Abstract: . Flood disaster mitigation strategies should be based on a comprehensive assessment of the flood risk combined with a thorough investigation of the uncertainties associated with the risk assessment procedure. Within the "German Research Network of Natural Disasters" (DFNK) the working group "Flood Risk Analysis" investigated the flood process chain from precipitation, runoff generation and concentration in the catchment, flood routing in the river network, possible failure of flood protection measures, inundation to economic damage. The working group represented each of these processes by deterministic, spatially distributed models at different scales. While these models provide the necessary understanding of the flood process chain, they are not suitable for risk and uncertainty analyses due to their complex nature and high CPU-time demand. We have therefore developed a stochastic flood risk model consisting of simplified model components associated with the components of the process chain. We parameterised these model components based on the results of the complex deterministic models and used them for the risk and uncertainty analysis in a Monte Carlo framework. The Monte Carlo framework is hierarchically structured in two layers representing two different sources of uncertainty, aleatory uncertainty (due to natural and anthropogenic variability) and epistemic uncertainty (due to incomplete knowledge of the system). The model allows us to calculate probabilities of occurrence for events of different magnitudes along with the expected economic damage in a target area in the first layer of the Monte Carlo framework, i.e. to assess the economic risks, and to derive uncertainty bounds associated with these risks in the second layer. It is also possible to identify the contributions of individual sources of uncertainty to the overall uncertainty. It could be shown that the uncertainty caused by epistemic sources significantly alters the results obtained with aleatory uncertainty alone. The model was applied to reaches of the river Rhine downstream of Cologne.

Abstract: A common way to account for uncertainty in inverse problems is to apply Bayes' rule and obtain a posterior distribution of the quantities of interest given a set of measurements. A conventional Bayesian treatment, however, requires assuming specific values for parameters of the prior distribution and of the distribution of the measurement errors (e.g., the standard deviation of the errors). In practice, these parameters are often poorly known a priori, and choosing a particular value is often problematic. Moreover, the posterior uncertainty is computed assuming that these parameters are fixed; if they are not well known a priori, the posterior uncertainties have dubious value.This paper describes extensions to the conventional Bayesian treatment that assign uncertainty to the parameters defining the prior distribution and the distribution of the measurement errors. These extensions are known in the statistical literature as “empirical Bayes” and “hierarchical Bayes.” We demonstrate the practical applicati...

Abstract: Static and dynamic aeroelasticity considerations are a particularly important component of airframe design because they often control safety and performance. Consequently, the impact of uncertainty on aeroelastic response prediction has begun to receive substantial attention in the research literature. In this paper, general sources of uncertainty that complicate airframe design and testing are briefly described. Recent applications of uncertainty quantification to various aeroelastic problems, for example, flutter flight testing, prediction of limit-cycle oscillations, and design optimization with aeroelastic constraints, are reviewed with an emphasis on new physical insights and promising paths toward improved design methods that have resulted from these studies. Several challenges and needs are explored to suggest future steps that will enable practical application of uncertainty quantification in aeroelasticity design and certification.

Abstract: This study compares two techniques for uncertainty quantification in chemistry computations, one based on sensitivity analysis and error propagation, and the other on stochastic analysis using polynomial chaos techniques. The two constructions are studied in the context of H{sub 2}-O{sub 2} ignition under supercritical-water conditions. They are compared in terms of their prediction of uncertainty in species concentrations and the sensitivity of selected species concentrations to given parameters. The formulation is extended to one-dimensional reacting-flow simulations. The computations are used to study sensitivities to both reaction rate pre-exponentials and enthalpies, and to examine how this information must be evaluated in light of known, inherent parametric uncertainties in simulation parameters. The results indicate that polynomial chaos methods provide similar first-order information to conventional sensitivity analysis, while preserving higher-order information that is needed for accurate uncertainty quantification and for assigning confidence intervals on sensitivity coefficients. These higher-order effects can be significant, as the analysis reveals substantial uncertainties in the sensitivity coefficients themselves.

Abstract: The random character of blade mistuning is a motivation to construct probability models of random uncertainties. Recently, a new approach known as a nonparametric model of random uncertainties, based on the entropy optimization principle, was introduced for modeling random uncertainties in linear and nonlinear elastodynamics. This paper presents an extension of this nonparametric model for vibration analysis of structures with cyclic geometry. In particular this probability model allows the blade eigenfrequencies uncertainties and the blade-modal-shape uncertainties to be modeled.

Abstract: The paper deals with an experimental validation of a nonparametric probabilistic model of nonhomogeneous uncertainties for dynamical systems. The theory used, recently introduced, allows model uncertainties and data uncertainties to be simultaneously taken into account. An experiment devoted to this validation was specifically developed. The experimental model is constituted of two simple dural rectangular plates connected together with a complex joint. In the mean mechanical model, the complex joint, which is constituted of two additional plates attached with 40 screw-bolts, is modeled by a homogeneous orthotropic continuous plate with constant thickness, as usual. Consequently, the mean model introduces a region (the joint) which has a high level of uncertainties. The objective of the paper is to present the experiment and the comparisons of the theoretical prediction with the experiments.

