negation of BPA in the evidence theory framework can model and reason uncertain information. However, how to address the uncertainty in the negation information modeled as the negation of BPA is still an open issue. Inspired by the uncertainty measures in Dempster–Shafer evidence theory, a method of measuring the uncertainty in the negation evidence is proposed. The belief entropy named Deng entropy, which has attracted a lot of attention among researchers, is adopted and improved for measuring the uncertainty of negation evidence. The proposed measure is defined based on the negation function of BPA and can quantify the uncertainty of the negation evidence. In addition, an improved method of multi-source information fusion considering uncertainty quantification in the negation evidence with the new measure is proposed. Experimental results on a numerical example and a fault diagnosis problem verify the rationality and effectiveness of the proposed method in measuring and fusing uncertain information.     

认知不确定二维声场的响应特性数值分析

针对声学参数存在认知不确定性的问题,为实现认知不确定声场声压响应的预测。提出了解决二维认知不确定声场的有限元法(Evidence Theory-based Finite Element Method,ETFEM),引入证据理论,采用焦元和基本可信度的概念来描述认知不确定参数,基于摄动法的区间分析技术,推导了认知不确定声场声压响应的标准差、期望求解公式。为验证本文方法的可行性。以认知不确定参数下的二维管道声场模型和某轿车二维声腔模型为例进行了数值计算,对比离散随机变量得到认知不确定参数的声场分析结果和相应的随机声场所得分析结果,研究表明:该方法能够有效的处理认知不确定参数下的二维声场,为工程问题中噪声预测提供可靠的分析模型。     

A collocation reliability analysis method for probabilistic and fuzzy mixed variables

The main bottleneck of the reliability analysis of structures with aleatory and epistemic uncertainties is the contradiction between the accuracy requirement and computational efforts.Existing methods are either computationally unaffordable or with limited application scope.Accordingly,a new technique for capturing the minimal and maximal point vectors instead of the extremum of the function is developed and thus a novel reliability analysis method for probabilistic and fuzzy mixed variables is proposed.The fuzziness propagation in the random reliability,which is the focus of the present study,is performed by the proposed method,in which the minimal and maximal point vectors of the structural random reliability with respect to fuzzy variables are calculated dimension by dimension based on the Chebyshev orthogonal polynomial approximation.First-Order,Second-Moment(FOSM)method which can be replaced by its counterparts is utilized to calculate the structural random reliability.Both the accuracy and efficiency of the proposed method are validated by numerical examples and engineering applications introduced in the paper.     

Numerical analysis of the 2D acoustic field with epistemic uncertainty

Aiming at the problem that the epistemic uncertain parameters exist in an acoustic field,an evidence theory-based finite element method(ETFEM) is proposed by introducing the evidence theory,in which the focal element and basic probability assignment(BPA) are used to describe the epistemic uncertainty.In order to reduce the computational cost,the interval analysis technique based on perturbation method is adopted to acquire the approximate sound pressure response bounds for each focal element.The corresponding formulations of intervals of expectation and standard deviation of the sound pressure response with epistemic uncertainty are deduced.The sound pressure response of a 2D acoustic tube and a 2D car acoustic cavity with epistemic uncertain parameters are analyzed by the proposed method.The proposed method is verified through the comparison of the analysis results of random acoustic field with that of epistemic uncertain acoustic field.Numerical analysis results show that the proposed method can analyze the 2D acoustic field with epistemic uncertainty effectively,and has good prospect of engineering application.     

Direct Computation of Stochastic Flow in Reservoirs with Uncertain Parameters

A direct method is presented for determining the uncertainty in reservoir pressure, flow, and net present value (NPV) using the time-dependent, one phase, two- or three-dimensional equations of flow through a porous medium. The uncertainty in the solution is modelled as a probability distribution function and is computed from given statistical data for input parameters such as permeability. The method generates an expansion for the mean of the pressure about a deterministic solution to the system equations using a perturbation to the mean of the input parameters. Hierarchical equations that define approximations to the mean solution at each point and to the field covariance of the pressure are developed and solved numerically. The procedure is then used to find the statistics of the flow and the risked value of the field, defined by the NPV, for a given development scenario. This method involves only one (albeit complicated) solution of the equations and contrasts with the more usual Monte-Carlo approach where many such solutions are required. The procedure is applied easily to other physical systems modelled by linear or nonlinear partial differential equations with uncertain data.     

