Bayesian Calibration Research Articles

Inverse Problems for Physics-Based Process Models

We describe and compare two formulations of inverse problems for a physics-based process model in the context of uncertainty and random variability: the Bayesian inverse problem and the stochastic inverse problem. We describe the foundations of the two problems in order to create a context for interpreting the applicability and solutions of inverse problems important for scientific and engineering inference. We conclude by comparing them to statistical approaches to related problems, including Bayesian calibration of computer models.

Applying approximate Bayesian computation to reduce uncertainty in multigroup 235-U cross-sections using ICSBEP experimental data

The multiplication factor (keff) and its uncertainty are critical design parameters in nuclear reactors. The keff uncertainty must be considered for operation, safety, and economic reasons. Consequently, reducing uncertainty in keff has been of interest to the nuclear industry for as long as nuclear reactors were designed. Two methods of reducing this uncertainty are in use – Generalized Linear Least Squares (GLLS) and A General Monte Carlo-Bayes Procedure for Improved Predictions of Integral Functions of Nuclear Data (MOCABA). Both methods have the ability to reduce the keff uncertainty by reducing uncertainty in cross-sections through the assimilation of measured critical systems’ or nuclear reactors’ operational data. The algorithms require means of simulating the experimental data and that the calibrated parameters are described by specific statistical distributions. GLLS is limited to linear models and multivariate normal prior and posterior, while MOCABA can use any (non-linear) model but is also limited to multivariate normal prior and posterior. This work implements a universal and rigorous algorithm called Sequential Monte Carlo – Approximate Bayesian Computation (SMC-ABC) for the same application. It is found that SMC-ABC, GLLS, and likely MOCABA give essentially the same results for the same problems. The paper also presents a reliable method for validating the results based on so-called “synthetic experiments”. Synthetic experiments are computationally generated data used in place of experiments. The method allows for verifying whether there is a risk of overfitting the calibrated parameters (cross-sections in case of this work) during the data assimilation. This novel method shows the potential of using SMC-ABC and other Bayesian calibration methods to generate improved general-purpose neutron cross-section libraries.

Bayesian parameter inference in hydrological modelling using a Hamiltonian Monte Carlo approach with a stochastic rain model

Abstract. Stochastic models in hydrology are very useful and widespread tools for making reliable probabilistic predictions. However, such models are only accurate at making predictions if model parameters are first of all calibrated to measured data in a consistent framework such as the Bayesian one, in which knowledge about model parameters is described through probability distributions. Unfortunately, Bayesian parameter calibration, a. k. a. inference, with stochastic models, is often a computationally intractable problem with traditional inference algorithms, such as the Metropolis algorithm, due to the expensive likelihood functions. Therefore, the prohibitive computational cost is often overcome by employing over-simplified error models, which leads to biased parameter estimates and unreliable predictions. However, thanks to recent advancements in algorithms and computing power, fully fledged Bayesian inference with stochastic models is no longer off-limits for hydrological applications. Our goal in this work is to demonstrate that a computationally efficient Hamiltonian Monte Carlo algorithm with a timescale separation makes Bayesian parameter inference with stochastic models feasible. Hydrology can potentially take great advantage of this powerful data-driven inference method as a sound calibration of model parameters is essential for making robust probabilistic predictions, which can certainly be useful in planning and policy-making. We demonstrate the Hamiltonian Monte Carlo approach by detailing a case study from urban hydrology. Discussing specific hydrological models or systems is outside the scope of our present work and will be the focus of further studies.

