A unified framework for multilevel uncertainty quantification in Bayesian inverse problems

Nagel, Joseph B.; Sudret, Bruno

doi:10.1016/j.probengmech.2015.09.007

Cited by 76 publications

(50 citation statements)

References 88 publications

(106 reference statements)

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“…The probabilistic shell that is shaped around the deterministic dam breach model can be represented by a Bayesian network (see Figure ). The Bayesian multilevel approach applied herein is adopted from Nagel and Sudret (). It provides “a natural framework for solving complex inverse problems in the presence of natural variability and epistemic uncertainty.” The multilevel character of the method at hand is given by the hierarchically composed submodels, such as the deterministic forward model itself, different categories of parameter uncertainty and/or variability described by the prior model, and prediction errors of the forward model specified in the residual model.…”

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

“…Directed acyclic graph representing the probabilistic multilevel modeling approach followed in this study: vertices represent known (

□

) or unknown ( ∘ ) quantities, whereas directed edges symbolize their deterministic (

\to

) or probabilistic (

--\to

) relations (adapted from Nagel & Sudret, ). (a) The general formulation and (b) the actual parameters of the dam breach model and their role in the multilevel framework.…”

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

“…The likelihood depends on the experiment specific knowns ( θ Z i , d i ) and is conditional on the unknowns ( m , θ U , Σ ε ). Here a marginalized (or integrated) formulation

\begin{matrix} right f (y_{i} ∣ m, θ_{U}, Σ_{ε}) = & left \int_{{scriptD}_{u}} \int_{{scriptD}_{z}} f_{E} (y_{i} - ℳ (m, u_{i}, z_{i}, d_{i}); Σ_{ε}) & right \\ right & left f_{U ∣ Θ_{U}} (u_{i} ∣ θ_{U}) f_{Z ∣ {boldΘ}_{bold-italicZ}} (z_{i} ∣ θ_{Z}) d u_{i} d z_{i} \end{matrix}

is used, wherein the aleatory uncertainty in the so‐called latent variables ( u , z ) is integrated out (Nagel & Sudret, ). This formulation focuses on global unknown parameters and therefore can be seen as a function of QoI ϕ = ( m , θ U , Σ ε ).…”

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

See 2 more Smart Citations

Development of Probabilistic Dam Breach Model Using Bayesian Inference

Peter

Siviglia

Nagel

et al. 2018

Water Resources Research

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Dam breach models are commonly used to predict outflow hydrographs of potentially failing dams and are key ingredients for evaluating flood risk. In this paper a new dam breach modeling framework is introduced that shall improve the reliability of hydrograph predictions of homogeneous earthen embankment dams. Striving for a small number of parameters, the simplified physics‐based model describes the processes of failing embankment dams by breach enlargement, driven by progressive surface erosion. Therein the erosion rate of dam material is modeled by empirical sediment transport formulations. Embedding the model into a Bayesian multilevel framework allows for quantitative analysis of different categories of uncertainties. To this end, data available in literature of observed peak discharge and final breach width of historical dam failures were used to perform model inversion by applying Markov chain Monte Carlo simulation. Prior knowledge is mainly based on noninformative distribution functions. The resulting posterior distribution shows that the main source of uncertainty is a correlated subset of parameters, consisting of the residual error term and the epistemic term quantifying the breach erosion rate. The prediction intervals of peak discharge and final breach width are congruent with values known from literature. To finally predict the outflow hydrograph for real case applications, an alternative residual model was formulated that assumes perfect data and a perfect model. The fully probabilistic fashion of hydrograph prediction has the potential to improve the adequate risk management of downstream flooding.

show abstract

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

“…Directed acyclic graph representing the probabilistic multilevel modeling approach followed in this study: vertices represent known (

□

) or unknown ( ∘ ) quantities, whereas directed edges symbolize their deterministic (

\to

) or probabilistic (

--\to

) relations (adapted from Nagel & Sudret, ). (a) The general formulation and (b) the actual parameters of the dam breach model and their role in the multilevel framework.…”

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

“…The likelihood depends on the experiment specific knowns ( θ Z i , d i ) and is conditional on the unknowns ( m , θ U , Σ ε ). Here a marginalized (or integrated) formulation

\begin{matrix} right f (y_{i} ∣ m, θ_{U}, Σ_{ε}) = & left \int_{{scriptD}_{u}} \int_{{scriptD}_{z}} f_{E} (y_{i} - ℳ (m, u_{i}, z_{i}, d_{i}); Σ_{ε}) & right \\ right & left f_{U ∣ Θ_{U}} (u_{i} ∣ θ_{U}) f_{Z ∣ {boldΘ}_{bold-italicZ}} (z_{i} ∣ θ_{Z}) d u_{i} d z_{i} \end{matrix}

