Large Number Of Uncertain Parameters Research Articles

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

Bayesian optimal sensor placement for parameter estimation under modeling and input uncertainties

A Bayesian optimal sensor placement (OSP) framework for parameter estimation in nonlinear structural dynamics models is proposed, based on maximizing a utility function built from appropriate measures of information contained in the input–output response time history data. The information gain is quantified using Kullback–Leibler divergence (KL-div) between the prior and posterior distribution of the model parameters. The design variables may include the type and location of sensors. Asymptotic approximations, valid for large number of data, provide valuable insight into the measure of information. Robustness to uncertainties in nuisance (non-updatable) parameters associated with modeling and excitation uncertainties is considered by maximizing the expected information gain over all possible values of the nuisance parameters. In particular, the framework handles the case where the excitation time history is measured by installed sensors but remains unknown at the experimental design phase. Introducing stochastic excitation models, the expected information gain is taken over the large number of uncertain parameters used to model the random variability in the input time histories. Monte Carlo or sparse grid methods estimate the multidimensional probability integrals arising in the formulation. Heuristic algorithms are used to solve the optimization problem. The effectiveness of the method is demonstrated for a multi-degree of freedom (DOF) spring–mass chain system with restoring elements that exhibit hysteretic nonlinearities.

Bayesian system identification for structures considering spatial and temporal correlation

Abstract The decreasing cost and improved sensor and monitoring system technology (e.g., fiber optics and strain gauges) have led to more measurements in close proximity to each other. When using such spatially dense measurement data in Bayesian system identification strategies, the correlation in the model prediction error can become significant. The widely adopted assumption of uncorrelated Gaussian error may lead to inaccurate parameter estimation and overconfident predictions, which may lead to suboptimal decisions. This article addresses the challenges of performing Bayesian system identification for structures when large datasets are used, considering both spatial and temporal dependencies in the model uncertainty. We present an approach to efficiently evaluate the log-likelihood function, and we utilize nested sampling to compute the evidence for Bayesian model selection. The approach is first demonstrated on a synthetic case and then applied to a (measured) real-world steel bridge. The results show that the assumption of dependence in the model prediction uncertainties is decisively supported by the data. The proposed developments enable the use of large datasets and accounting for the dependency when performing Bayesian system identification, even when a relatively large number of uncertain parameters is inferred.

Reliability estimation of a Tibetan heritage timber structure with inclination in its Que-Ti joints

The reliability of a Tibetan timber frame in a heritage building in Lasha, China is studied. The frame is featured with stochastic Que-Ti joints with inclination. A new method integrating the relative entropy-based probabilistic sensitivity analysis and the Probabilistic Density Evolution Method with absorbing boundary condition is developed for this study. The contributions of stochastic parameters to the probabilistic bearing capacity of the joint can be clearly defined from the differences between the probability density functions of the bearing capacity with different parameter perturbation. Results are obtained via sensitivity analysis with engineering considerations of the heritage frame structure with a large number of uncertain parameters. The Que-Ti inclination angle, column inclination angle, the parallel-to-grain shear modulus and perpendicular-to-grain elastic modulus, which have greater influence on the performances of the heritage frame structure are identified with acceptable computation requirements. A reduced-dimensioned model of the structure with selected variables is then verified and used for the reliability estimation. The structural reliability is noted efficiently implemented with the random variables of the frame system reduced from 13 to 4. The frames in this heritage building with stochastic inclined Que-Ti joints are relatively safe comparing to the recommended reliability limit in the design code.

Machine learning-based sensitivity of steel frames with highly imbalanced and high-dimensional data

The machine learning-based feature selection approach is presented to estimate the effect of uncertainties and identify failure modes of structures that incorporate a low failure probability and high-dimensional uncertainties. As structures are designed to have few failures, a dataset classified based on the failure status becomes imbalanced, which poses a challenge for the predictive modeling of machine learning classifiers. Moreover, in order to improve the accuracy and efficiency of the model performance, it is necessary to determine the critical factors and redundant factors, especially for a large feature set. This study benchmarks the novel method for sensitivity analysis by using datasets that exacerbate the problems involved in class imbalance and large number of input features. This study investigates two planar steel frames with spatially uncorrelated properties between structural members. Geometric and material properties are considered as uncertainties, such as material yield stress, Young’s modulus, frame sway, and residual stress. Six feature importance techniques including ANOVA, mRMR, Spearman’s rank, impurity-based, permutation, and SHAP are employed to measure the feature importance and identify parameters germane to the prediction of structural failures. Logistic regression and decision tree models are trained on the important feature set, and the predictive performance is evaluated. The use of the feature importance approach for structures with a low probability of failure and a large number of uncertain parameters is validated by showing identical results with the reliability-based sensitivity study and appropriate predictive accuracy.