Abstract: This paper deals with the transient response of a non-linear dynamical system with random uncertainties. The non-parametric probabilistic model of random uncertainties recently published and extended to nonlinear dynamical system analysis is used in order to model random uncertainties related to the linear part of the finite element model. The non-linearities are due to restoring forces whose parameters are uncertain and are modeled by the parametric approach. Jayne's maximum entropy principle with the constraints defined by the available information allows the probabilistic model of such random variables to be constructed. Therefore, a non-parametric-parametric formulation is developed in order to model all the sources of uncertainties in such a non-linear dynamical system. Finally, a numerical application for earthquake engineering analysis is proposed concerning a reactor cooling system under seismic loads.

Abstract: The endogenous complexity and spatial nature of the problems encountered in the urban water management environment present decision-makers with three major problems: (a) in the urban environment, every decision is site-specific, almost on a case-by-case basis, (b) the decision-maker must access, simultaneously, a large amount of information, increasing with rising spatial resolution and (c) the information to be evaluated is heterogeneous, including engineering, economical and social characteristics and constraints. The first two problems indicate that urban water management is an ideal field to develop and use spatial decision support systems (SDSS), while the latter promotes the use of fuzzy inference systems as a key mathematical framework. This research discusses the nature of uncertainty in environmental management in general and urban water management in particular, argues that fuzzy, rule-based, inference systems can be an invaluable tool for uncertainty quantification and presents the relevant elements of a prototype SDSS for urban water management. The examples presented in this paper are based on an application of the SDSS in water demand management.

Abstract: This paper discusses two practical strategies to quantify the uncertainty in production and injection forecasts for a field with a long and complex production history with poor quality measurements. These methods are applied to a large offshore field in Africa that has been on production for more than 30 years. The first method follows an advanced Experimental Design framework and requires the use of non-linear Response Surfaces such as kriging. The second method uses sensitivity coefficients; it can be considered as a first pass to evaluate uncertainty before embarking in a more comprehensive analysis. Both methods lead to multiple acceptable representations of the history of the reservoir. The range of outcomes obtained with production forecasts allows for uncertainty quantification. In this case, both methods delivered similar prediction results.

Abstract: The goal of a reservoir study is to help to decide the future development of a field based on technical and economic criteria. To reach this goal, one would like to quantify the impact of uncertainty on production and economic forecasts to take the decision while considering the risk. Practically it would correspond to supply to the manager the uncertainty distribution (or P10, 50 and 90) of the production forecasts associated to each scenario. The uncertainty on the production forecasts is linked to a specific scenario and to the knowledge of the reservoir. For a mature field two kinds of knowledge exist: - Static parameters used to build the numerical model: geological concept, variograms, correlation lengths, permeability and porosity distributions, etc. The static parameters are associated with "a priori uncertainties" defined by their probability distributions. - Dynamic data: measurements related to the dynamic behavior of the reservoir, such as measured pressure, oil/water/gas rates at the wells, 4d seismic, etc. The bayesian formalism enables to reduce, in a statistical framework, the static parameter uncertainties by taking into account the dynamic data. These "a posteriori" distributions of the static parameters can then be used to compute probabilistic production forecasts for each possible scenario honoring static and dynamic knowledge of the reservoir. However, this formalism involves the determination of the likelihood function, which can lead to a prohibitive cost in terms of reservoir simulations. To drastically decrease the amount of reservoir simulations needed to determine the "a posteriori" uncertainties we propose to approximate the likelihood function by a non-linear proxy model combining experimental design, universal kriging and dynamic training techniques. These "a posteriori" distributions can then be used into the classical experimental design approach to compute probabilistic production forecasts constrained by dynamic data. The proposed methodology will be illustrated on a field case.

Abstract: To support early design and design under risk, it is necessary to have methodologies to process the various forms of uncertainties. Three independent dimensions of uncertainty are identified in the paper as certainty of analysis (epistemic uncertainty), random variability (stochastic variability and design indecision) and type of variable. The type of variable is further categorized into six scales that are broadly grouped into quantitative and qualitative. Common engineering modelling tools used for design do not operate well on combinations of random variables, qualitative variables and imperfect knowledge. The hypothesis of this paper is that a modelling system could be developed to accommodate the multiple types of uncertainty that can exist during engineering design. This is worth doing as accommodating design uncertainty is an important part of risk management in engineering. The paper then proceeds to describe the way in which the design for system integrity (DSI) methodology meets these ob...

