Forward Uncertainty Quantification Research Articles

Development of higher order deterministic approaches of forward and inverse uncertainty quantification for nonlinear engineering systems

This paper introduces an innovative numerical scheme that may accurately quantify the parameter and response distributions with minimal computational costs, with specific applications to the numerical computation of multiphase flows under various fluid conditions. The probabilistic approach of uncertainty quantification, e.g., the Monte Carlo simulation, although known to be effective in estimating the parameter/response distribution for highly nonlinear problems, is usually not applicable to multi-physics and multi-scale integrated systems due to its high computational cost. The objective of this study is first, to reduce the computing demand for complex physical system calculations and, at the same time, to accurately predict actual probability distributions by developing a higher order deterministic approach to uncertainty quantification. To achieve this, instead of executing simulation codes multiple times to estimate the probability distribution, the first to fourth moments of the probability density function are derived directly by employing a second order Taylor expansion form of the responses along with Bayesian statistics. This characterizes the parameter and response distributions computed by inverse and forward uncertainty quantification based on their mean value, mode, covariance, skewness, and kurtosis. The parameter distribution is estimated by the numerically calculated higher moments of the parameters with a posteriori probability density function being derived by Bayesian inference. The parameter distribution is then mapped into the second order approximated response function to compute the characteristics of the response distribution through both analytical calculation and numerical approximation of the moments of the responses. The proposed approach is numerically demonstrated through simulations of two benchmark problems: MIT pressurizer and FEBA tests. The prediction of the responses is improved by inverse uncertainty quantification as numerical calculations with the a posteriori model parameters are in better agreement with the measured data over a wide range of transient periods. Additionally, the response distribution estimated by the proposed approach of forward uncertainty quantification accurately predicts the distribution computed by the Monte Carlo simulation.

Physics-based reduced order modeling for uncertainty quantification of guided wave propagation using Bayesian optimization

Guided wave propagation (GWP) is commonly employed for the design of SHM systems. However, GWP is sensitive to variations in the material properties, often leading to false alarms. To address this issue, uncertainty quantification (UQ) is employed to improve the reliability of the predictions. Computational structural mechanics is a useful tool for the simulation of GWP. Even so, the application of UQ methods requires numerous solutions, while large-scale, transient GWP simulations increase the computational cost. In this paper, we propose a machine learning (ML)-based reduced order model (ROM), annotated as BO-ML-ROM, to decrease the computational time of the GWP simulation. The ROM is integrated with a Bayesian optimization (BO) framework to adaptively sample the data for the ROM training. The results showed that BO outperforms one-shot sampling methods, both in terms of accuracy and speed. The developed ROM, which is orders of magnitude faster than the original solver, is then applied for forward uncertainty quantification of the GWP in an aluminum plate and a composite panel under varying material properties. The UQ analysis reveals the higher impact of the uncertainties in the wave reflections. Furthermore, to compute the influence of the material properties on the predictions, a sensitivity analysis (SA) is performed using the variance-based Sobol’ indices and the Shapley additive explanations, where it is shown that the uncertainties in the elastic properties have a varying influence across time. The predicted results reveal the efficiency of BO-ML-ROM for the simulation of the GWP and demonstrate its utility for UQ and SA.

Gaussian Process Emulation for High-Dimensional Coupled Systems

Complex coupled multiphysics simulations are ubiquitous in science and engineering. Evaluating these numerical simulators is often costly which limits our ability to run them sufficiently often for forward uncertainty quantification. Furthermore outputs are generally not scalar quantities but depend on space and/or time. Gaussian process emulators are statistical surrogates which can approximate the output of the complex computer models at untested inputs quickly while also providing uncertainty information about the accuracy of evaluating the emulator rather than the full physical model. GP emulators were originally developed in the context of scalar output from a single physical model but have since been extended to vector-valued quantities of interest (parallel partial emulators) and to coupled physics by connecting two independent emulators, one for each type of physics (linked emulation). The parallel partial linked GP emulator (PPLE) developed in this work combines the efficiency of a shared correlation structure with the accuracy of linked emulators to produce a new tool for emulating compositions of functions with vector-valued output. The PPLE applied to two numerical experiments out-performs direct emulation of the output composite function producing results with smaller average prediction error and variance. The authors report there are no competing interests to declare.