Section: Probabilistic Model Calibrationmentioning

confidence: 99%

See 1 more Smart Citation

Development of Probabilistic Dam Breach Model Using Bayesian Inference

Peter

Siviglia

Nagel

et al. 2018

Water Resources Research

View full text Add to dashboard Cite

show abstract

“…This leads to an extremely large computational cost for sampling from p( ψ|D), as well as estimating the evidence of the model, p(D). Current approaches include using conjugate pairs for analytical results [8], approximating the integrals with Laplace Asymptotic Approximation [29], or using some advanced Markov Chain Monte Carlo techniques [21]. These methods are still restrictive for either studying many complex systems or the efficiency is not very scalable to handle extra data sets.…”

Section: Efficient Approximation Of Hsmmentioning

confidence: 99%

Hierarchical Stochastic Model in Bayesian Inference for Engineering Applications: Theoretical Implications and Efficient Approximation

Angelikopoulos

Beck

et al. 2018

ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part B: Mechanical Engineering

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We classify two types of Hierarchical Bayesian Model found in the literature as Hierarchical Prior Model (HPM) and Hierarchical Stochastic Model (HSM). Then, we focus on studying the theoretical implications of the HSM. Using examples of polynomial functions, we show that the HSM is capable of separating different types of uncertainties in a system and quantifying uncertainty of reduced order models under the Bayesian model class selection framework. To tackle the huge computational cost for analyzing HSM, we propose an efficient approximation scheme based on Importance Sampling and Empirical Interpolation Method. We illustrate our method using two examples -a Molecular Dynamics simulation for Krypton and a pharmacokinetic/pharmacodynamic model for cancer drug.

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“…Recently, this framework has been used to quantify and propagate the uncertainty due to model errors, environmental variability, and vibration excitation amplitude [41][42][43][44]. Nagel and Sudret [45,46] have developed a hierarchical Bayesian framework for the particular case of having noise-free vibration data. Sedehi et al [47][48][49] have developed a hierarchical Bayesian framework for time-domain model updating and response predictions.…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Bayesian operational modal analysis: Theory and computations

Sedehi

Katafygiotis

Papadimitriou

2020

Mechanical Systems and Signal Processing

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This paper presents a hierarchical Bayesian modeling framework for the uncertainty quantification in modal identification of linear dynamical systems using multiple vibration data sets. This novel framework integrates the state-of-the-art Bayesian formulations into a hierarchical setting aiming to capture both the identification precision and the ensemble variability prompted due to modeling errors. Such cutting-edge developments have been absent from the modal identification literature, sustained as a long-standing problem at the research spotlight. Central to this framework is a Gaussian hyper probability model, whose mean and covariance matrix are unknown encapsulating the uncertainty of the modal parameters. Detailed computation of this hierarchical model is addressed under two major algorithms using Markov chain Monte Carlo (MCMC) sampling and Laplace asymptotic approximation methods. Since for a small number of data sets the hyper covariance matrix is often unidentifiable, a practical remedy is suggested through the eigenbasis transformation of the covariance matrix, which effectively reduces the number of unknown hyper-parameters. It is also proved that under some conditions the maximum a posteriori (MAP) estimation of the hyper mean and covariance coincide with the ensemble mean and covariance computed using the MAP estimations corresponding to multiple data sets. This interesting finding addresses relevant concerns related to the outcome of the mainstream Bayesian methods in capturing the stochastic variability from dissimilar data sets. Finally, the dynamical response of a prototype structure tested on a shaking table subjected to Gaussian white noise base excitation and the ambient vibration measurement of a cable footbridge are employed to demonstrate the proposed framework.

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A unified framework for multilevel uncertainty quantification in Bayesian inverse problems

Abstract: Cite this article as: Joseph B. Nagel and Bruno Sudret, A unified framework for multilevel uncertainty quantification in bayesian inverse problems, Probabilistic Engineering Mechanics, http://dx.

Cited by 76 publications

References 88 publications

Development of Probabilistic Dam Breach Model Using Bayesian Inference

Development of Probabilistic Dam Breach Model Using Bayesian Inference

Hierarchical Stochastic Model in Bayesian Inference for Engineering Applications: Theoretical Implications and Efficient Approximation

Hierarchical Bayesian operational modal analysis: Theory and computations

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