Global Sensitivity Analysis and Estimation of Model Error, Toward Uncertainty Quantification in Scramjet Computations

Optimal scenario reduction framework based on distance of uncertainty distribution and output performance: II. Sequential reduction

This paper presents a global sensitivity analysis based strategy for uncertainty quantification of the operating revenue and the carbon dioxide emissions in a crude oil distillation column. Aspen HYSYS is used for rigorous simulation of the column. A two-stage approach, which is executed by an interface between MATLAB and HYSYS, is implemented in this study. A multiplicative dimensional reduction method is applied to first identify which factors have the most influential to the model outputs, e.g. the operating revenue and the CO2 emissions. In the second stage, the uncertainty quantification regressed by Gaussian prediction regression method is exploited. As a result, there is a good agreement between the predicted results by the Gaussian prediction regression method with that of the conventional Quasi-Monte Carlo approach, which shows that the computational efforts was reduced significantly compared to the conventional study (about less than 10 times). Furthermore, the Kernel points estimated by the Gaussian prediction regression method are generated to highlight influential factors identified in the first stage.

Global sensitivity analysis and uncertainty quantification of soot formation in an n-dodecane spray flame

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.

Global sensitivity analysis with limited data via sparsity-promoting D-MORPH regression: Application to char combustion

Abstract. Integrated hydrological modeling of domains with complex subsurface features requires many highly uncertain parameters. Performing a global uncertainty analysis using an ensemble of model runs can help bring clarity as to which of these parameters really influence system behavior and for which high parameter uncertainty does not result in similarly high uncertainty of model predictions. However, already creating a sufficiently large ensemble of model simulation for the global sensitivity analysis can be challenging, as many combinations of model parameters can lead to unrealistic model behavior. In this work we use the method of active subspaces to perform a global sensitivity analysis. While building up the ensemble, we use the already-existing ensemble members to construct low-order meta-models based on the first two active-subspace dimensions. The meta-models are used to pre-determine whether a random parameter combination in the stochastic sampling is likely to result in unrealistic behavior so that such a parameter combination is excluded without running the computationally expensive full model. An important reason for choosing the active-subspace method is that both the activity score of the global sensitivity analysis and the meta-models can easily be understood and visualized. We test the approach on a subsurface-flow model including uncertain hydraulic parameters, uncertain boundary conditions and uncertain geological structure. We show that sufficiently detailed active subspaces exist for most observations of interest. The pre-selection by the meta-model significantly reduces the number of full-model runs that must be rejected due to unrealistic behavior. An essential but difficult part in active-subspace sampling using complex models is approximating the gradient of the simulated observation with respect to all parameters. We show that this can effectively and meaningfully be done with second-order polynomials.

Global Sensitivity Analysis of Arrhythmic Risk Biomarkers in Cardiac Single Cell Models

Multiple factors, many of them environmental, coalesce to inform agricultural decisions. Farm planning is often done months in advance. These decisions have to be made with the information available at the time, including current trends, historical data, or predictions of what future weather patterns may be. The effort described in this work is geared towards a flexible mathematical and software framework for simulating the impact of meteorological variability on future crop yield. Our framework is data driven and can easily be applied to any location with suitable historical observations. This will enable site-specific studies that are needed for rigorous risk assessments and climate adaptation planning. The framework combines a physics-based model of crop yield with stochastic process models for meteorological inputs. Combined with techniques from uncertainty quantification, global sensitivity analysis, and machine learning, this hybrid statistical–physical framework allows studying the potential impacts of meteorological uncertainty on future agricultural yields and identify the environmental variables that contribute the most to prediction uncertainty. To highlight the utility of our general approach, we studied the predicted yields of multiple crops in multiple scenarios constructed from historical data. Using global sensitivity analysis, we then identified the key environmental factors contributing to uncertainty in these scenarios’ predictions.

Copula-based methods for global sensitivity analysis with correlated random variables and stochastic processes under incomplete probability information

Abstract. We propose new metrics to assist global sensitivity analysis, GSA, of hydrological and Earth systems. Our approach allows assessing the impact of uncertain parameters on main features of the probability density function, pdf, of a target model output, y. These include the expected value of y, the spread around the mean and the degree of symmetry and tailedness of the pdf of y. Since reliable assessment of higher-order statistical moments can be computationally demanding, we couple our GSA approach with a surrogate model, approximating the full model response at a reduced computational cost. Here, we consider the generalized polynomial chaos expansion (gPCE), other model reduction techniques being fully compatible with our theoretical framework. We demonstrate our approach through three test cases, including an analytical benchmark, a simplified scenario mimicking pumping in a coastal aquifer and a laboratory-scale conservative transport experiment. Our results allow ascertaining which parameters can impact some moments of the model output pdf while being uninfluential to others. We also investigate the error associated with the evaluation of our sensitivity metrics by replacing the original system model through a gPCE. Our results indicate that the construction of a surrogate model with increasing level of accuracy might be required depending on the statistical moment considered in the GSA. The approach is fully compatible with (and can assist the development of) analysis techniques employed in the context of reduction of model complexity, model calibration, design of experiment, uncertainty quantification and risk assessment.

