Surrogate-assisted Bayesian inference inverse material identification method and application to advanced high strength steel

Proper definition of certain material properties is a paramount issue for accurate simulation. However, the values of a material parameter are commonly uncertain due to multiple factors in practice. To obtain reliable material parameters, parameter identification via Bayesian theory has become an attractive framework and received more attention recently. Based on this frame, the determination of likelihood function is critical for posterior probability. Unfortunately, it is commonly difficult to be determined directly, especially for complex engineering problems. In this study, Bayesian formulas for material parameter identification are given. To make it feasible for real engineering problems, the least square-support vector regression surrogate and Monte Carlo Simulation are integrated to obtain the maximum likelihood estimation of likelihood function. The uncertainty of parameter identification is quantified via the Bayesian method. In two benchmarks, two cases with single and multiple uncertainty sources are used to propagate and quantify uncertainties in material parameters based on Bayesian approach. Moreover, the proposed method is used to identify the material parameters of advanced high strength steel used in vehicle successfully.     

A Monte Carlo method for calculating Bayesian uncertainties in internal dosimetry

This paper presents a novel Monte Carlo method (WeLMoS, Weighted Likelihood Monte-Carlo sampling method) that has been developed to perform Bayesian analyses of monitoring data. The WeLMoS method randomly samples parameters from continuous prior probability distributions and then weights each vector by its likelihood (i.e. its goodness of fit to the measurement data). Furthermore, in order to quality assure the method, and assess its strengths and weaknesses, a second method (MCMC, Markov chain Monte Carlo) has also been developed. The MCMC method uses the Metropolis algorithm to sample directly from the posterior distribution of parameters. The methods are evaluated and compared using an artificially generated case involving an exposure to a plutonium nitrate aerosol. In addition to calculating the uncertainty on internal dose, the methods can also calculate the probability distribution of model parameter values given the observed data. In other words, the techniques provide a powerful tool to obtain the estimates of parameter values that best fit the data and the associated uncertainty on these estimates. Current applications of the methodology, including the determination of lung solubility parameters, from volunteer and cohort data, are also discussed.     

Accounting for Parameter Uncertainty in Simulation Input Modeling

We formulate and evaluate a Bayesian approach to probabilistic input modeling for simulation experiments that accounts for the parameter and stochastic uncertainties inherent in most simulations and that yields valid predictive inferences about outputs of interest. We use prior information to construct prior distributions on the parameters of the input processes driving the simulation. Using Bayes' rule, we combine this prior information with the likelihood function of sample data observed on the input processes to compute the posterior parameter distributions. In our Bayesian simulation replication algorithm, we estimate parameter uncertainty by independently sampling new values of the input-model parameters from their posterior distributions on selected simulation runs; and we estimate stochastic uncertainty by performing multiple (conditionally) independent runs with each set of parameter values. We formulate performance measures relevant to both Bayesian and frequentist input-modeling techniques, and we summarize an experimental performance evaluation demonstrating the advantages of the Bayesian approach.     

Bayesian analysis for estimating statistical parameter distributions of elasto-viscoplastic material models

High temperature design methods rely on constitutive models for inelastic deformation and failure typically calibrated against the mean of experimental data without considering the associated scatter. Variability may arise from the experimental data acquisition process, from heat-to-heat material property variations, or both and need to be accurately captured to predict parameter bounds leading to efficient component design. Applying the Bayesian Markov Chain Monte Carlo (MCMC) method to produce statistical models capturing the underlying uncertainty in the experimental data is an area of ongoing research interest. This work varies aspects of the Bayesian MCMC method and explores their effect on the posterior parameter distributions for a uniaxial elasto-viscoplastic damage model using synthetically generated reference data. From our analysis with the uniaxial inelastic model we determine that an informed prior distribution including different types of test conditions results in more accurate posterior parameter distributions. The parameter posterior distributions, however, do not improve when increasing the number of similar experimental data. Additionally, changing the amount of scatter in the data affects the quality of the posterior distributions, especially for the less sensitive model parameters. Moreover, we perform a sensitivity study of the model parameters against the likelihood function prior to the Bayesian analysis. The results of the sensitivity analysis help to determine the reliability of the posterior distributions and reduce the dimensionality of the problem by fixing the insensitive parameters. The comprehensive study described in this work demonstrates how to efficiently apply the Bayesian MCMC methodology to capture parameter uncertainties in high temperature inelastic material models. Quantifying these uncertainties in inelastic models will improve high temperature engineering design practices and lead to safer, more effective component designs.     