Uncertainty quantification for Fisher-Kolmogorov equation on graphs with application to patient-specific Alzheimer's disease

The Fisher-Kolmogorov equation is a diffusion-reaction PDE that is used to model the accumulation of prionic proteins, which are responsible for many different neurological disorders. Likely, the most important and studied misfolded protein in literature is the Amyloid-$\beta$, responsible for the onset of Alzheimer disease. Starting from medical images we construct a reduced-order model based on a graph brain connectome. The reaction coefficient of the proteins is modelled as a stochastic random field, taking into account all the many different underlying physical processes, which can hardly be measured. Its probability distribution is inferred by means of the Monte Carlo Markov Chain method applied to clinical data. The resulting model is patient-specific and can be employed for predicting the disease's future development. Forward uncertainty quantification techniques (Monte Carlo and sparse grid stochastic collocation) are applied with the aim of quantifying the impact of the variability of the reaction coefficient on the progression of protein accumulation within the next 20 years.

Quantifying uncertainties in direct numerical simulations of a turbulent channel flow

Direct numerical simulation (DNS) provides unrivalled levels of detail and accuracy for simulating turbulent flows. However, like all numerical methods, DNS is subject to uncertainties arising from the numerical scheme and input parameters (e.g. mesh resolution). While uncertainty quantification (UQ) techniques are being employed more and more to provide a systematic analysis of uncertainty for lower-fidelity models, their application to DNS is still relatively rare. In light of this, the aim of this work is to apply UQ and sensitivity analysis to the DNS of a canonical wall-bounded turbulent channel flow at low Reynolds number (Reτ=180). To compute the DNS, Incompact3d – a highly scalable open-source framework based on high-order compact finite differences and a spectral Poisson solver – is used as a black-box solver. Stochastic collocation is used to propagate the input uncertainties through Incompact3d to the output quantities of interest (QOIs). To facilitate the non-intrusive forward UQ analysis, the open-source EasyVVUQ package is used to provide integrated capability for sampling, pre-processing, execution, post-processing, and analysis of the computational campaign. Three separate UQ campaigns are conducted. The first two examine the effect of domain size and the numerical parameters (e.g. mesh resolution, time step, sample time), respectively, and adopt Gaussian quadrature rules combined via tensor products to sample the multi-dimensional input space. Finally, the third campaign investigates the performance of a dimension-adaptive sampling strategy that significantly reduces the computational cost compared to the full tensor product approach. The analysis focuses on the cross-channel statistical moments of the QOIs, as well as local and global sensitivity analyses to assess the sensitivity of each QOI with respect to each individual input. This enables an assessment of the robustness and sensitivity of DNS to the user-defined numerical parameters for wall-bounded turbulent flows, and provides an indication of suitable ranges for defining the values of these parameters.

Machine learning modeling of lung mechanics: Assessing the variability and propagation of uncertainty in respiratory-system compliance and airway resistance

Background and ObjectiveTraditional assessment of patient response in mechanical ventilation relies on respiratory-system compliance and airway resistance. Clinical evidence has shown high variability in these parameters, highlighting the difficulty of predicting them before the start of ventilation therapy. This motivates the creation of computational models that can connect structural and tissue features with lung mechanics. In this work, we leverage machine learning (ML) techniques to construct predictive lung function models informed by non-linear finite element simulations, and use them to investigate the propagation of uncertainty in the lung mechanical response. MethodsWe revisit a continuum poromechanical formulation of the lungs suitable for determining patient response. Based on this framework, we create high-fidelity finite element models of human lungs from medical images. We also develop a low-fidelity model based on an idealized sphere geometry. We then use these models to train and validate three ML architectures: single-fidelity and multi-fidelity Gaussian process regression, and artificial neural networks. We use the best predictive ML model to further study the sensitivity of lung response to variations in tissue structural parameters and boundary conditions via sensitivity analysis and forward uncertainty quantification. Codes are available for download at https://github.com/comp-medicine-uc/ML-lung-mechanics-UQ ResultsThe low-fidelity model delivers a lung response very close to that predicted by high-fidelity simulations and at a fraction of the computational time. Regarding the trained ML models, the multi-fidelity GP model consistently delivers better accuracy than the single-fidelity GP and neural network models in estimating respiratory-system compliance and resistance (R2∼0.99). In terms of computational efficiency, our ML model delivers a massive speed-up of ∼970,000× with respect to high-fidelity simulations. Regarding lung function, we observed an almost matched and non-linear behavior between specific structural parameters and chest wall stiffness with compliance. Also, we observed a strong modulation of airways resistance with tissue permeability. ConclusionsOur findings unveil the relevance of specific lung tissue parameters and boundary conditions in the respiratory-system response. Furthermore, we highlight the advantages of adopting a multi-fidelity ML approach that combines data from different levels to yield accurate and efficient estimates of clinical mechanical markers. We envision that the methods presented here can open the way to the development of predictive ML models of the lung response that can inform clinical decisions.