The main focus of this thesis is the uncertainty propagation (UP) and global sensitivity analysis (GSA) in complex hydrogeological numerical models. Various methods are presented with applications on numerical models for the groundwater flow and mean lifetime expectancy (MLE) in the scope of Andra's (French National Radioactive Waste Management Agency) project for the geological disposal of high-level and intermediate-level long-lived radioactive wastes in a highly impermeable layer from Callovo-Oxfordian age (COX) in France. <br> A state of the art is provided for the theory of uncertainty propagation and for the methodologies of sensitivity analyses in a broad sense. Methods for UP are provided from 2-levels Factorial Designs to quasi-random samplings. GSA techniques encompass screening methods such as the Morris Measures and the Derivative-based Global Sensitivity Measures (DGSM), and also the so-called Sobol' indices based upon the variance decomposition of the response of interest. Meta-modelling techniques such as polynomial regression and Polynomial Chaos Expansions (PCE) meta-models are employed as surrogate models for UP and GSA purpose at negligible computational-costs. A comparison of GSA techniques upon various complex analytical test-functions was undertaken with the purpose of determining a relevant method to be employed in the context of “screening” out unimportant variables in computer-intensive, high-dimensional models. <br> A numerical model of groundwater flow and lifetime expectancy is employed for assessing the effect of uncertain advection-dispersion parameters and their spatial distributions upon the MLE from a target zone inside the domain. The model is a 2-dimensions synthetic cross-section of the eastern region of the Paris Basin (Meuse/Haute-Marne sector). This model was used as an exploratory tool for sensitivity analysis methods applied upon numerous sets of uncertain hydrodynamic and dispersion parameters in 15 layers. The uncertainty characterizing the permeability-porosity values in aquifer formations encompassing the COX have proved to add much of variability to the MLE calculated from the target zone. The model also served at exploring the effect of the spatial variability of permeability-porosity parameters in two main aquifer sequences on the groundwater flow rates and MLE in the model. The variabilities of the output responses are mainly due to the uncertainty upon the means and variances of the permeability-porosity distributions, as well as the longitudinal correlation lengths, in each sequence. <br> Then, a 3-dimensions high-definition hydrogeological model representing the Meuse/Haute-Marne sector in the eastern region of the Paris Basin is a more comprehensive numerical model incorporating realistic geometries, fractures, heterogeneities and discontinuities encountered on field. A sensitivity analysis of the MLE from a given zone located in the middle of the COX layer was performed by perturbing the hydraulic conductivities and porosities values in fourteen hydrogeological formations. The uncertain permeability-porosity parameters in the aquifer formations from Bathonian in the Dogger sequence, and Rauracian-Sequanian in the Oxfordian sequence, have strong and rather non-linear effects on the variability of the output response of interest. <br> The methodologies for UP and GSA employed in the present thesis have proved to be very efficient when applied to large hydrogeological models of groundwater flow and MLE. In particular, quasi-random sampling methods offer a flexible frame for providing the uncertainty distribution of the output response of interest at low computational costs. Screening techniques provide a fast estimation for the overall contribution of each input variable to the variability of the output. Meta-modelling techniques such as PCE proved highly accurate in individualising the low- and high-order effects of each input variable upon the output response of interest.

Despite being a common procedure, abdominal hernia treatments still require improvement due to the number of relapses and other postoperative issues. In silico testing can be employed to predict the behavior of the complex abdominal wall and implant systems. Here, uncertainty quantification and sensitivity analysis are required in order to optimize the parameters of hernia repair models. This paper concerns the modeling of an abdominal wall and implant using the finite element method. A Gasser-Ogden-Holzapfel (GOH) material model is used for the abdominal wall and an orthotropic material model for the implant. The parameters of the GOH model and the orientation of the implant are assumed to be uncertain. Regression-based polynomial chaos expansion is used as a meta-modeling method for uncertainty propagation and global sensitivity analysis. The maximum force in the connection between the implant and native tissue is considered as the quantity of interest. A failure risk criterion is also defined and presented. It has been found that the significance of the material parameters depends on the type of implant that is analyzed. Likewise, the risk of connection failure varies considerably depending on the implant used. Models with different types of implant produce very diverse results. Moreover, these differences also appear in the global sensitivity index and the risk of connection failure. This would indicate that specific implant designs and material properties are crucial to the success of hernia repair surgery.

