Bayesian sparse polynomial chaos expansion for global sensitivity analysis

Shao, Qian; Younès, Anis; Fahs, Marwan; Mara, Thierry A.

doi:10.1016/j.cma.2017.01.033

Cited by 101 publications

(58 citation statements)

References 41 publications

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“…These indices reflect the contribution of each parameter to the variance of the model output [29]. The Sobol' indices were evaluated using a surrogate model approach based on Polynomial Chaos Expansion (PCE) [26,[41][42][43][44].…”

Section: Global Sensitivity Analysismentioning

confidence: 99%

Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

Younès

Shao

Mara

et al. 2020

Water

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Accurate simulation of flow and contaminant transport processes through unsaturated soils requires adequate knowledge of the soil parameters. This study deals with the hydraulic characterization of soils using laboratory experiments. A new strategy is developed by combining global sensitivity analysis (GSA) and Bayesian data-worth analysis (DWA) to obtain efficient data that ensure a good estimation of the soil properties. The strategy is applied for the estimation of soil properties from a laboratory infiltration experiment. Results of this study show that GSA allows identification of regions and periods of high sensitivity of each parameter and thereby, the observations prone to contain information for a successful calibration. Further, the sensitivity depicts a nonlinear behavior with regions of high influence and regions of weak influence inside the parameter space. Bayesian DWA, performed a priori, allows to quantify the improvement of the posterior uncertainty of the estimated parameters when adding a type of measurement. The results reveal that an accurate estimation of the soil properties can be obtained if the target parameter values are located in the regions of high influence in the parameter space.Water 2020, 12, 736 2 of 15 and evaporation experiments [13], unsaturated transport experiments [14][15][16][17], and hydrogeophysical experiments [18,19].Laboratory experiments are often time consuming, expensive, and tricky because of the risk of non-identifiability of the accepted parameters which depends on the measured data. Global sensitivity analysis (GSA) has been employed by Younes et al., [7] to help assess unsaturated soil hydraulic parameters. The GSA evaluates how output uncertainties are related to input parameter uncertainties with all the variation of inputs. In the last decades, GSA has been largely used for unsaturated flow in porous media. For instance, in Pan et al. [20], GSA has been conducted to assess the relative contribution of parameter uncertainties to the flow and transport uncertainties in a layered heterogeneous system. Brunetti et al. [21] used GSA to investigate the influence of soil hydraulic parameters on the behavior of a permeable pavement. Younes et al. [19] used GSA to investigate the influence of hydraulic and geophysical parameters on the streaming potential signals. In van Griensven et al. [22], GSA has been used to reduce the number of parameters and thus reduce over-parametrization of the numerical model.GSA allows to quantify the impact changes of parameters have on the model output. By this way, GSA enables the detection of the important parameters, their regions, and periods of influence [23]. This can facilitate the experimental procedure by identifying strategic measurements that could be relevant for soil characterization. However, collecting new output observations even when the output is highly sensitive to the parameters may not reduce parameter uncertainties if the information contained in the new observations are already contained in the previous observat...

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Section: Global Sensitivity Analysismentioning

confidence: 99%

Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

Younès

Shao

Mara

et al. 2020

Water

Self Cite

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“…Our DNNs are constructed using the open-source Keras [5] Python package which is built on top of the open-source Tensorflow DL library [1]. The two other methods are perhaps the most popular emulation techniques in engineering and the geosciences: Gaussian processes [GPs,32] and polynomial chaos expansion [PCE, see, e.g., 9,41,3,25,36, just to list a few]. In both cases we use state-of-the-art open source implementations: the GPflow Python package [29] for GPs, which is built on top of Tensorflow, and the Chaospy Python package [7] for PCE.…”

Section: Introductionmentioning

confidence: 99%

Emulation of CPU-demanding reactive transport models: a comparison of Gaussian processes, polynomial chaos expansion, and deep neural networks