An improved algorithm for synthesis of heat exchanger network with a large number of uncertain parameters

This paper presents an improved method for the Mixed Integer Nonlinear Programming (MINLP) synthesis of flexible Heat Exchanger Network with a large number of uncertain parameters. Typically, such a problem is written as a multi-scenario two-stage stochastic model with recourse which is difficult to solve because the size of the model grows exponentially with the number of uncertain parameters. The exponential growth could be avoided by decomposing the model into simpler problems that are solved sequentially in a small number of scenarios.In this work, the determination of the first-stage variables (process topology and unit sizes) and the second-stage variables (operating and control variables) is decomposed into several steps where they are determined separately. This is achieved by iteratively solving smaller two-scenario MINLP and one-scenario Nonlinear Programming (NLP) subproblems. In this way, the size of the problem remains independent of the number of uncertain parameters.The innovation of this study is the introduction of correction factors for the second-stage variables into the objective function when determining the first-stage variables. In this way, better trade-offs are found that mitigate the unpleasant effect of decomposition. The synthesis of the heat exchanger network shows that the implementation of correction factors improves the optimal result by 7.6%.

An adjoint‐assisted multilevel multifidelity method for uncertainty quantification and its application to turbomachinery manufacturing variability

AbstractIn this work we propose, analyze, and demonstrate an adjoint‐based multilevel multifidelity Monte Carlo (MLMF) framework called FastUQ. The framework is based on the MLMF of Geraci et al. and uses the Inexpensive Monte Carlo (IMC) method of Ghate as low‐fidelity surrogate. The setup cost of IMC‐1 surrogate in FastUQ requires just the adjoint solution at the input mean whose computational cost is independent of the number of input uncertainties making it suitable for solving problems with a large number of uncertain parameters. We demonstrate the robustness of the method to quantify uncertainties in aerodynamic parameters due to surface variations caused by the manufacturing processes for a highly loaded turbine cascade. A stochastic model for surface variations on the cascade is proposed and optimal dimensionality reduction of model parameters is realised using goal‐based principal component analysis using adjoint sensitivities of multiple quantities of interest. The proposed method achieves a 70% reduction in computational cost in predicting the mean quantities such as total‐pressure loss and mass flow rate compared to the state‐of‐art MLMC method.

Static Response Analysis of Uncertain Structures With Large-Scale Unknown-But-Bounded Parameters

This paper proposes an interval finite element method based on function decomposition for structural static response problems with large-scale unknown-but-bounded parameters. When there is a large number of uncertain parameters, it will lead to the curse of dimensionality. The existing Taylor expansion-based methods, which is often employed to deal with large-scale uncertainty problems, need the sensitivity information of response function to uncertain parameters. However, the gradient information may be difficult to obtain for some complicated structural problems. To overcome this drawback, univariate decomposition expression (UDE) and bivariate decomposition expression (BDE) are deduced by the higher-order Taylor series expansion. The original structure function with [Formula: see text]-dimensional interval parameters is decomposed into the sum of several low-dimensional response functions by UDE or BDE, each of which has only one or two interval parameters while the other interval parameters are replaced by their midpoint values. Therefore, solving the upper and lower bounds of the [Formula: see text]-dimensional function can be converted into solving those of the one- or two-dimensional functions, which savethe calculation costs and can be easily implemented. The accuracy and efficiency of the new method are verified by three numerical examples.