An Efficient Sampling Technique for Bayesian Inference With Computationally Demanding Models

In the environmental sciences, a large knowledge base is typically available on an investigated system or at least on similar systems. This makes the application of Bayesian inference techniques in environmental modeling very promising. However, environmental systems are often described by complex, computationally demanding simulation models. This strongly limits the application of Bayesian inference techniques, because numerical implementation of these techniques requires a very large number of simulation runs. The development of efficient sampling techniques that attempt to approximate the posterior distribution with a relatively small parameter sample can extend the range of applicability of Bayesian inference techniques to such models. In this article a sampling technique is presented that tries to achieve this goal. The proposed technique combines numerical techniques typically applied in Bayesian inference, including posterior maximization, local normal approximation, and importance sampling, with copula techniques for the construction of a multivariate distribution with given marginals and correlation structure and with low-discrepancy sampling. This combination improves the approximation of the posterior distribution by the sampling distribution and improves the accuracy of results for small sample sizes. The usefulness of the proposed technique is demonstrated for a simple model that contains the major elements of models used in the environmental sciences. The results indicate that the proposed technique outperforms conventional techniques (random sampling from simpler distributions or Markov chain Monte Carlo techniques) in cases in which the analysis can be limited to a relatively small number of parameters.     

1 on 1损伤阈值测量结果不确定度分析

针对ISO 21254中1 on1损伤阈值测试结果的测量不确定度评估需求,系统性分析测量结果不确定度的主要影响因素.采用公式法分析评定能量密度的测量不确定度以及损伤概率的测量不确定度分量.采用蒙特卡洛方法分析了线性拟合引入的测量不确定度,并对比分析了线性拟合过程中不同残差模型对损伤概率、能量密度线性拟合结果的影响.分析...     

基于Gibbs抽样的马尔科夫蒙特卡罗方法在结构物理参数识别及损伤定位中的研究

通过对结构动力特征方程进行的一系列变化,得到了线性结构识别模型,并由贝叶斯更新理论得到其后验分布形式.利用结构的模态参数,并考虑其随机性,应用基于Gibbs抽样的马尔科夫蒙特卡罗方法对线性结构识别模型中各参数的条件后验分布进行了抽样,成功地实现了结构物理参数识别及损伤定位.数值算例表明:Gibbs抽样结果可以以不同的方...     

三水源新安江模型参数不确定性分析PAM算法

针对水文模型参数不确定性分析常用方法 收敛速度缓慢,容易陷入参数空间局部最优区域等 问题,提出了PAM (parallel adaptive metropolis) 算法;对三水源新安江模型参数不确定性进行分析 研究。实例研究表明显著提高了计算速度和求解质 量,参数后验分布结果为区间预报提供了条件。     

Model-based process optimization in the presence of parameter uncertainty

In model-based process optimization one uses a mathematical model to optimize a certain criterion, for example the product yield of a chemical process. Models often contain parameters that have to be estimated from data. Typically, a point estimate (e.g. the least squares estimate) is used to fix the model for the optimization stage. However, parameter estimates are uncertain due to incomplete and noisy data. In this article, it is shown how parameter uncertainty can be taken into account in process optimization. To quantify the uncertainty, Markov Chain Monte Carlo (MCMC) sampling, an emerging standard approach in Bayesian estimation, is used. In the Bayesian approach, the solution to the parameter estimation problem is given as a distribution, and the optimization criteria are functions of that distribution. The formulation and implementation of the optimization is studied, and numerical examples are used to show that parameter uncertainty can have a large effect in optimization results.     