Efficient approximation of cardiac mechanics through reduced-order modeling with deep learning-based operator approximation.

Reducing the computational time required by high-fidelity, full-order models (FOMs) for the solution of problems in cardiac mechanics is crucial to allow the translation of patient-specific simulations into clinical practice. Indeed, while FOMs, such as those based on the finite element method, provide valuable information on the cardiac mechanical function, accurate numerical results can be obtained at the price of very fine spatio-temporal discretizations. As a matter of fact, simulating even just a few heartbeats can require up to hours of wall time on high-performance computing architectures. In addition, cardiac models usually depend on a set of input parameters that are calibrated in order to explore multiple virtual scenarios. To compute reliable solutions at a greatly reduced computational cost, we rely on a reduced basis method empowered with a new deep learning-based operator approximation, which we refer to as Deep-HyROMnet technique. Our strategy combines a projection-based POD-Galerkin method with deep neural networks for the approximation of (reduced) nonlinear operators, overcoming the typical computational bottleneck associated with standard hyper-reduction techniques employed in reduced-order models (ROMs) for nonlinear parametrized systems. This method can provide extremely accurate approximations to parametrized cardiac mechanics problems, such as in the case of the complete cardiac cycle in a patient-specific left ventricle geometry. In this respect, a 3D model for tissue mechanics is coupled with a 0D model for external blood circulation; active force generation is provided through an adjustable parameter-dependent surrogate model as input to the tissue 3D model. The proposed strategy is shown to outperform classical projection-based ROMs, in terms of orders of magnitude of computational speed-up, and to return accurate pressure-volume loops in both physiological and pathological cases. Finally, an application to a forward uncertainty quantification analysis, unaffordable if relying on a FOM, is considered, involving output quantities of interest such as, for example, the ejection fraction or the maximal rate of change in pressure in the left ventricle.

Bayesian uncertainty quantification of tristructural isotropic particle fuel silver release: Decomposing model inadequacy plus experimental noise and parametric uncertainties

Tristructural isotropic (TRISO) particle fuel is one of the most promising fuel concepts enabling high temperature and high burnup reactor operation. One dominant source of radioactivity released from the TRISO particles is silver (Ag), which is subject to a high release fraction and long decay life compared to other fission products. Previous modeling efforts using the fuel performance code BISON indicated nonnegligible uncertainties in modeling the diffusion process of fission products in TRISO compared to the Advanced Gas Reactor experiments. The overall uncertainties observed when modeling the fission product diffusion can result from uncertainties in model parameters, noisy experimental measurements, and deficiencies in the developed models. The three types of underlying uncertainties have not yet been properly quantified in open literature. This paper presents the Bayesian uncertainty quantification (UQ) using massively parallelizable Markov chain Monte Carlo samplers. The uncertainties due to model parameters, model inadequacy, and experimental measurement noise are quantified, with the σ term used to represent the sum of the model inadequacy and measurement noise uncertainties. It is worth noting that this is the first time the σ term is inferred for nuclear fuel experiments, as compared to using prescribed values for uncertainty quantification in previous work. The parallelizable Markov chain Monte Carlo samplers efficiently infer the model parameters and the σ term, giving insight into physical parameters like diffusion coefficients and the combined model discrepancy and measurement noise. A subsequent forward uncertainty quantification (UQ) is also performed based on the calibration results to generate more accurate predictions of the Ag release. The model inadequacy plus experimental noise is the most dominant source of uncertainty compared to the parametric uncertainty. All the UQ analyses presented in this work are based on the second series of the irradiation experiments in the Advanced Gas Reactor program.

