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Journal articles on the topic 'Sobol’s indices'

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Author: Grafiati

Published: 6 September 2023

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1

Cousin, A., A. Janon, V. Maume-Deschamps, and I. Niang. "On the consistency of Sobol indices with respect to stochastic ordering of model parameters." ESAIM: Probability and Statistics 23 (2019): 387–408. http://dx.doi.org/10.1051/ps/2018001.

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In the past decade, Sobol’s variance decomposition has been used as a tool to assess how the output of a model is affected by the uncertainty on its input parameters. We show some links between global sensitivity analysis and stochastic ordering theory. More specifically, we study the influence of inputs’ distributions on Sobol indices in relation with stochastic orders. This gives an argument in favor of using Sobol’s indices in uncertainty quantification, as one indicator among others.

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2

Wang, Zhixiang, Yongjun Lei, Dapeng Zhang, Guanri Liu, and Jie Wang. "Data-driven global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells." Advances in Mechanical Engineering 14, no. 4 (April 2022): 168781322210901. http://dx.doi.org/10.1177/16878132221090105.

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The influence of input parameters on the load-carrying capacity of the large-diameter stiffened cylindrical shell has not been satisfactory understood. To obtain the global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells, a novel data-driven sensitivity analysis method is presented for the efficient calculation of Sobol’s indices. In the work, the analytical expressions to compute Sobol’s indices are derived in detail based on the Radial Basis Function (RBF) metamodel. Then, the collapse mode of stiffened cylindrical shells with different sections of stringers are simulated, and the sensitivity of the load-carrying performance to geometric parameters are analyzed. The results show that for the considered stiffened cylindrical shell: (1) The coupled flexural-torsional induced deformation becomes the main factor leading to the overall collapse of the large-diameter and heavy-load stiffened cylindrical shells. (2) The flange thickness of stringers has the strongest influence on the load-bearing capacity, followed by the web thickness of stringers. (3) The interaction influences between stringers and other components in stiffened cylindrical shells are insignificant and negligible compared to the individual influence of stringers.

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3

Wang, Zhixiang, Yongjun Lei, Dapeng Zhang, Guanri Liu, and Jie Wang. "Data-driven global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells." Advances in Mechanical Engineering 14, no. 4 (April 2022): 168781322210901. http://dx.doi.org/10.1177/16878132221090105.

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The influence of input parameters on the load-carrying capacity of the large-diameter stiffened cylindrical shell has not been satisfactory understood. To obtain the global sensitivity indices for the load-carrying capacity of large-diameter stiffened cylindrical shells, a novel data-driven sensitivity analysis method is presented for the efficient calculation of Sobol’s indices. In the work, the analytical expressions to compute Sobol’s indices are derived in detail based on the Radial Basis Function (RBF) metamodel. Then, the collapse mode of stiffened cylindrical shells with different sections of stringers are simulated, and the sensitivity of the load-carrying performance to geometric parameters are analyzed. The results show that for the considered stiffened cylindrical shell: (1) The coupled flexural-torsional induced deformation becomes the main factor leading to the overall collapse of the large-diameter and heavy-load stiffened cylindrical shells. (2) The flange thickness of stringers has the strongest influence on the load-bearing capacity, followed by the web thickness of stringers. (3) The interaction influences between stringers and other components in stiffened cylindrical shells are insignificant and negligible compared to the individual influence of stringers.

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4

Melillo, Nicola, and Adam S. Darwich. "A latent variable approach to account for correlated inputs in global sensitivity analysis." Journal of Pharmacokinetics and Pharmacodynamics 48, no. 5 (May 25, 2021): 671–86. http://dx.doi.org/10.1007/s10928-021-09764-x.