A fast Bayesian approach using adaptive densifying approximation technique accelerated MCMC

A novel algorithm is presented in this study to improve the efficiency and accuracy of Bayesian approach for fast sampling of posterior distributions of the unknown structure parameters. This algorithm can save a computational cost by resolving the efficiency problem in Bayesian identifications. In this algorithm, an approximation model based on the radial basis function is first used to replace the actual joint posterior distribution of the unknown parameters. An adaptive densifying technique is then introduced to increase the accuracy of the approximation model by reconstructing them with densified samples. Finally, the marginal posterior distributions for each parameter with fine accuracy can be efficiently achieved using the Markov Chain Monte Carlo method based on the present densified approximation model. Two numerical examples are investigated to demonstrate that the present algorithm can achieve significant computational gains without sacrificing the accuracy.     

Sequential Designs Based on Bayesian Uncertainty Quantification in Sparse Representation Surrogate Modeling

A numerical method, called overcomplete basis surrogate method (OBSM), was recently proposed, which employs overcomplete basis functions to achieve sparse representations. While the method can handle nonstationary response without the need of inverting large covariance matrices, it lacks the capability to quantify uncertainty in predictions. We address this issue by proposing a Bayesian approach that first imposes a normal prior on the large space of linear coefficients, then applies the Markov chain Monte Carlo (MCMC) algorithm to generate posterior samples for predictions. From these samples, Bayesian credible intervals can then be obtained to assess prediction uncertainty. A key application for the proposed method is the efficient construction of sequential designs. Several sequential design procedures with different infill criteria are proposed based on the generated posterior samples. Numerical studies show that the proposed schemes are capable of solving problems of positive point identification, optimization, and surrogate fitting.     

Using Markov Chain Monte Carlo methods to solve full Bayesian modeling of PWR vessel flaw distributions

We present a hierarchical Bayesian method for estimating the density and size distribution of subclad-flaws in French Pressurized Water Reactor (PWR) vessels. This model takes into account in-service inspection (ISI) data, a flaw size-dependent probability of detection (different functions are considered) with a threshold of detection, and a flaw sizing error distribution (different distributions are considered). The resulting model is identified through a Markov Chain Monte Carlo (MCMC) algorithm. The article includes discussion for choosing the prior distribution parameters and an illustrative application is presented highlighting the model's ability to provide good parameter estimates even when a small number of flaws are observed.     

结合近似贝叶斯计算和改进群体蒙特卡洛抽样的结构损伤识别

为避免陷入低概率区抽样并提高抽样效率,改进了群体蒙特卡洛(PMC)抽样算法,再结合近似贝叶斯计算(ABC)和随机响应面(SRS)提出一种概率损伤识别方法。首先将ABC和改进PMC算法进行嵌套,利用每个迭代步的样本方差来搅动粒子群和求取自适应权重系数,再构造衡量仿真和实测样本间相似度的误差函数,用于替代似然函数;然后使用SRS建立结构随机响应的显式表达式,大幅提高响应统计特征值的计算效率;最后将求得的参数后验概率分布统计特征值作为损伤指标,根据损伤前后指标值的变化来判断损伤位置和程度。对试验钢筋混凝土梁的单、多工况损伤进行了识别,验证了所提出方法在保证参数后验分布估计精度的条件下,可以有效提高贝叶斯推断过程的计算效率。     

Inference for dependence competing risks with partially observed failure causes from bivariate Gompertz distribution under generalized progressive hybrid censoring

Competing risks model is considered with dependence causes of failure in this paper. When the latent failure times are distributed by a bivariate Gompertz model, statistical inference for the unknown model parameters is studied from classical and Bayesian approaches, respectively. Under a generalized progressive hybrid censoring, maximum likelihood estimators of the unknown parameters together with the associated existence and uniqueness are established, and the approximate confidence intervals are also obtained based on asymptotic likelihood theory via the observed Fisher information matrix. Moreover, Bayes estimates and the highest posterior density credible intervals of the unknown parameters are also provided based on a flexible Gamma–Dirichlet prior, and Monte Carlo sampling method is also derived to compute associated estimates. Finally, simulation studies and a real-life example are given for illustration purposes.     