Impact of Digitalization on the Way of Working and Skills Development in Hydrocarbon Production Forecasting and Project Decision Analysis

SummaryHydrocarbon (re-)development projects need to be evaluated under uncertainty. Forecasting oil and gas production needs to capture the ranges of the multitude of uncertain parameters and their impact on the forecast to maximize the value of the project for the company. Several authors showed, however, that the oil and gas industry has challenges in adequately assessing the distributions of hydrocarbon production forecasts.The methods for forecasting hydrocarbon production developed with digitalization from using analytical solutions to numerical models with an increasing number of gridblocks (“digital twins”) toward ensembles of models covering the uncertainty of the various parameters. Analytical solutions and single numerical models allow calculation of incremental production for a single case. However, neither the uncertainty of the forecasts nor the question in which the distribution of various outcomes the single model is located can be determined. Ensemble-based forecasts are able to address these questions, but they need to be able to cover a large number of uncertain parameters and the amount of data that is generated accordingly.Theory-guided data science (TGDS) approaches have recently been used to overcome these challenges. Such approaches make use of the scientific knowledge captured in numerical models to generate a sufficiently large data set to apply data science approaches. These approaches can be combined with economics to determine the desirability of a project for a company (expected utility). Quantitative decision analysis, including a value of information (VoI) calculation, can be done addressing the uncertainty range but also the risk hurdles as required by the decision-maker (DM). The next step is the development of learning agent systems (agent: autonomous, goal-directed entity that observes and acts upon an environment) that are able to cope with the large amount of data generated by sensors and to use them for conditioning models to data and use the data in decision analysis.Companies need to address the challenges of data democratization to integrate and use the available data, organizational agility, and the development of data science skills but making sure that the technical skills, which are required for the TGDS approach, are kept.

Efficient uncertainty quantification and management in the early stage design of composite applications

One of the key enablers of valued early decision-making in composite material designs is the ability to account for different aspects/scale of uncertainties within models and design processes. In order to enable well-judged decisions and to improve the trust of industrial decision-makers, measures of uncertainty, risk, and cost involved in materials are crucial. In this work, we provide i) efficient uncertainty analysis (UA) and ii) sensitivity analysis (SA) in composite structures to quantify the influence of input parameters on the output of interest accounting for the stochastic nature in multi-scale modeling with a large number of uncertain parameters. UA also provides a statistical distribution of the output of interest. The influences of different input parameters on the system responses can be estimated by conducting the global sensitivity analysis on the multi-scale models. To this end, a data-driven model approximating the relationship between the inputs and outputs is constructed by using an adaptive Sparse Polynomial Chaos Expansion (SPCE) approach. The sensitivities of the input factors on the system performances are computed analytically from the constructed data-driven model without any additional computational cost. To demonstrate the sensitivity and uncertainty management, two different test cases (composite leafspring and aircraft fuselage airframe) are considered for the stochastic structural analysis in multi-scale composite modeling. In both cases, the combined effects of multi-scale uncertainties are evaluated on the structural performances. Input parameters include the material microstructure (micro-scale uncertainties), composite layers stacking sequence (mesoscale uncertainties), and structural loading (uncertainty in macro-scale). The responses are provided in terms of their variations and probability distributions.

Uncertainty modeling and analysis of the European X-ray free electron laser cavities manufacturing process

This paper reports on comprehensive efforts on uncertainty quantification and global sensitivity analysis for accelerator cavity design. As a case study object the TESLA shaped superconducting cavities, as produced for the European X-ray Free Electron Laser (EXFEL), are selected. The choice for these cavities is explained by the available measurement data that can be leveraged to substantiate the simulation model. Each step of the manufacturing chain is documented together with the involved uncertainties. Several of these steps are mimicked on the simulation side, e.g., by introducing a random eigenvalue problem. The uncertainties are then quantified numerically and in particular the sensitivities give valuable insight into the system behavior. We also compare these findings to purely statistical studies carried out for the manufactured cavities. More advanced, adaptive, surrogate modeling techniques are adopted, which are crucial to incorporate a large number of uncertain parameters. The main contribution is the detailed comparison and fusion of measurement results for the EXFEL cavities on the one hand and simulation based uncertainty studies on the other hand. After introducing the quantities of physical interest for accelerator cavities and the Maxwell eigenvalue problem, the details on the manufacturing of the EXFEL cavities and measurements are reported. This is followed by uncertainty modeling with quantification studies.

