Uncertainty Quantification Techniques Research Articles

Validation of a Mathematical Model for Green Algae (Raphidocelis Subcapitata) Growth and Implications for a Coupled Dynamical System with Daphnia Magna

Toxicity testing in populations probes for responses in demographic variables to anthropogenic or natural chemical changes in the environment. Importantly, these tests are primarily performed on species in isolation of adjacent tropic levels in their ecosystem. The development and validation of coupled species models may aid in predicting adverse outcomes at the ecosystems level. Here, we aim to validate a model for the population dynamics of the green algae Raphidocelis subcapitata, a planktonic species that is often used as a primary food source in toxicity experiments for the fresh water crustacean Daphnia magna. We collected longitudinal data from three replicate population experiments of R. subcapitata. We used this data with statistical model comparison tests and uncertainty quantification techniques to compare the performance of four models: the Logistic model, the Bernoulli model, the Gompertz model, and a discretization of the Logistic model. Overall, our results suggest that the logistic model is the most accurate continuous model for R. subcapitata population growth. We then implement the numerical discretization showing how the continuous logistic model for algae can be coupled to a previously validated discrete-time population model for D. magna.

Composite laminate failure parameter optimization through four-point flexure experimentation and analysis

Abstract Fiber-reinforced composite materials offer light-weight solutions to many structural challenges. In the development of high-performance composite structures, a thorough understanding is required of the composite materials themselves as well as methods for the analysis and failure prediction of the relevant composite structures. However, the mechanical properties required for the complete constitutive definition of a composite material can be difficult to determine through experimentation. Therefore, efficient methods are necessary that can be used to determine which properties are relevant to the analysis of a specific structure and to establish a structure's response to a material parameter that can only be defined through estimation. The objectives of this study deal with demonstrating the potential value of sensitivity and uncertainty quantification techniques during the failure analysis of loaded composite structures; and the proposed methods are applied to the simulation of the four-point flexural characterization of a carbon fiber composite material. Utilizing a recently implemented, phenomenological orthotropic material model that is capable of predicting progressive composite damage and failure, a sensitivity analysis is completed to establish which material parameters are truly relevant to a simulation's outcome. Then, a parameter study is completed to determine the effect of the relevant material properties' expected variations on the simulated four-point flexural behavior as well as to determine the value of an unknown material property. This process demonstrates the ability to formulate accurate predictions in the absence of a rigorous material characterization effort. The presented results indicate that a sensitivity analysis and parameter study can be used to streamline the material definition process as the described flexural characterization was used for model validation.

Probabilistic fatigue damage prognosis using surrogate models trained via three-dimensional finite element analysis

Utilizing inverse uncertainty quantification techniques, structural health monitoring (SHM) can be integrated with damage progression models to form a probabilistic prediction of a structure’s remaining useful life (RUL). However, damage evolution in realistic structures is physically complex. Accurately representing this behavior requires high-fidelity models which are typically computationally prohibitive. In this paper, high-fidelity fatigue crack growth simulation times are reduced by three orders of magnitude using a model based on a set of surrogate models trained via three-dimensional finite element analysis. The developed crack growth modeling approach is experimentally validated using SHM-based damage diagnosis data. A probabilistic prediction of RUL is formed for a metallic, single-edge notch tension specimen with a fatigue crack growing under mixed-mode conditions.

Uncertainty quantification and sensitivity analysis of volcanic columns models: Results from the integral model PLUME-MoM