Mathematical Methods of Subjective Modeling in Scientific Research. 2: Applications

This article considers applications of the formalism of subjective modeling proposed in [36], based on modeling of uncertainty reflecting unreliability of subjective information and fuzziness common of its content. A subjective model of probabilistic randomness is defined and studied. It is shown that a researcher–modeler (RM) defines a subjective model of a discrete probability space as a space with plausibility and believability that de facto turns out to be a subjective model of the class of subjectively equivalent probability spaces that model an arbitrary evolving stochastic object, and the same space with plausibility and believability is its subjective model. This enables us to empirically recover a subjective model of an evolving stochastic object accurately and using a finite number of event observations, while its probabilistic model cannot be empirically recovered. A similar connection is established between equivalence classes of plausibility and believability distributions and classes of subjectively equivalent absolutely continuous probability densities. For two versions of plausibility and believability measures, entropies of plausibility and believability distributions of the values of an uncertain element (UCE) \(\tilde x\)that model RM’s subjective judgments as characteristics of the information content and uncertainty of his judgments are considered. It is shown that in the first version entropies have properties that are formally similar to those of Shannon entropy but due to absence of the law of large numbers (LLN) their interpretation fundamentally differs from the interpretation of Shannon entropy. In the third version there is an analog of the LLN, and its connection to the Shannon entropy was obtained for the expected value of subjective informational content/uncertainty. A subjective model M(\(\tilde x\))=(Ω,3(Ω), P ^ζ,? (·,·;\(\tilde x\)), N ^ζ,? (·,·;\(\tilde x\)) of an uncertain fuzzy element is considered, and an optimal subjective rule of identification of its states using observation data is obtained and studied. Methods of expert-aided reconstruction of fuzzy and uncertain fuzzy element models are also considered.     

基于多项式混沌方法的场线耦合响应不确定度量化

在传输线场线耦合的计算中,由于辐射场可能从不同方向入射,入射方位角和入射仰角会在一定范围内变化,可以将其看作不确定变量,因此传输线的响应也呈现出不确定性。针对输入参数服从非典型分布的情况,应用多项式混沌方法对传输线场线耦合频域响应进行不确定度量化。结合入射方位角和仰角的物理意义,给出其服从的分布类型并构建相应的正交多项式基底,并对该模型的传输线方程进行多项式混沌展开。最后结合含两个不确定参数的传输线场线耦合算例,给出远端电流响应的统计信息,对比蒙特卡罗方法,验证了该方法的正确性和高效性。     

A probabilistic study of the influence of parameter uncertainty on solutions of the radiative transfer equation

The influence of uncertainty in the absorption and scattering coefficients on the solution and associated parameters of the radiative transfer equation is studied using polynomial chaos theory. The uncertainty is defined by means of uniform and log-uniform probability distributions. By expanding the radiation intensity in a series of polynomial chaos functions we may reduce the stochastic transfer equation to a set of coupled deterministic equations, analogous to those that arise in multigroup neutron transport theory, with the effective multigroup transfer scattering coefficients containing information about the uncertainty. This procedure enables existing transport theory computer codes to be used, with little modification, to solve the problem. Applications are made to a transmission problem and a constant source problem in a slab. In addition, we also study the rod model for which exact analytical solutions are readily available. In all cases, numerical results in the form of mean, variance and sensitivity are given that illustrate how absorption and scattering coefficient uncertainty influences the solution of the radiative transfer equation.     