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AbstractIn drug development decision-making is often supported through model-based methods, such as physiologically-based pharmacokinetics (PBPK). Global sensitivity analysis (GSA) is gaining use for quality assessment of model-informed inference. However, the inclusion and interpretation of correlated factors in GSA has proven an issue. Here we developed and evaluated a latent variable approach for dealing with correlated factors in GSA. An approach was developed that describes the correlation between two model inputs through the causal relationship of three independent factors: the latent variable and the unique variances of the two correlated parameters. The latent variable approach was applied to a set of algebraic models and a case from PBPK. Then, this method was compared to Sobol’s GSA assuming no correlations, Sobol’s GSA with groups and the Kucherenko approach. For the latent variable approach, GSA was performed with Sobol’s method. By using the latent variable approach, it is possible to devise a unique and easy interpretation of the sensitivity indices while maintaining the correlation between the factors. Compared methods either consider the parameters independent, group the dependent variables into one unique factor or present difficulties in the interpretation of the sensitivity indices. In situations where GSA is called upon to support model-informed decision-making, the latent variable approach offers a practical method, in terms of ease of implementation and interpretability, for applying GSA to models with correlated inputs that does not violate the independence assumption. Prerequisites and limitations of the approach are discussed.

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Sun, Xifu, Barry Croke, Stephen Roberts, and Anthony Jakeman. "Investigation of determinism-related issues in the Sobol′ low-discrepancy sequence for producing sound global sensitivity analysis indices." ANZIAM Journal 62 (December 7, 2021): C84—C97. http://dx.doi.org/10.21914/anziamj.v62.16094.

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A computationally efficient and robust sampling scheme can support a sensitivity analysis of models to discover their behaviour through Quasi Monte Carlo approximation. This is especially useful for complex models, as often occur in environmental domains when model runtime can be prohibitive. The Sobol' sequence is one of the most used quasi-random low-discrepancy sequences as it can explore the parameter space significantly more evenly than pseudo-random sequences. The built-in determinism of the Sobol' sequence assists in achieving this attractive property. However, the Sobol' sequence tends to deteriorate in the sense that the estimated errors are distributed inconsistently across model parameters as the dimensions of a model increase. By testing multiple Sobol' sequence implementations, it is clear that the deterministic nature of the Sobol' sequence occasionally introduces relatively large errors in sensitivity indices produced by well-known global sensitivity analysis methods, and that the errors do not diminish by averaging through multiple replications. Problematic sensitivity indices may mistakenly guide modellers to make type I and II errors in trying to identify sensitive parameters, and this will potentially impact model reduction attempts based on these sensitivity measurements. This work investigates the cause of the Sobol' sequence's determinism-related issues. References I. A. Antonov and V. M. Saleev. An economic method of computing LPτ-sequences. USSR Comput. Math. Math. Phys. 19.1 (1979), pp. 252–256. doi: 10.1016/0041-5553(79)90085-5 P. Bratley and B. L. Fox. Algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 14.1 (1988), pp. 88–100. doi: 10.1145/42288.214372 J. Feinberg and H. P. Langtangen. Chaospy: An open source tool for designing methods of uncertainty quantification. J. Comput. Sci. 11 (2015), pp. 46–57. doi: 10.1016/j.jocs.2015.08.008 on p. C90). S. Joe and F. Y. Kuo. Constructing Sobol sequences with better two-dimensional projections. SIAM J. Sci. Comput. 30.5 (2008), pp. 2635–2654. doi: 10.1137/070709359 S. Joe and F. Y. Kuo. Remark on algorithm 659: Implementing Sobol’s quasirandom sequence generator. ACM Trans. Math. Soft. 29.1 (2003), pp. 49–57. doi: 10.1145/641876.641879 W. J. Morokoff and R. E. Caflisch. Quasi-random sequences and their discrepancies. SIAM J. Sci. Comput. 15.6 (1994), pp. 1251–1279. doi: 10.1137/0915077 X. Sun, B. Croke, S. Roberts, and A. Jakeman. Comparing methods of randomizing Sobol’ sequences for improving uncertainty of metrics in variance-based global sensitivity estimation. Reliab. Eng. Sys. Safety 210 (2021), p. 107499. doi: 10.1016/j.ress.2021.107499 S. Tarantola, W. Becker, and D. Zeitz. A comparison of two sampling methods for global sensitivity analysis. Comput. Phys. Com. 183.5 (2012), pp. 1061–1072. doi: 10.1016/j.cpc.2011.12.015 S. Tezuka. Discrepancy between QMC and RQMC, II. Uniform Dist. Theory 6.1 (2011), pp. 57–64. url: https://pcwww.liv.ac.uk/~karpenk/JournalUDT/vol06/no1/5Tezuka11-1.pdf I. M. Sobol′. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7.4 (1967), pp. 86–112. doi: 10.1016/0041-5553(67)90144-9 I. M. Sobol′. Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp 1.4 (1993), pp. 407–414.