A Perturbative Stochastic Galerkin Method for the Uncertainty Quantification of Linear Circuits

This paper presents an iterative and decoupled perturbative stochastic Galerkin (SG) method for the variability analysis of stochastic linear circuits with a large number of uncertain parameters. State-of-the-art implementations of polynomial chaos expansion and SG projection produce a large deterministic circuit that is fully coupled, thus becoming cumbersome to implement and inefficient to solve when the number of random parameters is large. In a perturbative approach, component variability is interpreted as a perturbation of its nominal value. The relaxation of the resulting equations and the application of a SG method lead to a decoupled system of equations, corresponding to a modified equivalent circuit in which each stochastic component is replaced by the nominal element equipped with a parallel current source accounting for the effect of variability. The solution of the perturbation problem is carried out in an iterative manner by suitably updating the equivalent current sources by means of Jacobi- or Gauss-Seidel strategies, until convergence is reached. A sparse implementation allows avoiding the refinement of negligible coefficients, yielding further efficiency improvement. Moreover, for time-invariant circuits, the iterations are effectively performed in post-processing after characterizing the circuit in time or frequency domain by means of a limited number of simulations. Several application examples are used to illustrate the proposed technique and highlight its performance and computational advantages.

Challenges in calibration of water distribution network: a case study of Ramnagar Elevated Service Reservoir command area in Nagpur City, India

Abstract Calibration of a hydraulic model is a challenging task as it considers the involvement of a large number of uncertain parameters. There are some parameters like length and diameter of pipes, for which fairly accurate values can be obtained. As with all hydraulic models, water demands are one of the main parameters that cause the most uncertainty in the model outputs. The calibration of the water demands is usually not feasible, which is attributable to the limited quantity of available measurements in most real water networks. However, some parameters like nodal demands and pipe roughness coefficients are estimated close to the actual values. Various types of valves are used for flow control by throttling. Hence, their setting in the field is also an important input to the model. Having more precise data helps in reducing time and results in better calibration as presented in the case study of one hydraulic zone served from Ramnagar Elevated Service Reservoir (ESR) (Zone II) in Nagpur City, Maharashtra State, India. This paper aims at presenting the complexities and challenges involved in calibration of the study area. It further describes the entire process from collection of the required data to the calibration of the network.

Multistage adaptive optimization using hybrid scenario and decision rule formulation

AbstractScenario‐based stochastic programming and linear decision rule (LDR)‐based robust optimization are prevalent methods for solving multistage adaptive optimization (MSAP) problems. In practical applications such as capacity expansion planning of chemical processes, often multiple sources of uncertainty affect the problem which introduces challenges to traditional stochastic optimization methods. While a large number of uncertain parameters exist in the problem, using scenario‐based method results in very large problem size and the solution becomes computationally expensive. In addition, when the constraints include multiplication of uncertain parameters and adaptive variables, the constraints are not linear with respect to uncertain parameters when the LDR method is used. In order to address these challenges, we propose two different hybrid methods where scenario and decision rule methods are combined to solve the MSAP problem. The article demonstrates the computational performance of the proposed hybrid methods using two chemical process planning examples.

An effective optimization-based parameterized interval analysis approach for static structural response with multiple uncertain parameters

The framework of parameterized interval analysis (PIA) and optimization and anti-optimization problems (OAP) was recently proposed for considerably reducing or eliminating the overestimation of the interval solution arising in the classical interval analysis. However, reducing the computational effort of this framework still remains a challenging task, especially for large-scale structures with a large number of uncertain parameters. In this regard, an effective approach formulated in the framework is proposed for evaluating the static structural response with multiple uncertain-but-bounded parameters. First, the PIA is used to describe uncertain input properties incorporated into the interval stiffness matrix and the interval load vector of finite element model. Subsequently, the parametric inverse of the stiffness matrix is handled using a Neumann series expansion. To efficiently solve the objective function of OAP, a robust optimization solver known as lightning attachment procedure optimization algorithm is applied in the field of IFEM for the first time. Finally, the numerical investigations on three kinds of structures concerning truss, frame and plate structures with multiple uncertain parameters are presented to demonstrate the accuracy and effectiveness of the proposed approach.

Active subspace-based dimension reduction for chemical kinetics applications with epistemic uncertainty

We focus on an efficient approach for quantification of uncertainty in complex chemical reaction networks with a large number of uncertain parameters and input conditions. Parameter dimension reduction is accomplished by computing an active subspace that predominantly captures the variability in the quantity of interest (QoI). In the present work, we compute the active subspace for a H2/O2 mechanism that involves 19 chemical reactions, using an efficient iterative strategy. The active subspace is first computed for a 19-parameter problem wherein only the uncertainty in the pre-exponents of the individual reaction rates is considered. This is followed by the analysis of a 36-dimensional case wherein the activation energies and initial conditions are also considered uncertain. In both cases, a 1-dimensional active subspace is observed to capture the uncertainty in the QoI, which indicates enormous potential for efficient statistical analysis of complex chemical systems. In addition, we explore links between active subspaces and global sensitivity analysis, and exploit these links for identification of key contributors to the variability in the model response.