Conditional Karhunen–Loève regression model with Basis Adaptation for high-dimensional problems: Uncertainty quantification and inverse modeling

We propose a methodology for improving the accuracy of surrogate models of the observable response of physical systems as a function of the systems’ spatially heterogeneous parameter fields, with applications to uncertainty quantification and parameter estimation in high-dimensional problems. Practitioners often formulate finite-dimensional representations of spatially heterogeneous parameter fields using truncated unconditional Karhunen–Loève expansions (KLEs) for a certain choice of unconditional covariance kernel and construct surrogate models of the observable response with respect to the KLE coefficients. When direct measurements of the parameter fields are available, we propose improving the accuracy of these surrogate models by representing the parameter fields via conditional Karhunen-Loève expansions (CKLEs). CKLEs are constructed by conditioning the covariance kernel of the unconditional expansion on the direct measurements of the parameter field via Gaussian process regression, and then truncating the corresponding KLE.We apply the proposed methodology to constructing surrogate models via the Basis Adaptation (BA) method of the stationary hydraulic head response, measured at spatially discrete observation locations, of a groundwater flow model of the Hanford Site, as a function of the 1000-dimensional representation of the model’s log-transmissivity field. We find that BA surrogate models of the hydraulic head based on CKLEs are more accurate than BA surrogate models based on unconditional expansions for forward uncertainty quantification tasks. Furthermore, we find that inverse estimates of the hydraulic transmissivity field computed using CKLE-based BA surrogate models are more accurate than those computed using unconditional BA surrogate models.

An LSTM RNN proposal for surrogate modeling the dynamic response of buried structures to earthquake plane waves in soil half-spaces

This paper presents a variant of the long short-term memory (LSTM) recurrent neural network for surrogate modeling and predicting the seismic response of buried structures to in-plane inclined shear waves in heterogeneous soil profiles. The proposed model is designed to accept scalar and time-dependent random variables as input features to facilitate Monte Carlo (MC) based uncertainty quantification tasks where both the earthquake wave field and system properties are sources of uncertainty. To this end, it employs short-duration synthetic time histories for training instead of actual earthquake records to ensure efficiency. We discuss several configurations to evaluate the performance of the proposed methodology in predicting the tunnel response at different locations. We use two application examples concerning forward uncertainty quantification and sensitivity analysis to assess further the efficacy and accuracy of the proposed surrogate modeling approach.

On the random fractional Bateman equations

We study the random fractional Bateman equations for a radioactive decay chain, in the Caputo-derivative sense. On the one hand, a fractional order entails memory effects in the system, and on the other hand, randomness accounts for uncertainties on the physical quantities. Results on the deterministic fractional Bateman equations and on the random ordinary Bateman equations were recently published. The proposed stochastic fractional model extends these previous works. We construct the stochastic solution; first, in the pathwise sense by using the Laplace-transform method, and second, in the mean-square sense by applying Banach fixed-point theorem. The probability density function, that is associated to the solution of a chain of length three, is computed in the present work by a transformation method. Continuity of this density with respect to the fractional order, in the pointwise and in the total variation sense, is investigated. Numerical results are included for forward uncertainty quantification, with the purpose of illustrating the developed theory.

Multi-output multilevel best linear unbiased estimators via semidefinite programming

Multifidelity forward uncertainty quantification (UQ) problems often involve multiple quantities of interest and heterogeneous models (e.g., different grids, equations, dimensions, physics, surrogate and reduced-order models). While computational efficiency is key in this context, multi-output strategies in multilevel/multifidelity methods are either sub-optimal or non-existent. In this paper we extend multilevel best linear unbiased estimators (MLBLUE) to multi-output forward UQ problems and we present new semidefinite programming formulations for their optimal setup. Not only do these formulations yield the optimal number of samples required, but also the optimal selection of low-fidelity models to use. While existing MLBLUE approaches are single-output only and require a non-trivial nonlinear optimization procedure, the new multi-output formulations can be solved reliably and efficiently. We demonstrate the efficacy of the new methods and formulations in practical UQ problems with model heterogeneity.

Sparse-grids uncertainty quantification of part-scale additive manufacturing processes

The present paper aims at applying uncertainty quantification methodologies to process simulations of powder bed fusion of metal. In particular, for a part-scale thermomechanical model of an Inconel 625 super-alloy beam, we study the uncertainties of three process parameters, namely the activation temperature, the powder convection coefficient and the gas convection coefficient. First, we perform a variance-based global sensitivity analysis to study how each uncertain parameter contributes to the variability of the beam displacements. The results allow us to conclude that the gas convection coefficient has little impact and can therefore be fixed to a constant value for subsequent studies. Then, we conduct an inverse uncertainty quantification analysis, based on a Bayesian approach on synthetic displacement data, to quantify the uncertainties of the two remaining parameters, namely the activation temperature and the powder convection coefficient. Finally, we use the results of the inverse uncertainty quantification analysis to perform a data-informed forward uncertainty quantification analysis of the residual strains. Crucially, we make use of surrogate models based on sparse grids to keep to a minimum the computational burden of every step of the uncertainty quantification analysis. The proposed uncertainty quantification workflow allows us to substantially ease the typical trial-and-error approach used to calibrate power bed fusion part-scale models, and to greatly reduce uncertainties on the numerical prediction of the residual strains. In particular, we demonstrate the possibility of using displacement measurements to obtain a data-informed probability density function of the residual strains, a quantity much more complex to measure than displacements.