Abstract: Response surface experimental designs provide a framework for evaluating sensitivities and assessing uncertainties in reservoir-production forecasts for continuous parameters (i.e. permeability, flow rate, etc.). In this paper, the method is extended in order to integrate both continuous and discrete parameters (i.e. fault status: open/close, injection scheme: SWAG/WAG, etc.). This paper presents an appropriate experimental designs approach, notably the regression models associated with, and the statistical interpretation (sensitivity study, Monte Carlo simulations, etc.). The method has been successfully applied to a reservoir oil-production simulation problem. The objective was to define the best production scheme by optimizing the well-completion level. This application has highlighted the advantages of this new approach, both in terms of decreasing simulation cost and improving the interpretation quality.

Abstract: Natural porous media are highly heterogeneous and characterized by parameters that are often uncertain due to the lack of sufficient data. This uncertainty (randomness) occurs on a multiplicity of scales. We focus on geologic formations with the two dominant scales of uncertainty: a large-scale uncertainty in the spatial arrangement of geologic facies and a small-scale uncertainty in the parameters within each facies. We propose an approach that combines random domain decompositions (RDD) and polynomial chaos expansions (PCE) to account for the large- and small-scales of uncertainty, respectively. We present a general fremework and use a one-dimensional flow example to demonstrate that our combined approach provides robust, non-perturbative approximations for the statistics of the system states.

Abstract: In manufacturing processes, it is widely accepted that uncertainty plays an important role and should be taken into account during analysis and design processes. However, uncertainty quantification of its effects on an end-product is a very challenging task, especially when an expensive computational effort is already needed in deterministic models such as sheet metal forming simulations. In this paper, we focus our work on the variance estimation of the system response. A weighted three-point-based strategy is proposed to efficiently and effectively estimate the variance of the system response. Three first-order derivatives for each variable are used to estimate the nonlinear behavior and variance of the system. The details of the derivation of the approach are presented in the paper. The optimal locations of the three points along each axis in the standard normal space and weights for input variables following normal distributions are proposed as (-1.8257,0.0,+1.8257) and (0.075,0.850,0.075), respectively. For input variables following uniform distributions U(-1,1), the optimal locations and weights are proposed as (-0.84517, 0.0,+0.84517) and (0.04667,0.90666,0.04667), respectively. The proposed approach is applicable to nonlinear and multivariable systems as well as problems having no explicit function such as those design simulations based on finite element methods. The significant accuracy improvement over the traditional first-order approximation is demonstrated with a number of test problems. The proposed method requires significantly less computational effort compared with the Monte Carlo simulations. Discussions and conclusions of this work are given at the end of the paper.

Abstract: : There is a growing interest in understanding how uncertainties in flight conditions and structural parameters affect the character of a limit cycle oscillation (LCO) response, leading to failure of an aeroelastic system. Uncertainty quantification of a stochastic system (parametric uncertainty) with stochastic inputs (initial condition uncertainty) has traditionally been analyzed with Monte Carlo simulations (MCS). Probability density functions (PDF) of the LCO response are obtained from the MCS to estimate the probability of failure. A candidate approach to efficiently estimate the PDF of an LCO response is the stochastic projection method. The objective of this research is to extend the stochastic projection method to include the construction of B-spline surfaces in the stochastic domain. The multivariate B-spline problem is solved to estimate the LCO response surface. An MCS is performed on this response surface to estimate the PDF of the LCO response. The probability of failure is then computed from the PDF. This method is applied to the problem of estimating the PDF of a subcritical LCO response of a nonlinear airfoil in inviscid transonic flow. The stochastic algorithm provides a conservative estimate of the probability of failure of this aeroelastic system two orders of magnitude more efficiently than performing an MCS on the governing equations.

Abstract: Uncertainty quantification and forecasting is a problem of growing importance in reservoir production forecasting. The goal is no longer to produce a single best history matched model, but to generate multiple models, which are “good enough” to honour the production history data and at the same time include the effects of uncertainty in the model components. We have developed an uncertainty quantification framework for reservoir performance forecasting based on the Neighbourhood Algorithm (NA), which aims to provide a rigorous way to generate an ensemble of parameterised reservoir models with performance close to the observed history data. NA is a stochastic search algorithm, which uses geometrical features of Voronoi polygons in high dimensions to provide an efficient way of describing and searching multidimensional parameter space. A Gibbs sampler is used to estimate the posterior probabilities of the models allowing accurate quantification of uncertainty in prediction. We applied this framework to the PUNQ-S3 problem, which is a well-known synthetic benchmark case study for comparison and validation of uncertainty quantification methodologies. By running NA and generating an ensemble of solutions which agree with the observed data we are able to quantify the uncertainty by computing the posterior mean prediction and the posterior confidence intervals around the mean. We obtain good agreement with the true solution, with an uncertainty spread comparable with one of the other original entries. A key question for any uncertainty estimation method is to ask how good are the uncertain predictions. Just because the true solution lies within our confidence bands does not mean that we have a good method. To address this, we estimated the underlying uncertainty by running multiple stochastic realisations using the truth case model parameters. We then use this as our estimate of the true uncertainty and compute a “calibration curve” to show how close our estimate is to the true uncertainty.