Bayesian Nonlocal Operator Regression: A Data-Driven Learning Framework of Nonlocal Models with Uncertainty Quantification

We consider the problem of modeling heterogeneous materials where microscale dynamics and interactions affect global behavior. In the presence of heterogeneities in material microstructure it is often impractical, if not impossible, to provide quantitative characterization of material response. The goal of this work is to develop a Bayesian framework for uncertainty quantification (UQ) in material response prediction when using nonlocal models. Our approach combines the nonlocal operator regression (NOR) technique and Bayesian inference. Specifically, additive independent identically distributed Gaussian noise is employed to model the discrepancy between the nonlocal model and the data. Then, we use a Markov chain Monte Carlo (MCMC) method to sample the posterior probability distribution on parameters involved in the nonlocal constitutive law and associated modeling discrepancies relative to higher-fidelity computations. As an application, we consider the propagation of stress waves through a one-dimensional heterogeneous bar with randomly generated microstructure. Several numerical tests illustrate the construction, enabling UQ in nonlocal model predictions. Although nonlocal models have become popular means for homogenization, their statistical calibration with respect to high-fidelity models has not been presented before. This work is a first step in this direction, focused on Bayesian parameter calibration.

Bayesian Model Calibration Using High-Fidelity Simulations of a Mach 8 Scramjet Isolator and Combustor

ABSTRACT The development of complex physical systems, such as scramjets, requires comprehensive assessments of the relevant performance envelopes. Because high-fidelity simulations and experiments are generally too expensive to perform over a wide range of operating conditions, reduced models can be especially useful for exploring the design space. However, these models often involve empirical parameters which must be calibrated using high-fidelity data sets. The present work develops a calibration procedure for a one-dimensional model of the wall pressure in a scramjet flowpath. Using wall pressure measurements from three-dimensional simulations, the six model parameters are tuned using a Bayesian methodology. The results show that the tuned model is largely able to capture the mean wall pressure in both the isolator and combustor. The calibration procedure is able to converge for both uniform and Gaussian priors, provided that the Gaussian priors are sufficiently wide. Physically-consistent correlations between parameters are obtained, which could inform further reductions in the model’s dimensionality. Application of the model to additional flow conditions shows good agreement to simulations not used in the calibration procedure.

Machine learning based approach to predict ductile damage model parameters for polycrystalline metals

Damage models for ductile materials typically need to be parameterized, often with the appropriate parameters changing for a given material depending on the loading conditions. This can make parameterizing these models computationally expensive, since an inverse problem must be solved for each loading condition. Using standard inverse modeling techniques typically requires hundreds or thousands of high-fidelity computer simulations to estimate the optimal parameters. Additionally, the time of a human expert is required to set up the inverse model. Machine learning has recently emerged as an alternative approach to inverse modeling in these settings, where the machine learning model is trained in an offline manner and new parameters can be quickly generated on the fly, after training is complete. This work utilizes such a workflow to enable the rapid parameterization of a ductile damage model called TEPLA with a machine learning inverse model. The machine learning model can efficiently estimate the model parameters much faster, as compared to previously employed methods, such as Bayesian calibration. The results demonstrate good accuracy on a synthetic test dataset and is validated against experimental data.

A surrogate-assisted uncertainty-aware Bayesian validation framework and its application to coupling free flow and porous-medium flow

Existing model validation studies in geoscience often disregard or partly account for uncertainties in observations, model choices, and input parameters. In this work, we develop a statistical framework that incorporates a probabilistic modeling technique using a fully Bayesian approach to perform a quantitative uncertainty-aware validation. A Bayesian perspective on a validation task yields an optimal bias-variance trade-off against the reference data. It provides an integrative metric for model validation that incorporates parameter and conceptual uncertainty. Additionally, a surrogate modeling technique, namely Bayesian Sparse Polynomial Chaos Expansion, is employed to accelerate the computationally demanding Bayesian calibration and validation. We apply this validation framework to perform a comparative evaluation of models for coupling a free flow with a porous-medium flow. The correct choice of interface conditions and proper model parameters for such coupled flow systems is crucial for physically consistent modeling and accurate numerical simulations of applications. We develop a benchmark scenario that uses the Stokes equations to describe the free flow and considers different models for the porous-medium compartment and the coupling at the fluid–porous interface. These models include a porous-medium model using Darcy’s law at the representative elementary volume scale with classical or generalized interface conditions and a pore-network model with its related coupling approach. We study the coupled flow problems’ behaviors considering a benchmark case, where a pore-scale resolved model provides the reference solution. With the suggested framework, we perform sensitivity analysis, quantify the parametric uncertainties, demonstrate each model’s predictive capabilities, and make a probabilistic model comparison.