Laloy

Jacques

2019

Comput Geosci

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This paper presents a detailed comparison between 3 methods for emulating CPU-intensive reactive transport models (RTMs): Gaussian processes (GPs), polynomial chaos expansion (PCE) and deep neural networks (DNNs). State-of-the-art open-source libraries are used for each emulation method while the CPU-time incurred by one forward run of the considered RTMs varies from 1h to between 1h30 and 5 days. Besides direct emulation of the simulated uranium concentration time series, replacing the original RTM by its emulator is also investigated for global sensitivity analysis (GSA), uncertainty propagation and probabilistic calibration using Markov chain Monte Carlo (MCMC) sampling. The selected DNN is found to be superior to both GPs and PCE in reproducing the input -output behavior of the considered 8-dimensional and 13-dimensional CPU-intensive RTMs. This even though the used training sets are small: from 75 to 500 samples. Furthermore, the two used PCE variants: standard PCE and sparse PCE (sPCE) appear to always provide the least accuracy while not differing much in performance. As a consequence of its better emulation capabilities, the DNN method outperforms the two other methods for uncertainty propagation. Regarding GSA, the DNN and GP methods offer equally good approximations to the true first-order and total-order Sobol' sensitivity indices while PCE does less well. Most surprisingly, despite its superior emulation skills the DNN approach leads to the worst solution of the considered synthetic inverse problem which involves 1224 measurement data with low noise. This apparently contradicting behavior of the used DNN is at least partially due to the small but complicated deterministic noise that affects the DNN-based predictions. Indeed, this complex error structure can drive the emulated solutions far away from the true posterior distribution in case of high quality measurement data. Among the considered 3 methods only the GP method allows for retrieving emulated posterior solutions that jointly (1) fit the high-quality measurement data to the appropriate noise level (log-likelihood value) and (2) most closely fit the true model parameter values. Overall, our findings indicate that when the available training set is relatively small (75 -to 500 input -output examples) and fixed beforehand, PCE is not the best choice for emulating CPU-intensive RTMs. Instead GPs or DNNs should be preferred. However, a DNN can deliver overly biased model calibration results. In contrast, the GP method performs fairly well across all considered tasks: direct emulation, global sensitivity analysis, uncertainty propagation, and calibration.

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“…Another way of finding the coefficients is by regression method, which utilizes a finite set of MCS results to evaluate the unknown coefficients. This process is known as nonintrusive method . However, in both the methods, as the number of random variables or the order of expansion increases, the size of the system matrix increases exponentially.…”

Section: Introductionmentioning

confidence: 99%

“…Blatman and Sudret considered a hyperbolic PC, where the higher‐order cross terms between two random variables are omitted. Blatman and Sudret and Shao et al considered a sensitivity analysis by allowing a tolerance limit to evaluate the dominant terms of PC expansion and obtain a global sparsity. Thus, construction of sparse PC is not a straightforward procedure and requires additional postprocessing.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

An iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non‐Gaussian randomness

Nath

Dutta

Hazra

2019

Numerical Meth Engineering

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Summary Stochastic analysis of structure with non‐Gaussian material property and loading in the framework of polynomial chaos (PC) is considered. A new approach for the solution of stochastic mechanics problem with random coefficient is presented. The major focus of the method is to consider reduced size of expansion in an iterative manner to overcome the problem of large system matrix in conventional PC expansion. The iterative method is based on orthogonal expansion of stochastic responses and generation of an iterative PC based on the responses of the previous iteration. The polynomials are evaluated using Gram‐Schmidt orthogonalization process. The numbers of random variables in PC expansion are reduced by considering only the dominant components of the response characteristics, which is evaluated using Karhunen‐Loève (KL) expansion. In case of random material field problem, the KL expansion is used to discretize and simulate the non‐Gaussian random field. Independent component analysis (ICA) is carried out on the non‐Gaussian KL random variables to minimize statistical dependence. The usefulness of the proposed method in terms of accuracy and computational efficiency is examined. From the numerical analysis of three different types of structural mechanics problems, the proposed iterative method is observed to be computationally more efficient and accurate than conventional PC method for solution of linear elastostatic structural mechanics problems.

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Bayesian sparse polynomial chaos expansion for global sensitivity analysis

Cited by 101 publications

References 41 publications

Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

Use of Global Sensitivity and Data-Worth Analysis for an Efficient Estimation of Soil Hydraulic Properties

Emulation of CPU-demanding reactive transport models: a comparison of Gaussian processes, polynomial chaos expansion, and deep neural networks

An iterative polynomial chaos approach toward stochastic elastostatic structural analysis with non‐Gaussian randomness

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