The goal of the present PhD research is to establish a methodology of nuclear data uncertainty quantification (NDUQ) for MCNPX, the continuous-energy Monte-Carlo (M-C) code. The high fidelity (continuous-energy treatment and flexible geometry modelling) of MCNPX makes it the choice of routine criticality safety calculations at PSI/LRS, but also raises challenges for NDUQ by conventional sensitivity/uncertainty (S/U) methods. The methodology developed during this PhD research is fundamentally different from the conventional S/U approach: nuclear data are treated as random variables and sampled in accordance to presumed probability distributions. When sampled nuclear data are used in repeated model calculations, the output variance is attributed to the collective uncertainties of nuclear data. The NUSS (Nuclear data Uncertainty Stochastic Sampling) tool based on this sampling approach, is implemented to work with MCNPX's ACE format of nuclear data, which also gives NUSS compatibility with MCNP and SERPENT M-C codes. In contrast, multigroup uncertainties are used for the sampling of ACE-formatted pointwise-energy nuclear data in a groupwise manner due to the more limited quantity and quality of nuclear data uncertainties. Conveniently, the usage of multigroup nuclear data uncertainties allows consistent comparison between NUSS and other methods (both S/U and sampling-based) that employ the same nuclear data uncertainty format. The first stage of NUSS development focuses on applying simple random sampling algorithm for uncertainty quantification. The effect of combining multigroup and ACE format on the propagated nuclear data uncertainties is assessed. It is found that the number of energy groups has minor impact on the precision of the multiplication factor (k-eff) uncertainty as long as the group structure reflects the neutron flux spectrum. Successful verification of the NUSS tool for propagating nuclear data uncertainties through MCNPX and quantifying MCNPX output parameter uncertainties is obtained. The second stage of NUSS development is motivated by the need for an efficient sensitivity analysis methodology based on global sampling and coupled with MCNPX. For complex systems, the computing time for obtaining a breakdown of the total uncertainty contribution by individual inputs becomes prohibitive when many MCNPX runs are required. The capability of determining simultaneously the total uncertainty and individual nuclear data uncertainty contributions is thus researched and implemented into the NUSS-RF tool. It is based on the Random Balance Design algorithm and is validated by three mathematical test cases for both linear and nonlinear models and correlated inputs. NUSS-RF is then applied to demonstrate the efficient decomposition of total uncertainty by individual nuclear data. However an attempt to decompose total uncertainty into individual contributions using the conventional S/U method shows different decomposition results when the inputs are correlated. The investigation and findings of this PhD work are valuable because of the introduction of global sensitivity analysis into the existing repertoire of nuclear data uncertainty quantification methods. The NUSS tool is expected to be useful for expanding the types of MCNPX-related applications, such as an upgrade to the current PSI criticality safety assessment methodology for Swiss application, for which nuclear data uncertainty contributions can be quantified.

Uncertainty quantification has always been an important topic in model reduction and simulation of complex systems. In this aspect, global sensitivity analysis (GSA) methods such as Fourier amplitude sensitivity test (FAST) are well recognized as effective algorithms. Recently, some data-based metamodeler such as Random Forest (RF) also developed their own variable importance selection solutions for parameters with perturbations. This paper proposes a visual comparison of these two uncertainty quantification methods, using datasets retrieved from vibroacoustic models. Their results have a lot in common and are capable to explain many results. The remarkable agreement between methods under fundamentally different definitions can potentially improve their compatibility in various occasions.

Modeling the impact of space weather events such as coronal mass ejections (CMEs) is crucial to protecting critical infrastructure. The Space Weather Modeling Framework is a state‐of‐the‐art framework that offers full Sun‐to‐Earth simulations by computing the background solar wind, CME propagation, and magnetospheric impact. However, reliable long‐term predictions of CME events require uncertainty quantification (UQ) and data assimilation. We take the first steps by performing global sensitivity analysis (GSA) and UQ for background solar wind simulations produced by the Alfvén Wave Solar atmosphere Model (AWSoM) for two Carrington rotations: CR2152 (solar maximum) and CR2208 (solar minimum). We conduct GSA by computing Sobol' indices that quantify contributions from model parameter uncertainty to the variance of solar wind speed and density at 1 au, both crucial quantities for CME propagation and strength. Sobol' indices also allow us to rank and retain only the most important parameters, which aids in the construction of smaller ensembles for the reduced‐dimension parameter space. We present an efficient procedure for computing the Sobol' indices using polynomial chaos expansion surrogates and space‐filling designs. The PCEs further enable inexpensive forward UQ. Overall, we identify three important model parameters: the multiplicative factor applied to the magnetogram, Poynting flux per magnetic field strength constant used at the inner boundary, and the coefficient of the perpendicular correlation length in the turbulent cascade model in AWSoM.