Dynamic comfort criteria – A possible solution to the conflict between heat balance and adaptive thermal comfort models

Physiologists typically assume that optimal thermal comfort can only be achieved by minimizing thermoregulatory efforts. People are thus believed to be thermally comfortable with a mean skin temperature of 33–34 °C and minimized sweating rate at a low metabolic rate. Many heat balance comfort models are established based on these fixed comfort criteria. However, occupants’ thermal requirements are found to be more flexible and dynamic because of physiological and psychological adaptions. In this study, dynamic comfort criteria (DCC) are proposed to consider the adaption in the PMV model. Occupants tend to accept a lower skin temperature in a cold climate and a slightly higher sweating rate in a hot climate. We provide two methods to infer DCC based on ASHRAE thermal comfort database II. One is to develop a generalized DCC that can be applied globally. Another is to use Bayesian calibration to develop a customized DCC that further localizes the thermal requirements. With DCC, the deficiency of the PMV and the adaptive model can be largely eliminated. Unlike PMV, the model with DCC performs consistently in different operation modes. DCC is conceptually different from actual physiological responses, although they can sometimes be close to each other in values. The study paves the road for unifying comfort theories and equipping traditional comfort models with the ability to re-learn.

Which demographic processes control competitive equilibria? Bayesian calibration of a size‐structured forest population model

AbstractIn forest communities, light competition is a key process for community assembly. Species' differences in seedling and sapling tolerance to shade cast by overstory trees is thought to determine species composition at late‐successional stages. Most forests are distant from these late‐successional equilibria, impeding a formal evaluation of their potential species composition. To extrapolate competitive equilibria from short‐term data, we therefore introduce the JAB model, a parsimonious dynamic model with interacting size‐structured populations, which focuses on sapling demography including the tolerance to overstory competition. We apply the JAB model to a two‐“species” system from temperate European forests, that is, the shade‐tolerant species Fagus sylvatica L. and the group of all other competing species. Using Bayesian calibration with prior information from external Slovakian national forest inventory (NFI) data, we fit the JAB model to short time series from the German NFI. We use the posterior estimates of demographic rates to extrapolate that F. sylvatica will be the predominant species in 94% of the competitive equilibria, despite only predominating in 24% of the initial states. We further simulate counterfactual equilibria with parameters switched between species to assess the role of different demographic processes for competitive equilibria. These simulations confirm the hypothesis that the higher shade tolerance of F. sylvatica saplings is key for its long‐term predominance. Our results highlight the importance of demographic differences in early life stages for tree species assembly in forest communities.

BEYONDPLANCK

We present a Bayesian calibration algorithm for cosmic microwave background (CMB) observations as implemented within the global end-to-end BEYONDPLANCKframework and applied to thePlanckLow Frequency Instrument (LFI) data. Following the most recentPlanckanalysis, we decomposed the full time-dependent gain into a sum of three nearly orthogonal components: one absolute calibration term, common to all detectors, one time-independent term that can vary between detectors, and one time-dependent component that was allowed to vary between one-hour pointing periods. Each term was then sampled conditionally on all other parameters in the global signal model through Gibbs sampling. The absolute calibration is sampled using only the orbital dipole as a reference source, while the two relative gain components were sampled using the full sky signal, including the orbital and Solar CMB dipoles, CMB fluctuations, and foreground contributions. We discuss various aspects of the data that influence gain estimation, including the dipole-polarization quadrupole degeneracy and processing masks. Comparing our solution to previous pipelines, we find good agreement in general, with relative deviations of −0.67% (−0.84%) for 30 GHz, 0.12% (−0.04%) for 44 GHz and −0.03% (−0.64%) for 70 GHz, compared toPlanckPR4 andPlanck2018, respectively. We note that the BEYONDPLANCKcalibration was performed globally, which results in better inter-frequency consistency than previous estimates. Additionally, WMAP observations were used actively in the BEYONDPLANCKanalysis, which both breaks internal degeneracies in thePlanckdata set and results in an overall better agreement with WMAP. Finally, we used a Wiener filtering approach to smoothing the gain estimates. We show that this method avoids artifacts in the correlated noise maps as a result of oversmoothing the gain solution, which is difficult to avoid with methods like boxcar smoothing, as Wiener filtering by construction maintains a balance between data fidelity and prior knowledge. Although our presentation and algorithm are currently oriented toward LFI processing, the general procedure is fully generalizable to other experiments, as long as the Solar dipole signal is available to be used for calibration.