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Botshekan, Meshkat, Mazdak P. Tootkaboni, and Arghavan Louhghalam. "Global Sensitivity of Roughness-Induced Fuel Consumption to Road Surface Parameters and Car Dynamic Characteristics." Transportation Research Record: Journal of the Transportation Research Board 2673, no. 2 (February 2019): 183–93. http://dx.doi.org/10.1177/0361198118821318.

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Pavement roughness is one of the key contributors to rolling resistance and thus vehicle fuel consumption. Roughness-induced fuel consumption is the result of energy dissipation in the suspension system of vehicles and therefore depends on both road surface characteristics and vehicle dynamic properties. In this paper, the sensitivity of roughness-induced excess fuel consumption to all involving factors, i.e., road roughness metrics, vehicle dynamic properties, and speed is investigated, and the dominant factors affecting fuel consumption are identified. This is achieved by using the Sobol’s method—a robust analysis of variance (ANOVA)-based technique for global sensitivity analysis. To this end, Monte-Carlo (MC) simulation is performed by generating realizations of all input parameters according to their probability distributions and estimating the energy consumption via a mechanistic roughness model. The results of the simulation are then used to obtain global Sobol sensitivity indices. Finally, the comparison between the Sobol sensitivity indices and the previously employed indices based on Spearman Rank Correlation Coefficient (SRCC) is illustrated. It is found that roughness metrics, i.e. the International Roughness Index (IRI) and the waviness number, account for 88–93% of the total variations in energy dissipation and are the most influential factors affecting the excess fuel consumption. It is also observed that among vehicle dynamic properties, the stiffness of tire is the most important parameter accounting for 2–7% of the total variance of the excess energy consumption.

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Nguyen, Trong-Ha, and Duy-Duan Nguyen. "Reliability Assessment of Steel-Concrete Composite Beams considering Metal Corrosion Effects." Advances in Civil Engineering 2020 (December 8, 2020): 1–15. http://dx.doi.org/10.1155/2020/8817809.

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Steel-concrete composite (SCC) beams have been widely used in civil engineering and industrial structures. This kind of structure has some advantages such as fast fabrication time and optimized weight. However, designers are often concerned about the initial reliability, while over time the structural reliability will be reduced, especially due to metal corrosion. The objective of the paper is to assess the structural reliability of corroded SCC beams, in which the input parameters are considered as random variables. The SCC beam has been designed according to Eurocode-4 (EC-4), in which input parameters consist of cross-sectional dimensions of the beam, material properties, and applied loads. The effects of the random input variables on the reliability of structures are evaluated by sensitivity analyses, which are calculated by the global sensitivity analysis using Sobol’s method and Monte Carlo simulation. The developed reliability analysis algorithm in this study is verified with previous studies, highlighting the capability of the used method. Four different corrosion levels, which are pristine, 10-year, 20-year, and 50-year, are considered in the sensitivity analyses of the SCC beam. Finally, a series of first-order and total-order Sobol’s indices are obtained for measuring the sensitivity of input parameters with four corrosion levels.