New evolution equations for the joint response-excitation probability density function of stochastic solutions to first-order nonlinear PDEs

By using functional integral methods we determine new evolution equations satisfied by the joint response-excitation probability density function (PDF) associated with the stochastic solution to first-order nonlinear partial differential equations (PDEs). The theory is presented for both fully nonlinear and for quasilinear scalar PDEs subject to random boundary conditions, random initial conditions or random forcing terms. Particular applications are discussed for the classical linear and nonlinear advection equations and for the advection–reaction equation. By using a Fourier–Galerkin spectral method we obtain numerical solutions of the proposed response-excitation PDF equations. These numerical solutions are compared against those obtained by using more conventional statistical approaches such as probabilistic collocation and multi-element probabilistic collocation methods. It is found that the response-excitation approach yields accurate predictions of the statistical properties of the system. In addition, it allows to directly ascertain the tails of probabilistic distributions, thus facilitating the assessment of rare events and associated risks. The computational cost of the response-excitation method is order magnitudes smaller than the one of more conventional statistical approaches if the PDE is subject to high-dimensional random boundary or initial conditions. The question of high-dimensionality for evolution equations involving multidimensional joint response-excitation PDFs is also addressed.     

A Decision Probability Transformation Method Based on the Neural Network

When the Dempster–Shafer evidence theory is applied to the field of information fusion, how to reasonably transform the basic probability assignment (BPA) into probability to improve decision-making efficiency has been a key challenge. To address this challenge, this paper proposes an efficient probability transformation method based on neural network to achieve the transformation from the BPA to the probabilistic decision. First, a neural network is constructed based on the BPA of propositions in the mass function. Next, the average information content and the interval information content are used to quantify the information contained in each proposition subset and combined to construct the weighting function with parameter r. Then, the BPA of the input layer and the bias units are allocated to the proposition subset in each hidden layer according to the weight factors until the probability of each single-element proposition with the variable is output. Finally, the parameter r and the optimal transform results are obtained under the premise of maximizing the probabilistic information content. The proposed method satisfies the consistency of the upper and lower boundaries of each proposition. Extensive examples and a practical application show that, compared with the other methods, the proposed method not only has higher applicability, but also has lower uncertainty regarding the transformation result information.     

概率盒框架下混合认知不确定声场的响应预测

针对含概率盒-证据混合认知不确定参数声场的响应预测问题,提出了一种概率盒框架下的改进区间蒙特卡洛方法。该方法首先将混合认知不确定参数转换为纯概率盒形式,然后结合有限元方法推导出混合认知不确定声场的盖根鲍尔多项式代理模型,再采用蒙特卡洛方法求解代理模型得到声压响应。以含概率盒-证据混合认知不确定参数的二维管道声场模型和卡车乘客舱声腔模型为例,计算结果表明混合认知不确定参数影响下的声压响应为概率盒形式,其包括声压响应极值和相应的概率信息,并且所提方法较常规混合离散方法效率更优,较基于一阶摄动法的区间蒙特卡洛方法准确性更高。研究结果表明:所提方法可以有效预测混合认知不确定声场的声压响应,并可进行声学性能的风险和保守估计。     

Uncertainty quantification in the catalytic partial oxidation of methane

This work focuses on uncertainty quantification of eight random parameters required as input for 1D modelling of methane catalytic partial oxidation within a highly dense foam reactor. Parameters related to geometrical properties, reactor thermophysics and catalyst loading are taken as uncertain. A widely applied 1D heterogeneous mathematical model that accounts for proper transport and surface chemistry steps is considered for the evaluation of deterministic samples. The non-intrusive spectral projection approach based on polynomial chaos expansion is applied to determine the stochastic temperature and species profiles along the reactor axial direction as well as their ensemble mean and error bars with a confidence interval of 95%. Probability density functions of relevant variables in specific reactor sections are also analysed. A different contribution is noticed from each random input to the total uncertainty range. Porosity, specific surface area and catalyst loading appear as the major sources of uncertainty to bulk gas and surface temperature and species molar profiles. Porosity and the mean pore diameter have an important impact on the pressure drop along the whole reactor as expected. It is also concluded that any trace of uncertainty in the eight input random variables can be almost dissipated near the catalyst outlet section for a long-enough catalyst, mainly due to the approximation to thermodynamic equilibrium.