Bayesian Calibration of MEMS Accelerometers

This study aims to investigate the utilization of Bayesian techniques for the calibration of micro-electro-mechanical systems (MEMS) accelerometers. These devices have garnered substantial interest in various practical applications and typically require calibration through error-correcting functions. The parameters of these error-correcting functions are determined during a calibration process. However, due to various sources of noise, these parameters cannot be determined with precision, making it desirable to incorporate uncertainty in the calibration models. Bayesian modeling offers a natural and complete way of reflecting uncertainty by treating the model parameters as variables rather than fixed values. Additionally, Bayesian modeling enables the incorporation of prior knowledge, making it an ideal choice for calibration. Nevertheless, it is infrequently used in sensor calibration. This study introduces Bayesian methods for the calibration of MEMS accelerometer data in a straightforward manner using recent advances in probabilistic programming.

A new modelling framework for predator-prey interactions: A case study of an aphid-ladybeetle system

The significance of the relationship between data and models in ecology cannot be overstated. Over the past few decades, numerous techniques have been developed and applied in the study of ecological systems. One such technique is the Bayesian calibration of model parameters, which draws from Bayes' theorem to incorporate prior knowledge in estimating parameter values and quantifying their uncertainty. In this study, we establish a Bayesian framework for analysing ecological time series and apply it to a specific case of aphid-ladybeetle predation. To accomplish this, we construct and assess six mathematical models consisting of ordinary differential equations. These models encompass three phenomenological models and three data-driven models. To gain insights into the behaviour of the models concerning a specific quantity of interest, namely aphid abundance, we conduct a sensitivity analysis of the parameters. Subsequently, we compare the outputs of the models by employing model selection techniques. Based on our analysis, the best-performing model is a phenomenological one that incorporates a Holling's type II functional response. We believe that this framework has the potential for application and extension to other ecological systems, thereby enhancing our understanding of these intricate systems.

Improving the Predictivity of a Steam Generator Clogging Numerical Model by Global Sensitivity Analysis and Bayesian Calibration Techniques

In pressurized nuclear reactors, steam generators are massive tubular heat exchangers transferring heat from the primary to the secondary fluid to produce the steam needed by the turbines. After several years of operation, because of deposit, their tube support plates (TSPs) can undergo clogging that may cause important economic and safety issues in case of nonpreventive actions. To understand and predict this phenomenon, several nondestructive examinations can generally be gathered at various times during the heat exchanger operation. A numerical mechanistic model has been recently developed and implemented in a dedicated computer code. The objective of this work is to improve the modeling of the clogging phenomenon to increase the predictive capability of the computer code. A global sensitivity analysis, based on Sobol’ indices, is first performed by the use of a metamodel that is learned on several runs of the computer code. Such an analysis, cast under a physical perspective, helps the identification of the most influential physical parameters and paves the way to a better understanding of TSP clogging. A Bayesian calibration of an epistemic calibration model parameter is then applied to fit the simulation results to experimental data. The additional information coming from the experimental data is then transferred to the calibration parameter with a mathematical model (artificial neural network). The resulting hybrid model thus compensates some lacks of the initial physical model on the considered data set.