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8

Marcuccio, Gabriele, Elvio Bonisoli, Stefano Tornincasa, John E. Mottershead, Edoardo Patelli, and Weizhuo Wang. "Image decomposition and uncertainty quantification for the assessment of manufacturing tolerances in stress analysis." Journal of Strain Analysis for Engineering Design 49, no. 8 (May 27, 2014): 618–31. http://dx.doi.org/10.1177/0309324714533694.

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This article presents a methodology for the treatment of uncertainty in nonlinear, interference-fit, stress analysis problems arising from manufacturing tolerances. Image decomposition is applied to the uncertain stress field to produce a small number of shape descriptors that allow for variability in the location of high-stress points when geometric parameters (dimensions) are changed within tolerance ranges. A meta-model, in this case based on the polynomial chaos expansion, is trained using a full finite element model to provide a mapping from input geometric parameters to output shape descriptors. Global sensitivity analysis using Sobol’s indices provides a design tool that enables the influence of each input parameter on the observed variances of the outputs to be quantified. The methodology is illustrated by a simplified practical design problem in the manufacture of automotive wheels.

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Hariri-Ardebili, Mohammad, Golsa Mahdavi, Azam Abdollahi, and Ali Amini. "An RF-PCE Hybrid Surrogate Model for Sensitivity Analysis of Dams." Water 13, no. 3 (January 26, 2021): 302. http://dx.doi.org/10.3390/w13030302.

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Quantification of structural vibration characteristics is an essential task prior to perform any dynamic health monitoring and system identification. Anatomy of vibration in concrete arch dams (especially tall dams with un-symmetry shape) is very complicated and requires special techniques to solve the eigenvalue problem. The situation becomes even more complicated if the material distribution is assumed to be heterogeneous within the dam body (as opposed to conventional isotropic homogeneous relationship). This paper proposes a hybrid Random Field (RF)–Polynomial Chaos Expansion (PCE) surrogate model for uncertainty quantification and sensitivity assessment of dams. For different vibration modes, the most sensitive spatial locations within dam body are identified using both Sobol’s indices and correlation rank methods. Results of the proposed hybrid model is further validated using the classical random forest regression method. The outcome of this study can improve the results of system identification and dynamic analysis by properly determining the vibration characteristics.

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10

Nariman, Nazim Abdul. "Control efficiency optimization and Sobol’s sensitivity indices of MTMDs design parameters for buffeting and flutter vibrations in a cable stayed bridge." Frontiers of Structural and Civil Engineering 11, no. 1 (March 2017): 66–89. http://dx.doi.org/10.1007/s11709-016-0356-8.

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11

Ou, Liuli, Lei Liu, Shuai Dong, and Yongji Wang. "Robust Stability Clearance of Flight Control Law Based on Global Sensitivity Analysis." Journal of Applied Mathematics 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/153602.

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To validate the robust stability of the flight control system of hypersonic flight vehicle, which suffers from a large number of parametrical uncertainties, a new clearance framework based on structural singular value (μ) theory and global uncertainty sensitivity analysis (SA) is proposed. In this framework, SA serves as the preprocess of uncertain model to be analysed to help engineers to determine which uncertainties affect the stability of the closed loop system more slightly. By ignoring these unimportant uncertainties, the calculation ofμcan be simplified. Instead of analysing the effect of uncertainties onμwhich involves solving optimal problems repeatedly, a simpler stability analysis function which represents the effect of uncertainties on closed loop poles is proposed. Based on this stability analysis function, Sobol’s method, the most widely used global SA method, is extended and applied to the new clearance framework due to its suitability for system with strong nonlinearity and input factors varying in large interval, as well as input factors subjecting to random distributions. In this method, the sensitive indices can be estimated via Monte Carlo simulation conveniently. An example is given to illustrate the efficiency of the proposed method.

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12

Sysoev, Anton. "Sensitivity Analysis of Mathematical Models." Computation 11, no. 8 (August 14, 2023): 159. http://dx.doi.org/10.3390/computation11080159.