Ablative thermal protection system under uncertainties including pyrolysis gas composition

Spacecrafts such as Stardust (NASA, 2006) are protected by an ablative Thermal Protection System (TPS) for their hypersonic atmospheric entry. A new generation of TPS material, called Phenolic Impregnated Carbon Ablator (PICA), has been introduced with the Stardust mission. This new generation of low density carbon-phenolic composites is now widely used in the aerospace industry. Complex heat and mass transfer phenomena coupled to phenolic pyrolysis and pyrolysis gas chemistry occur in the material during atmospheric entry. Computer programs, as the Porous material Analysis Toolbox based on OpenFoam (PATO) released open source by NASA, allow to study the material response. In this study, a non-intrusive Anchored Analysis of Variance (Anchored-ANOVA) method has been interfaced with PATO to perform low-cost sensitivity analysis on this problem featuring a large number of uncertain parameters. Then, a Polynomial-Chaos method has been employed in order to compute the statistics of some quantities of interest for the atmospheric entry of the Stardust capsule, by taking into account uncertainties on effective material properties and pyrolysis gas composition. This first study including pyrolysis gas composition uncertainties shows their key contribution to the variability of the quantities of interest.

Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

Abstract. Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable Markov chain Monte Carlo (MCMC) methods are computationally expensive. Sequential ensemble methods such as ensemble Kalman filters and particle filters provide a favorable alternative. However, ensemble Kalman filter has an assumption of Gaussianity. Ensemble transform particle filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimations in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ ensemble transform particle filter (ETPF) and ensemble transform Kalman filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of particular interest for subsurface reservoir modeling as it allows us to parameterize permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (one and five) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble, while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen–Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands knowledge of which mode to truncate.

Toward a Comprehensive and Efficient Robust Optimization Framework for (Bio)chemical Processes

Model-based design principles have received considerable attention in biotechnology and the chemical industry over the last two decades. However, parameter uncertainties of first-principle models are critical in model-based design and have led to the development of robustification concepts. Various strategies have been introduced to solve the robust optimization problem. Most approaches suffer from either unreasonable computational expense or low approximation accuracy. Moreover, they are not rigorous and do not consider robust optimization problems where parameter correlation and equality constraints exist. In this work, we propose a highly efficient framework for solving robust optimization problems with the so-called point estimation method (PEM). The PEM has a fair trade-off between computational expense and approximation accuracy and can be easily extended to problems of parameter correlations. From a statistical point of view, moment-based methods are used to approximate robust inequality and equality constraints for a robust process design. We also apply a global sensitivity analysis to further simplify robust optimization problems with a large number of uncertain parameters. We demonstrate the performance of the proposed framework with two case studies: (1) designing a heating/cooling profile for the essential part of a continuous production process; and (2) optimizing the feeding profile for a fed-batch reactor of the penicillin fermentation process. According to the derived results, the proposed framework of robust process design addresses uncertainties adequately and scales well with the number of uncertain parameters. Thus, the described robustification concept should be an ideal candidate for more complex (bio)chemical problems in model-based design.

Ancillary Services Bidding for Uncertain Bidirectional V2G Using Fuzzy Linear Programming

The operation of bidirectional V2G participating in ancillary services markets is particularly challenging due to various uncertainties. This work proposes an algorithm to optimize the uncertain operation of bidirectional V2G for an electric vehicle (EV) aggregator. The proposed algorithm maximizes the profits of the aggregator while providing additional system flexibility and low charging costs to the EV owners. The proposed algorithm considers electricity market and EV mobility uncertainties using fuzzy linear programming. These uncertainties include those of the regulation and responsive reserve prices, deployment signals, and the energy used for EV trips. Unlike other works in the literature, the proposed algorithm is capable of considering a large number of uncertain parameters without greatly impacting the problem's complexity or simulation time. Also, the algorithm does not result in conservative solutions. The simulation results show the superiority of the proposed fuzzy algorithm over its deterministic counterpart in terms of higher realized profits. Also, the proposed approach results in lower real-time penalties, which ensures its superiority to provide the EVs with the required trip energy at the required time. In addition, it results in lower battery degradation costs, which helps increase the life time of the EVs.