How spatio-temporal resolution impacts urban energy calibration

Building Energy Modeling tools help forecast the energy performance of buildings. Urban energy models (UBEMs) emerged as important instruments to analyze the energy performance of buildings aggregated at different spatial resolutions, from the building level to the district level. They heavily rely on available data on geometries and measurements to create accurately calibrated energy models. However, limited research has been conducted to understand the impact of spatial and temporal resolution on the simulation results because of the difficulty of comparing results and not having a standardized procedure to report simulation errors. We review the literature on UBEM validation compared to measured energy data and show the discrepancies in the reporting accuracy. We articulate the need for consistent reporting on model accuracy and introduce a multi-dimensional Level of Detail (LoD) specification for UBEM, including geometry, thermal zoning, and spatio-temporal resolution of the measured data used to calibrate the models. Using a university campus with 70 buildings as an extensive case study, we demonstrate the performance of Bayesian calibration from the building level to the aggregated level. Our results suggest that the accuracy of urban energy prediction with annual temporal resolution can be significantly increased if calibration is performed by using building-level data. However, whenever privacy is a concern, then the data should be provided by aggregating them based on primary use type. Additionally, using monthly data to calibrate uncertain input parameters is not improving the accuracy of the models because the obtained posterior distributions for the selected parameters are not informative for monthly data. To improve this shortcoming, we suggest seasonal calibration, which is computationally costly.

A new initial point search algorithm for bayesian calibration with insufficient statistical information: greedy stochastic section search

A new initial point search algorithm for bayesian calibration with insufficient statistical information: greedy stochastic section search

Bayesian calibration and reliability analysis of ultra high-performance fibre reinforced concrete beams exposed to fire

A study of uncertainty quantification and reliability of ultra-high-performance fibre reinforced concrete (UHPFRC) beams exposed to fire is presented. The study sets forth two methodologies, the first involving the stochastic calibration of temperature-dependent thermal properties of UHPFRC via Bayesian inversion, and the second concerning the computation of the reliability of UHPFRC beams in fire via Monte Carlo Simulation (MCS). Using the calibrated posterior probability distributions of temperature-dependent thermal properties of UHPFRC in stochastic heat transfer analyses results in predictive temperature distributions that reasonably track (within 95% confidence interval) the temperature response reported on UHPFRC beams specimen subjected to 80 and 60 minutes standard fire. Stochastic temperature distributions based on the calibrated thermal properties were sequentially transferred via a time series surrogate model, into a structural-fire reliability analysis of UHPFRC beams designed to the specifications of the ACI 544. Reliability analysis is conducted and the obtained reliability index generally reduced with increasing fire exposure for the UHPFRC beam considered and fire resistance is estimated based on a reliability target of 0.7.

Bayesian model calibration and damage detection for a digital twin of a bridge demonstrator

AbstractUsing digital twins for decision making is a very promising concept which combines simulation models with corresponding experimental sensor data in order to support maintenance decisions or to investigate the reliability. The quality of the prognosis strongly depends on both the data quality and the quality of the digital twin. The latter comprises both the modeling assumptions as well as the correct parameters of these models. This article discusses the challenges when applying this concept to real measurement data for a demonstrator bridge in the lab, including the data management, the iterative development of the simulation model as well as the identification/updating procedure using Bayesian inference with a potentially large number of parameters. The investigated scenarios include both the iterative identification of the structural model parameters as well as scenarios related to a damage identification. In addition, the article aims at providing all models and data in a reproducible way such that other researcher can use this setup to validate their methodologies.