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The construction of a mathematical model of a complicated system is often associated with the evaluation of inputs’ (arguments, factors) influence on the output (response), the identification of important relationships between the variables used, and reduction of the model by decreasing the number of its inputs. These tasks are related to the problems of Sensitivity Analysis of mathematical models. The author proposes an alternative approach based on applying Analysis of Finite Fluctuations that uses the Lagrange mean value theorem to estimate the contribution of changes to the variables of a function to the output change. The article investigates the presented approach on an example of a class of fully connected neural network models. As a result of Sensitivity Analysis, a set of sensitivity measures for each input is obtained. For their averaging, it is proposed to use a point-and-interval estimation algorithm using Tukey’s weighted average. The comparison of the described method with the computation of Sobol’s indices is given; the consistency of the proposed method is shown. The computational robustness of the procedure for finding sensitivity measures of inputs is investigated. Numerical experiments are carried out on the neuraldat data set of the NeuralNetTools library of the R data processing language and on data of the healthcare services provided in the Lipetsk region.

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Servin-Palestina, Miguel, Irineo L. López-Cruz, Jorge A. Zegbe-Domínguez, Agustín Ruiz-García, Raquel Salazar-Moreno, and Guillermo Medina-García. "Spatiotemporal Uncertainty and Sensitivity Analysis of the SIMPLE Model Applied to Common Beans for Semi-Arid Climate of Mexico." Agronomy 12, no. 8 (July 30, 2022): 1813. http://dx.doi.org/10.3390/agronomy12081813.

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Simulation models are used to estimate, forecast, optimize and identify limiting factors and analyze changes in crop production. In order to obtain a functional and reliable mathematical model, it is necessary to know the source of uncertainty and identify the most influential parameters. This study aimed to carry out an uncertainty analysis (UA) and a global spatiotemporal sensitivity analysis (SA) for the parameters of the SIMPLE model, which uses 13 parameters, has two state variables and uses daily weather data to simulate crop growth and development. A Monte Carlo simulation was performed for the UA, and Sobol’s method was used for the SA. Four automatic weather stations representing the climatic conditions of the different bean-producing areas in Zacatecas, Mexico, and a four-year historical series of each station for irrigated and rainfed common bean crops were analyzed. From the UA the coefficients of variation (CV) for thermal time were 11.49% and 11.47%, for biomass the CV were 47.94% and 37.80% and for yield the CV were 49.52% and 39.70% for irrigated and rainfed beans, respectively. From the SA, the most influential parameters for irrigated beans were Tsum > Swater > Tbase > I50A > Topt and for rainfed beans, Tsum > Tbase > I50A > Topt > Swater, according to indices calculated on biomass and thermal time. In conclusion, UA was able to accurately quantify the uncertainty of the biomass, and SA allowed the identification of the most influential of the parameters of the SIMPLE model applied to a common bean crop.

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Gamannossi, Andrea, Alberto Amerini, Lorenzo Mazzei, Tommaso Bacci, Matteo Poggiali, and Antonio Andreini. "Uncertainty Quantification of Film Cooling Performance of an Industrial Gas Turbine Vane." Entropy 22, no. 1 (December 22, 2019): 16. http://dx.doi.org/10.3390/e22010016.

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Computational Fluid Dynamics (CFD) results are often presented in a deterministic way despite the uncertainties related to boundary conditions, numerical modelling, and discretization error. Uncertainty quantification is the field studying how these phenomena affect the numerical result. With these methods, the results obtained are directly comparable with the experimental ones, for which the uncertainty related to the measurement is always shown. This work presents an uncertainty quantification approach applied to CFD: the test case consists of an industrial prismatic gas turbine vane with standard film cooling shaped holes system on the suction side only. The vane was subject of a previous experimental test campaign which had the objective to evaluate the film cooling effectiveness through pressure-sensitive paint technique. CFD analyses are conducted coherently with the experiments: the analogy between heat and mass transfer is adopted to draw out the adiabatic film effectiveness, solving an additional transport equation to track the concentration of CO2 used as a coolant fluid. Both steady and unsteady simulations are carried out: the first one using a RANS approach with k-ω SST turbulence model the latter using a hybrid LES-RANS approach. Regarding uncertainty quantification, three geometrical input parameters are chosen: the hole dimension, the streamwise inclination angle of the holes, and the inlet fillet radius of the holes. Polynomial-chaos approach in conjunction with the probabilistic collocation method is used for the analysis: a first-order polynomial approximation was adopted which required eight evaluations only. RANS approach is used for the uncertainty quantification analysis in order to reduce the computational cost. Results show the confidence interval for the analysis as well as the probabilistic output. Moreover, a sensitivity analysis through Sobol’s indices was carried out which prove how these input parameters contribute to the film cooling effectiveness, in particular, when dealing with the additive manufacturing process.