Forward uncertainty quantification in random differential equation systems with delta‐impulsive terms: Theoretical study and applications

This contribution aims at studying a general class of random differential equations with Dirac‐delta impulse terms at a finite number of time instants. Our approach directly addresses calculating the so‐called first probability density function, from which all the relevant statistical information about the solution, a stochastic process, can be extracted. We combine the Liouville partial differential equation and the random variable transformation method to conduct our study. Finally, all our theoretical findings are illustrated on two stochastic models, widely used in mathematical modeling, for which numerical simulations are carried out.

Seismic fragility analysis using stochastic polynomial chaos expansions

Within the performance-based earthquake engineering (PBEE) framework, the fragility model plays a pivotal role. Such a model represents the probability that the engineering demand parameter (EDP) exceeds a certain safety threshold given a set of selected intensity measures (IMs) that characterize the earthquake load. The-state-of-the art methods for fragility computation rely on full non-linear time–history analyses. Within this perimeter, there are two main approaches: the first relies on the selection and scaling of recorded ground motions; the second, based on random vibration theory, characterizes the seismic input with a parametric stochastic ground motion model (SGMM). The latter case has the great advantage that the problem of seismic risk analysis is framed as a forward uncertainty quantification problem. However, running classical full-scale Monte Carlo simulations is intractable because of the prohibitive computational cost of typical finite element models. Therefore, it is of great interest to define fragility models that link an EDP of interest with the SGMM parameters — which are regarded as IMs in this context. The computation of such fragility models is a challenge on its own and, despite a few recent studies, there is still an important research gap in this domain. This comes with no surprise as classical surrogate modeling techniques cannot be applied due to the stochastic nature of SGMM. This study tackles this computational challenge by using stochastic polynomial chaos expansions to represent the statistical dependence of EDP on IMs. More precisely, this surrogate model estimates the full conditional probability distribution of EDP conditioned on IMs. We compare the proposed approach with some state-of-the-art methods in two case studies. The numerical results show that the new method prevails over its competitors in estimating both the conditional distribution and the fragility functions.

Global Sensitivity Analysis and Uncertainty Quantification for Background Solar Wind Using the Alfvén Wave Solar Atmosphere Model

AbstractModeling 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.

OptEx: An integrated framework for experimental design and combustion kinetic model optimization

Computational fluid dynamics (CFD) simulation in the design of combustion devices imposes increased demand on combustion kinetic models with acceptable uncertainties. Model optimization is often utilized to constrain the model parameters with experimental data to reduce the prediction uncertainties. Since it is unaffordable to conduct experiments under all the concerned conditions, experimental design approaches are proposed to find the most valuable experiments to be conducted. An integrated computational framework, OptEx (Optimal Experiments), is proposed to facilitate applications of experimental design, data clustering, and model optimization with optimal experimental data. Specifically, this framework integrates the functions of dimension reduction, global sensitivity analysis, forward uncertainty quantification, model-analysis-based experimental design, and model optimization. The share of data and surrogate models between different modules significantly improves the computational efficiencies of model analysis, experimental design and model optimization. Two case studies of a methanol system are utilized to demonstrate the functionalities of OptEx. First, experimental designs are performed based on sensitivity entropy and surrogate model similarity analysis to find out informative while independent experiments. Second, experimental data clustering with OptEx is demonstrated by grouping massive experiments and selecting optimal experiments for each group. For both cases, a small size of optimal experimental datasets are utilized for model optimization, yielding an optimized model with smaller uncertainty bounds. Meanwhile, the input parameter uncertainties are significantly reduced and parameter correlations are identified via the joint probability distributions of optimized parameters.