The behavior of plumes associated with explosive volcanic eruptions is complex and dependent on eruptive source parameters (e.g. exit velocity, gas fraction, temperature and grain-size distribution). It is also well known that the atmospheric environment interacts with volcanic plumes produced by explosive eruptions in a number of ways. The wind field can bend the plume but also affect atmospheric air entrainment into the column, enhancing its buoyancy and in some cases, preventing column collapse. In recent years, several numerical simulation tools and observational systems have investigated the action of eruption parameters and wind field on volcanic column height and column trajectory, revealing an important influence of these variables on plume behavior. In this study, we assess these dependencies using the integral model PLUME-MoM, whereby the continuous polydispersity of pyroclastic particles is described using a quadrature-based moment method, an innovative approach in volcanology well-suited for the description of the multiphase nature of magmatic mixtures. Application of formalized uncertainty quantification and sensitivity analysis techniques enables statistical exploration of the model, providing information on the extent to which uncertainty in the input or model parameters propagates to model output uncertainty. In particular, in the framework of the IAVCEI Commission on tephra hazard modeling inter-comparison study, PLUME-MoM is used to investigate the parameters exerting a major control on plume height, applying it to a weak plume scenario based on 26 January 2011 Shinmoe-dake eruptive conditions and a strong plume scenario based on the climatic phase of the 15 June 1991 Pinatubo eruption.

An overview of uncertainty quantification techniques with application to oceanic and oil‐spill simulations

AbstractWe give an overview of four different ensemble‐based techniques for uncertainty quantification and illustrate their application in the context of oil plume simulations. These techniques share the common paradigm of constructing a model proxy that efficiently captures the functional dependence of the model output on uncertain model inputs. This proxy is then used to explore the space of uncertain inputs using a large number of samples, so that reliable estimates of the model's output statistics can be calculated. Three of these techniques use polynomial chaos (PC) expansions to construct the model proxy, but they differ in their approach to determining the expansions' coefficients; the fourth technique uses Gaussian Process Regression (GPR). An integral plume model for simulating the Deepwater Horizon oil‐gas blowout provides examples for illustrating the different techniques. A Monte Carlo ensemble of 50,000 model simulations is used for gauging the performance of the different proxies. The examples illustrate how regression‐based techniques can outperform projection‐based techniques when the model output is noisy. They also demonstrate that robust uncertainty analysis can be performed at a fraction of the cost of the Monte Carlo calculation.

A hybrid differential evolution algorithm approach towards assisted history matching and uncertainty quantification for reservoir models

Abstract History matching is an important process in the reservoir model development. In the process of history matching, the most significant uncertain model parameters are identified and adjusted to get an acceptable match between the simulated production with the historical field production data. In the past decade, many population based algorithms have been applied for history matching. In this paper, a novel population based stochastic algorithm called hybrid differential evolution (HDE) is applied for the assisted history matching process. An adaptive mechanism for the control parameters is incorporated in the algorithm which automatically adjusts the control parameters according to the problem. The performance of the algorithm is tested on a 3-D reservoir model called PUNQ-S3 which is a benchmark model for the comparison of different history matching and uncertainty quantification techniques. Since history matching is an inverse problem, multiple models can give good match. So, prediction using a single history matched model involves more risk because of the parameter uncertainty. One of the methods to solve this problem is to quantify the uncertainty in the predictions. In this paper, the neighbourhood approximation Bayes (NAB) algorithm is applied to quantify the uncertainty in reservoir forecast which is a Bayesian extension of neighbourhood algorithm. The NAB algorithm quantifies the uncertainty in the predictions using multiple models generated during history matching phase and this does not require additional simulations. The main focus of this paper is to study about how HDE algorithm can be used when coupling with the NAB algorithm in predicting the true forecast with minimum uncertainty range under limited number of simulations. The influence of population size on the performance of the algorithm in history matching and forecast is analyzed. The HDE provides wide sampling of the search space and the truth case was comfortably included within the predicted confidence bounds. The results show that HDE can be used as a promising tool for assisted history matching of the reservoir models.