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15

Kenig, Carlos E., Gustavo Ponce, and Luis Vega. "Sobolev spaces of negative indices." Duke Mathematical Journal 71, no. 1 (July 1993): 1–21. http://dx.doi.org/10.1215/s0012-7094-93-07101-3.

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16

Owen, A. B., J. Dick, and S. Chen. "Higher order Sobol' indices." Information and Inference 3, no. 1 (March 1, 2014): 59–81. http://dx.doi.org/10.1093/imaiai/iau001.

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17

Lamboni, Matieyendou. "Derivative-based generalized sensitivity indices and Sobol’ indices." Mathematics and Computers in Simulation 170 (April 2020): 236–56. http://dx.doi.org/10.1016/j.matcom.2019.10.017.

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18

Lu, Rong, Danxin Wang, Min Wang, and Grzegorz A. Rempala. "Estimation of Sobol's sensitivity indices under generalized linear models." Communications in Statistics - Theory and Methods 47, no. 21 (November 20, 2017): 5163–95. http://dx.doi.org/10.1080/03610926.2017.1388397.

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19

Owen, Art B. "Sobol' Indices and Shapley Value." SIAM/ASA Journal on Uncertainty Quantification 2, no. 1 (January 2014): 245–51. http://dx.doi.org/10.1137/130936233.

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20

Owen, Art B. "Variance Components and Generalized Sobol' Indices." SIAM/ASA Journal on Uncertainty Quantification 1, no. 1 (January 2013): 19–41. http://dx.doi.org/10.1137/120876782.

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21

Jiménez Rugama, Lluís Antoni, and Laurent Gilquin. "Reliable error estimation for Sobol’ indices." Statistics and Computing 28, no. 4 (June 14, 2017): 725–38. http://dx.doi.org/10.1007/s11222-017-9759-1.

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22

Glen, Graham, and Kristin Isaacs. "Estimating Sobol sensitivity indices using correlations." Environmental Modelling & Software 37 (November 2012): 157–66. http://dx.doi.org/10.1016/j.envsoft.2012.03.014.

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23

Kala, Zdeněk. "Global Sensitivity Analysis of Quantiles: New Importance Measure Based on Superquantiles and Subquantiles." Symmetry 13, no. 2 (February 4, 2021): 263. http://dx.doi.org/10.3390/sym13020263.

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The article introduces quantile deviation l as a new sensitivity measure based on the difference between superquantile and subquantile. New global sensitivity indices based on the square of l are presented. The proposed sensitivity indices are compared with quantile-oriented sensitivity indices subordinated to contrasts and classical Sobol sensitivity indices. The comparison is performed in a case study using a non-linear mathematical function, the output of which represents the elastic resistance of a slender steel member under compression. The steel member has random imperfections that reduce its load-carrying capacity. The member length is a deterministic parameter that significantly changes the sensitivity of the output resistance to the random effects of input imperfections. The comparison of the results of three types of global sensitivity analyses shows the rationality of the new quantile-oriented sensitivity indices, which have good properties similar to classical Sobol indices. Sensitivity indices subordinated to contrasts are the least comprehensible because they exhibit the strongest interaction effects between inputs. However, using total indices, all three types of sensitivity analyses lead to approximately the same conclusions. The similarity of the results of two quantile-oriented and Sobol sensitivity analysis confirms that Sobol sensitivity analysis is empathetic to the structural reliability and that the variance is one of the important characteristics significantly influencing the low quantile of resistance.