基于多链差分进化的贝叶斯有限元模型修正方法

针对传统贝叶斯算法在高维参数下采样效率低且收敛难的问题,建立了基于多链差分进化算法的贝叶斯有限元模型修正方法。在标准马尔可夫链蒙特卡罗（MCMC）方法的基础上,引入差分进化算法,通过多条马氏链间的随机差分运算来自适应选择条件分布的大小和方向以快速逼近目标分布;引入子空间采样算法,通过自适应选择优良的参数维度进行采样以提高采样效率;引入异常链检测算法,通过在采样的非平稳期对马氏链进行异常检测与剔除以提高在平稳期的采样效率。简支梁理论模型和实验室4层框架结构的模型修正结果表明:该方法修正精度较高,且具有良好的抗噪性,在高阶频率以及振型下的修正效果均优于DRAM算法,为解决不确定性模型修正中的计算精度提供了一种新手段。     

Circular prediction regions for miss distance models under heteroskedasticity

Circular prediction regions are used in ballistic testing to express the uncertainty in shot accuracy. We compare two modeling approaches for estimating circular prediction regions for the miss distance of a ballistic projectile. The miss distance response variable is bivariate normal and has a mean and variance that can change with one or more experimental factors. The first approach fits a heteroskedastic linear model using restricted maximum likelihood, and uses the Kenward-Roger statistic to estimate circular prediction regions. The second approach fits an analogous Bayesian model with unrestricted likelihood modifications, and computes circular prediction regions by sampling from the posterior predictive distribution. The two approaches are applied to an example problem, and are compared using simulation.     

计量中回归模型参数值及其不确定度评估

针对计量领域中广泛应用的数据回归处理方法,阐述了在基于正态分布噪声条件下,最小二乘法与贝叶斯推断法用于回归模型参数估计以及相应不确定度评估的过程。GUM系列不确定度评估准则中没有明确指出如何对回归参数进行不确定度评估,同时有些回归模型也无法唯一地转化为相应的测量方程。通过计量校准的实例说明了如何处理相应参数的确定等问题,以此说明2种方法的相同与不同之处。最小二乘方法计算简单直接且便于使用;而基于贝叶斯推断的方法则能充分利用计量校准中的经验和历史数据等信息,但由于其参数后验分布计算通常较为复杂,需采用马尔科夫链-蒙特卡罗(MCMC)法通过数值计算得到关注参数的结果。     

Sensitivity estimations for Bayesian inference models solved by MCMC methods

The advent of Markov Chain Monte Carlo (MCMC) methods to simulate posterior distributions has virtually revolutionized the practice of Bayesian statistics. Unfortunately, sensitivity analysis in MCMC methods is a difficult task. In this paper, a computationally low-cost method to estimate local parametric sensitivities in Bayesian models is proposed. The sensitivity measure considered here is the gradient vector of a posterior quantity with respect to the parameter. The gradient vector components are estimated by using a result based on the integral/derivative interchange. The MCMC simulations used to estimate the posterior quantity can be re-used to estimate the sensitivity measures and their errors, avoiding the need for further sampling. The proposed method is easy to apply in practice as it is shown with an illustrative example.     

Bayesian inference in probabilistic risk assessment—The current state of the art

Markov chain Monte Carlo (MCMC) approaches to sampling directly from the joint posterior distribution of aleatory model parameters have led to tremendous advances in Bayesian inference capability in a wide variety of fields, including probabilistic risk analysis. The advent of freely available software coupled with inexpensive computing power has catalyzed this advance. This paper examines where the risk assessment community is with respect to implementing modern computational-based Bayesian approaches to inference. Through a series of examples in different topical areas, it introduces salient concepts and illustrates the practical application of Bayesian inference via MCMC sampling to a variety of important problems.     

A Wiener‐based degradation model with logistic distributed measurement errors and remaining useful life estimation

Degradation data analysis, which investigates degradation processes of products to extrapolate the lifetime properties, is an effective method for reliability analysis. But degradation data that reflect a product's inherent randomness of degradation are often contaminated by measurement errors. To deal with the problem, this paper proposes a Wiener‐based model with an assumption of logistic distributed measurement errors and adopts the Monte Carlo expectation‐maximization method together with the Gibbs sampling for parameter estimation. Based on the model and parameter estimates, an efficient algorithm is proposed for a quick calculation of maximum likelihood value. Also, the estimation of remaining useful lifetime is discussed. Simulation results show that the proposed model is relatively better and more robust in comparison with the Wiener process with Gaussian noises. Finally, the application of the proposed model is illustrated by an example.