Symmetry breaking of azimuthal thermoacoustic modes: the UQ perspective

Since its introduction in the late 19th century, symmetry breaking has been found to play a crucial role in physics. In particular, it appears as one key phenomenon controlling hydrodynamic and acoustic instabilities in problems with rotational symmetries. A previous paper investigated its desired potential application to the control of circumferential thermoacoustic modes in one annular cavity coupled with multiple flames (Bauerheimet al.,J. Fluid Mech., vol. 760, 2014, pp. 431–465). The present paper focuses on a similar problem when symmetry breaking appears unintentionally, for example when uncertainties due to tolerances are taken into account. It yields a large uncertainty quantification (UQ) problem containing numerous uncertain parameters. To tackle this well-known ‘curse of dimensionality’, a novel UQ methodology is used. It relies on the active subspace approach to construct a reduced set of input variables. This strategy is applied on two annular cavities coupled by 19 flames to determine its modal risk factor, i.e. the probability of an azimuthal acoustic mode being unstable. Since each flame is modelled by two uncertain parameters, it leads to a large UQ problem involving 38 parameters. An acoustic network model is then derived, which yields a nonlinear dispersion relation for azimuthal modes. This nonlinear problem, subject to bifurcations, is solved quasi-analytically. Results show that the dimension of the probabilistic problem can be drastically reduced, from 38 uncertain parameters to only 3. Moreover, it is found that the three active variables are related to physical quantities, which unveils underlying phenomena controlling the stability of the two coupled cavities. The first active variable is associated with a coupling strength controlling the bifurcation of the system, while the two others correspond to a symmetry-breaking effect induced by the uncertainties. Thus, an additional destabilization effect appear caused by the non-uniform pattern of the uncertainty distribution, which breaks the initial rotating symmetry of the annular cavities. Finally, the active subspace is exploited by fitting the response surface with polynomials (linear, quadratic and cubic). By comparing accuracy and cost, results prove that 5 % error can be achieved with only 30 simulations on the reduced space, whereas 2000 are required on the complete initial space. It exemplifies that this novel UQ technique can accurately predict the risk factor of an annular configuration at low cost as well as unveil key parameters controlling the stability.

An adaptive strategy on the error of the objective functions for uncertainty-based derivative-free optimization

In this work, a strategy is developed to deal with the error affecting the objective functions in uncertainty-based optimization. We refer to the problems where the objective functions are the statistics of a quantity of interest computed by an uncertainty quantification technique that propagates some uncertainties of the input variables through the system under consideration. In real problems, the statistics are computed by a numerical method and therefore they are affected by a certain level of error, depending on the chosen accuracy. The errors on the objective function can be interpreted with the abstraction of a bounding box around the nominal estimation in the objective functions space. In addition, in some cases the uncertainty quantification methods providing the objective functions also supply the possibility of adaptive refinement to reduce the error bounding box. The novel method relies on the exchange of information between the outer loop based on the optimization algorithm and the inner uncertainty quantification loop. In particular, in the inner uncertainty quantification loop, a control is performed to decide whether a refinement of the bounding box for the current design is appropriate or not. In single-objective problems, the current bounding box is compared to the current optimal design. In multi-objective problems, the decision is based on the comparison of the error bounding box of the current design and the current Pareto front. With this strategy, fewer computations are made for clearly dominated solutions and an accurate estimate of the objective function is provided for the interesting, non-dominated solutions. The results presented in this work prove that the proposed method improves the efficiency of the global loop, while preserving the accuracy of the final Pareto front. The proposed method improves the efficiency of the uncertainty-based optimization by reducing the number of objective evaluations.The method relies on the presence of a derivative-free optimization method, and an adaptive uncertainty quantification method.The proposed method is conceived for multi-objective problems, but it applies to single-objective optimization as well.Results prove that the proposed method saves computations on the dominated solutions, and predicts with accuracy the Pareto front.

Uncertainty Quantification of Large-Eddy Spray Simulations

Two uncertainty quantification (UQ) techniques, latin-hypercube sampling (LHS) and polynomial chaos expansion (PCE), have been used in an initial UQ study to calculate the effect of boundary condition uncertainty on Large-eddy spray simulations. Liquid and vapor penetration as well as multidimensional liquid and vapor data were used as response variables. The Morris one-at-a-time (MOAT) screening method was used to identify the most important boundary conditions. The LHS and PCE methods both predict the same level of variability in the response variables, which was much larger than the corresponding experimental uncertainty. Nested grids were used in conjunction with the PCE method to examine the effects of subsets of boundary condition variables. Numerical modeling parameters had a much larger effect on the resulting spray predictions; the uncertainty in spray penetration or multidimensional spray contours from physically derived boundary conditions was close to the uncertainty of the measurements.