A new framework for the assessment of model probabilities of the different crystal plasticity models for lamellar grains in Titanium alloys

This paper presents a novel framework for assessing the relative accuracy of the different slip transfer criterion in modeling the constitutive response of the lamellar morphology grains in Titanium alloys. The main steps involved in the proposed framework include: (i) rigorous statistical evaluation of the salient morphological characteristics of the lamellar grains using image processing techniques, (ii) modeling the effects of dislocation slip transfer as additive penalties to the critical resolved shear strengths of the different slip systems in each phase, (iii) consideration of multiple geometric slip transfer criteria for the numerical evaluation of the different penalty terms, and (iv) the Bayesian calibration of the material parameters in the constitutive descriptions using spherical nanoindentation experimental data on a polycrystalline Ti-6Al-4V sample. Step (iv) described above involves the construction and refinement of a Gaussian process (GP) surrogate model for the indentation measurements, which is trained on crystal plasticity finite element simulations incorporating the constitutive descriptions selected in Steps (ii) and (iii). The GP surrogates are then combined with Markov Chain Monte Carlo sampling techniques to calibrate the parameters in the various constitutive models studied in this work. Additionally, we have computed the posterior model probabilities for all the constitutive models considered and identified the most plausible slip transfer criterion. This specific constitutive model is then utilized to present a homogenized crystal plasticity model for the lamellar morphology. The proposed framework presents a robust scale-bridging protocol for incorporating the uncertain information obtained from lower scales.

Bayesian calibration with summary statistics for the prediction of xenon diffusion in UO2 nuclear fuel

The evolution and release of fission gas impacts the performance of UO2 nuclear fuel. We have created a Bayesian framework to calibrate a novel model for fission gas transport that predicts diffusion rates of uranium and xenon in UO2 under both thermal equilibrium and irradiation conditions. Data sets are taken from historical diffusion, gas release, and thermodynamic experiments. These data sets consist invariably of summary statistics, including a measurement value with an associated uncertainty. Our calibration strategy uses synthetic data sets in order to estimate the parameters in the model, such that the resulting model predictions agree with the reported summary statistics. In doing so, the reported uncertainties are effectively reflected in the inferred uncertain parameters. Furthermore, to keep our approach computationally tractable, we replace the fission gas evolution model by a polynomial surrogate model with a reduced number of parameters, which are identified using global sensitivity analysis. We discuss the efficacy of our calibration strategy, and investigate how the contribution of the different data sets, taken from multiple sources in the literature, can be weighted in the likelihood function constructed as part of our Bayesian calibration setup, in order to account for the different number of data points in each set of data summaries. Our results indicate a good match between the calibrated diffusivity and non-stoichiometry predictions and the given data summaries. We demonstrate a good agreement between the calibrated xenon diffusivity and the established fit from Turnbull et al. (1982), indicating that the dominant uranium vacancy diffusion mechanism in the model is able to capture the trends in the data.

Combining noisy well data and expert knowledge in a Bayesian calibration of a flow model under uncertainties: an application to solute transport in the Ticino basin

Groundwater flow modeling is commonly used to calculate groundwater heads, estimate groundwater flow paths and travel times, and provide insights into solute transport processes within an aquifer. However, the values of input parameters that drive groundwater flow models are often highly uncertain due to subsurface heterogeneity and geologic complexity in combination with lack of measurements/unreliable measurements. This uncertainty affects the accuracy and reliability of model outputs. Therefore, parameters’ uncertainty must be quantified before adopting the model as an engineering tool. In this study, we model the uncertain parameters as random variables and use a Bayesian inversion approach to obtain a posterior, data-informed, probability density function (pdf) for them: in particular, the likelihood function we consider takes into account both well measurements and our prior knowledge about the extent of the springs in the domain under study. To keep the modelistic and computational complexities under control, we assume Gaussianity of the posterior pdf of the parameters. To corroborate this assumption, we run an identifiability analysis of the model: we apply the inversion procedure to several sets of synthetic data polluted by increasing levels of noise, and we determine at which levels of noise we can effectively recover the “true value” of the parameters. We then move to real well data (coming from the Ticino River basin, in northern Italy, and spanning a month in summer 2014), and use the posterior pdf of the parameters as a starting point to perform an uncertainty quantification analysis on groundwater travel-time distributions.