Combined effect of experimental and kinetic uncertainties on NO predictions in low-pressure premixed laminar H2/CH4/CO-air and H2/CH4/CO/C6H6-air flames

Accurate predictions of nitrogen oxides (NOx) are crucial for the design of low-emissions combustion technologies and require a truthful description of chemical kinetics and underlying thermo-chemical field. Nowadays, uncertainty remains in the reaction rates of different pathways of NOx formation. Experiments in simple systems, such as 1D flat premixed laminar flames, are necessary to shed light on the NOx formation process. However, experimental uncertainty cannot be avoided and influences the subsequent NOx formation modelling. In this context, forward propagation of experimental and kinetic uncertainties can be beneficial to assess their impact on the predictions of NOx emissions. In this work, forward Uncertainty Quantification (UQ) is performed via Polynomial Chaos Expansion (PCE), based on a sparse grid, to quantify the influence of the variability of temperature measurements and kinetic parameters on NO predictions in H2/CH4/CO-air 1D premixed laminar flames at low pressure, without and with 1.12% of benzene (C6H6) in the fuel blend. A significant uncertainty affects the prediction of NO for all the investigated flames. The measured temperature, with its uncertainty, is proven to be one of the most sensitive factor affecting the prediction of NO, particularly for the flames operated in lean and stoichiometric conditions. The results demonstrate that an accurate quantification of the uncertainty affecting the temperature measurements is critical for the development of predictive NOx formation models.

Comparing multi-index stochastic collocation and multi-fidelity stochastic radial basis functions for forward uncertainty quantification of ship resistance

This paper presents a comparison of two multi-fidelity methods for the forward uncertainty quantification of a naval engineering problem. Specifically, we consider the problem of quantifying the uncertainty of the hydrodynamic resistance of a roll-on/roll-off passenger ferry advancing in calm water and subject to two operational uncertainties (ship speed and payload). The first four statistical moments (mean, variance, skewness, and kurtosis), and the probability density function for such quantity of interest (QoI) are computed with two multi-fidelity methods, i.e., the Multi-Index Stochastic Collocation (MISC) and an adaptive multi-fidelity Stochastic Radial Basis Functions (SRBF). The QoI is evaluated via computational fluid dynamics simulations, which are performed with the in-house unsteady Reynolds-Averaged Navier–Stokes (RANS) multi-grid solver chinavis. The different fidelities employed by both methods are obtained by stopping the RANS solver at different grid levels of the multi-grid cycle. The performance of both methods are presented and discussed: in a nutshell, the findings suggest that, at least for the current implementation of both methods, MISC could be preferred whenever a limited computational budget is available, whereas for a larger computational budget SRBF seems to be preferable, thanks to its robustness to the numerical noise in the evaluations of the QoI.

Implementation of a Gaussian Markov random field sampler for forward uncertainty quantification in the Ice-sheet and Sea-level System Model v4.19

Abstract. Assessing the impact of uncertainties in ice-sheet models is a major and challenging issue that needs to be faced by the ice-sheet community to provide more robust and reliable model-based projections of ice-sheet mass balance. In recent years, uncertainty quantification (UQ) has been increasingly used to characterize and explore uncertainty in ice-sheet models and improve the robustness of their projections. A typical UQ analysis first involves the (probabilistic) characterization of the sources of uncertainty, followed by the propagation and sensitivity analysis of these sources of uncertainty. Previous studies concerned with UQ in ice-sheet models have generally focused on the last two steps but have paid relatively little attention to the preliminary and critical step of the characterization of uncertainty. Sources of uncertainty in ice-sheet models, like uncertainties in ice-sheet geometry or surface mass balance, typically vary in space and potentially in time. For that reason, they are more adequately described as spatio-(temporal) random fields, which account naturally for spatial (and temporal) correlation. As a means of improving the characterization of the sources of uncertainties for forward UQ analysis within the Ice-sheet and Sea-level System Model (ISSM), we present in this paper a stochastic sampler for Gaussian random fields with Matérn covariance function. The class of Matérn covariance functions provides a flexible model able to capture statistical dependence between locations with different degrees of spatial correlation or smoothness properties. The implementation of this stochastic sampler is based on a notable explicit link between Gaussian random fields with Matérn covariance function and a certain stochastic partial differential equation. Discretization of this stochastic partial differential equation by the finite-element method results in a sparse, scalable and computationally efficient representation known as a Gaussian Markov random field. In addition, spatio-temporal samples can be generated by combining an autoregressive temporal model and the Matérn field. The implementation is tested on a set of synthetic experiments to verify that it captures the desired spatial and temporal correlations well. Finally, we illustrate the interest of this stochastic sampler for forward UQ analysis in an application concerned with assessing the impact of various sources of uncertainties on the Pine Island Glacier, West Antarctica. We find that larger spatial and temporal correlations lengths will both likely result in increased uncertainty in the projections.