Analytics for understanding customer behavior in the energy and utility industry

Understanding customer behavior is becoming critical for electric grid operations. Grid operators need to model customer behavior from multiple perspectives, in part due to recent changes of the customer role from passive loads to prosumers. Customers are active agents rather than “passive loads,” and this change in behavior comes with a variety of challenges. Solving these challenges requires a 360-degree view of the customer, which calls for machine learning techniques for classification, time-series analysis, and uncertainty quantification. This allows utilities to actively work in the challenging landscape of active customer behavior. In this paper, we describe several important techniques for practical modeling of active customer behavior, together with corresponding areas of behavior modeling: energy savings potential, adoption of sustainable products and services, prediction of photovoltaic adoption, and fraud detection. For each of these applications, we briefly survey machine learning tools that have allowed demonstrable practical impact. Where possible, we illustrate results using datasets from Alliander N.V. (a Dutch energy company), as well as other companies. We present quantitative results, describe their qualitative impact on the companies concerned, and provide some practical insight into model building and validation.

Stochastic Collocation Methods for Nonlinear Parabolic Equations with Random Coefficients

We evaluate the performance of global stochastic collocation methods for solving nonlinear parabolic and elliptic problems (e.g., transient and steady nonlinear diffusion) with random coefficients. The robustness of these and other strategies based on a spectral decomposition of stochastic state variables depends on the regularity of the system's response in outcome space. The latter is affected by statistical properties of the input random fields. These include variances of the input parameters, whose effect on the computational efficiency of this class of uncertainty quantification techniques has remained unexplored. Our analysis shows that if random coefficients have low variances and large correlation lengths, stochastic collocation strategies outperform Monte Carlo simulations (MCS). As variance increases, the regularity of the stochastic response decreases, which requires higher-order quadrature rules to accurately approximate the moments of interest and increases the overall computational cost abov...

Uncertainty Quantification of a Non-linear Rotating Plate Behavior in Compressible Fluid Medium

The paper focuses on application of uncertainty quantification techniques to a fluid structure interaction (FSI) problem involving a nonlinear rotating disc with surrounding fluid at high pressure without any dissipations. The deterministic dynamics shows strong influence of system parameters on the coupled FSI behavior. The nonlinear FSI problem has been discussed in detail and the instability mechanism has been presented. Such instabilities are known as flutter in closed medium and has significance in the turbomachinery industries. This gives way to a sudden large amplitude oscillation, known as the Hamiltonian Hopf bifurcation in such FSI systems. Uncertain variations in system parameters have been considered in order to see the effect on the flutter instability. Investigation with spectral uncertainty quantification tool has given the propagation of the input uncertainties on the instability behavior.

CORRELATION OF PARAMETER ESTIMATORS FOR MODELS ADMITTING MULTIPLE PARAMETRIZATIONS

When estimating parameters using noisy data, uncertainty quan- tification methods provide a way to investigate the confidence one has in the parameter estimates, as well as to obtain information on the possible depen- dence of parametric estimators on one another. In this note, we consider un- certainty quantification techniques that allow visualization of the distributions of these parameter estimators for evidence of possible correlation. We consider three mathematical models (the logistic curve, the Richards curve, and the spring equation), which permit multiple parametrizations, and compare the cor- responding parameter estimators for possible dependence/independence. The uncertainty quantification techniques we employ include the correlation coeffi- cients, asymptotic as well as exact confidence regions or ellipsoids, and Monte Carlo plots generated by the DRAM algorithm.

Hierarchical calibration and validation of computational fluid dynamics models for solid sorbent-based carbon capture

To quantify the predictive confidence of a device scale solid sorbent-based carbon capture design where there is no direct experimental data available, a hierarchical validation methodology is first proposed. In this hierarchy, a sequence of increasingly complex “unit problems” are validated using a statistical calibration framework. This paper describes the computational fluid dynamics (CFD) multi-phase reactive flow simulations and the associated data flows within each unit problem. Each validation requires both simulated and physical data, so the bench-top experiments used in each increasingly complex stage were carefully designed to follow the same operating conditions as the simulation scenarios. A Bayesian calibration procedure is employed and the posterior model parameter distributions obtained at one unit-problem level are used as prior distributions for the same parameters in the next-tier simulations. Overall, the results have demonstrated that the calibrated multiphase reactive flow models within MFIX can be used to capture the bed pressure, temperature, CO2 capture capacity, and kinetics with quantitative accuracy. The CFD modeling methodology and associated uncertainty quantification techniques presented herein offer a solid framework for estimating the predictive confidence in the virtual scale up of a larger carbon capture device.

Damage detection with small data set using energy-based nonlinear features

This study proposes a new algorithm for damage detection in structures. The algorithm employs an energy-based method to capture linear and nonlinear effects of damage on structural response. For more accurate detection, the proposed algorithm combines multiple damage sensitive features through a distance-based method by using Mahalanobis distance. Hypothesis testing is employed as the statistical data analysis technique for uncertainty quantification associated with damage detection. Both the distance-based and the data analysis methods have been chosen to deal with small size data sets. Finally, the efficacy and robustness of the algorithm are experimentally validated by testing a steel laboratory prototype, and the results show that the proposed method can effectively detect and localize the defects. Copyright © 2015 John Wiley & Sons, Ltd.

Comparison of two stochastic techniques for reliable urban runoff prediction by modeling systematic errors

AbstractIn urban rainfall‐runoff, commonly applied statistical techniques for uncertainty quantification mostly ignore systematic output errors originating from simplified models and erroneous inputs. Consequently, the resulting predictive uncertainty is often unreliable. Our objective is to present two approaches which use stochastic processes to describe systematic deviations and to discuss their advantages and drawbacks for urban drainage modeling. The two methodologies are an external bias description (EBD) and an internal noise description (IND, also known as stochastic gray‐box modeling). They emerge from different fields and have not yet been compared in environmental modeling. To compare the two approaches, we develop a unifying terminology, evaluate them theoretically, and apply them to conceptual rainfall‐runoff modeling in the same drainage system. Our results show that both approaches can provide probabilistic predictions of wastewater discharge in a similarly reliable way, both for periods ranging from a few hours up to more than 1 week ahead of time. The EBD produces more accurate predictions on long horizons but relies on computationally heavy MCMC routines for parameter inferences. These properties make it more suitable for off‐line applications. The IND can help in diagnosing the causes of output errors and is computationally inexpensive. It produces best results on short forecast horizons that are typical for online applications.

A Novel Weakly-Intrusive Non-linear Multiresolution Framework for Uncertainty Quantification in Hyperbolic Partial Differential Equations

In this paper, a novel multiresolution framework, namely the Truncate and Encode (TE) approach, previously proposed by some of the authors (Abgrall et al. in J Comput Phys 257:19---56, 2014. doi:10.1016/j.jcp.2013.08.006), is generalized and extended for taking into account uncertainty in partial differential equations (PDEs). Innovative ingredients are given by an algorithm permitting to recover the multiresolution representation without requiring the fully resolved solution, the possibility to treat a whatever form of pdf and the use of high-order (even non-linear, i.e. data-dependent) reconstruction in the stochastic space. Moreover, the spatial-TE method is introduced, which is a weakly intrusive scheme for uncertainty quantification (UQ), that couples the physical and stochastic spaces by minimizing the computational cost for PDEs. The proposed scheme is particularly attractive when treating moving discontinuities (such as shock waves in compressible flows), even if they appear during the simulations as it is common in unsteady aerodynamics applications. The proposed method is very flexible since it can easily coupled with different deterministic schemes, even with high-resolution features. Flexibility and performances of the present method are demonstrated on various numerical test cases (algebraic functions and ordinary differential equations), including partial differential equations, both linear and non-linear, in presence of randomness. The efficiency of the proposed strategy for solving stochastic linear advection and Burgers equation is shown by comparison with some classical techniques for UQ, namely Monte Carlo or the non-intrusive polynomial chaos methods.

Nonintrusive-Polynomial-Chaos-Based Kinematic Reliability Analysis for Mechanisms with Mixed Uncertainty

Due to the scarcity of statistical data, epistemic uncertainties are inevitable in the mechanism. As a promising uncertainty quantification technique, polynomial chaos has advantages over other methods in terms of accuracy and efficiency. In this paper, an improved nonintrusive polynomial chaos method is proposed for the kinematic reliability analysis of the mechanism with fuzzy and random variables as well as fuzzy failure/safety states. Klir log-scale transformation is applied to unify the fuzzy and random variables. Meanwhile, the polynomial-chaos-based probability formula of the fuzzy event is developed to characterize the fuzzy failure/safety states. The proposed method is applied to the reliability analysis of a retractable mechanical system.The results show good accuracy and efficiency of the proposed method when compared with the response surface method (RSM), Kriging method, and Monte Carlo simulation (MCS).

A stochastically and spatially adaptive parallel scheme for uncertain and nonlinear two-phase flow problems

Simulating in flow problems in porous media often requires techniques for uncertainty quantification in order to represent parameter values that are not given exactly. Straightforward Monte-Carlo (MC) methods have a limited efficiency due to slow convergence. Better convergence in low-parametric problems can be achieved with polynomial chaos expansion (PCE) techniques. The PCE approach yields a coupled deterministic system to be solved. The degree of coupling increases with the non-linearity of the considered equations and with the order of polynomial expansion. This fact increases the computational effort of PCE and significantly reduces the scalability in parallelization. We present an application of the hybrid stochastic Galerkin finite volume (HSG-FV) method to a two-phase flow problem in two space dimensions. The method extends the classical polynomial chaos expansion by a multi-element discretization in the probability space of the parameters. It leads to a deterministic system that is coupled to a lesser degree than in element-free PCE versions, respectively, fully decoupled in stochastic elements (SE). Therefore, the HSG-FV method allows for more efficient parallelization. For the further reduction of complexity, we present a new stochastic adaptivity method. We present numerical examples in two spatial dimensions with linear and nonlinear fractional flow functions in the two-phase flow problem. The flow functions might depend in a discontinuous manner on the unknown spatial position of porous-medium heterogeneities. Finally, we discuss the interplay of the new method with spatial adaptivity per SE for these problems.

Global Sensitivity Analysis and Uncertainty Quantification for a Hypersonic Shock Interaction Flow

Hypersonic gas flows over concentric double-cone configurations are commonly used to investigate shock–shock and shock wave/boundary-layer interaction phenomena, and are characterized by the formation of interacting flow structures that are highly sensitive to models used in flow simulation. The work presented here is intended to clarify the influence of several modeling parameters and approximations on simulation output quantities through a more general and systematic approach than has previously been employed for this type of flow, with the ultimate goal of improved model validation for simulation of hypersonic shock interactions. Global sensitivity analysis and uncertainty quantification techniques are integrated with a direct simulation Monte Carlo gas flow simulation code, and a large number of these Monte Carlo simulations are performed for a representative hypersonic double-cone flow. Aleatory and epistemic uncertainties are considered in sensitivity analysis/uncertainty quantification calculations, both independently and in combination, through a variety of probabilistic sampling procedures. These procedures include a novel uncertainty quantification technique that combines elements of importance sampling sensitivity analysis with Latin hypercube sampling, as an efficient surrogate-free means of simultaneously considering mixed uncertainty types. Following a strong engineering interest in surface heating augmentation due to hypersonic shock wave/boundary-layer interaction, this paper focuses on surface heat flux at a shock impingement point as an output quantity in sensitivity analysis/uncertainty quantification calculations; other output quantities of interest include impingement point pressure and the integrated force and heat transfer rate over the model surface.