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Kozhevnikova, L. M. "On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices." Contemporary Mathematics. Fundamental Directions 63, no. 3 (December 15, 2017): 475–93. http://dx.doi.org/10.22363/2413-3639-2017-63-3-475-493.

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For a certain class of second-order anisotropic elliptic equations with variable nonlinearity indices and L1 right-hand side we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable indices.

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25

Fabec, R. C. "Multipliers Between Sobolev Spaces." Canadian Mathematical Bulletin 34, no. 4 (December 1, 1991): 465–73. http://dx.doi.org/10.4153/cmb-1991-075-7.

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AbstractA sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.

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26

Wei, Pengfei, Zhenzhou Lu, and Jingwen Song. "Regional and parametric sensitivity analysis of Sobol׳ indices." Reliability Engineering & System Safety 137 (May 2015): 87–100. http://dx.doi.org/10.1016/j.ress.2014.12.012.

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27

Owen, Art B. "Better estimation of small sobol' sensitivity indices." ACM Transactions on Modeling and Computer Simulation 23, no. 2 (May 2013): 1–17. http://dx.doi.org/10.1145/2457459.2457460.

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28

Arriagada, Waldo. "Matuszewska–Orlicz indices of the Sobolev conjugate Young function." Partial Differential Equations in Applied Mathematics 3 (June 2021): 100029. http://dx.doi.org/10.1016/j.padiff.2021.100029.

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29

das Neves Carneiro, Gonçalo, and Carlos Conceição António. "Sobol’ indices as dimension reduction technique in evolutionary-based reliability assessment." Engineering Computations 37, no. 1 (August 21, 2019): 368–98. http://dx.doi.org/10.1108/ec-03-2019-0113.

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Purpose In the reliability assessment of composite laminate structures with multiple components, the uncertainty space defined around design solutions easily becomes over-dimensioned, and not all of the random variables are relevant. The purpose of this study is to implement the importance analysis theory of Sobol’ to reduce the dimension of the uncertainty space, improving the efficiency toward global convergence of evolutionary-based reliability assessment. Design/methodology/approach Sobol’ indices are formulated analytically for implicit structural response functions, following the theory of propagation of moments and without violating the fundamental principles presented by Sobol’. An evolutionary algorithm capable of global convergence in reliability assessment is instrumented with the Sobol’ indices. A threshold parameter is introduced to identify the important variables. A set of optimal designs of a multi-laminate composite structure is evaluated. Findings Importance analysis shows that uncertainty is concentrated in the laminate where the critical stress state is found. Still, it may also be reasonable in other points of the structure. An accurate and controlled reduction of the uncertainty space significantly improves the convergence rate, while maintaining the quality of the reliability assessment. Practical implications The theoretical developments assume independent random variables. Originality/value Applying Sobol’ indices as an analytical dimension reduction technique is a novelty. The proposed formulation only requires one adjoint system of equilibrium equations to be solved once. Although a local estimate of a global measure, this analytical formulation still holds because, in structural design, uncertainty is concentrated around the mean-values.

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Jaxa-Rozen, Marc, Astu Sam Pratiwi, and Evelina Trutnevyte. "Variance-based global sensitivity analysis and beyond in life cycle assessment: an application to geothermal heating networks." International Journal of Life Cycle Assessment 26, no. 5 (May 2021): 1008–26. http://dx.doi.org/10.1007/s11367-021-01921-1.

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Abstract Purpose Global sensitivity analysis increasingly replaces manual sensitivity analysis in life cycle assessment (LCA). Variance-based global sensitivity analysis identifies influential uncertain model input parameters by estimating so-called Sobol indices that represent each parameter’s contribution to the variance in model output. However, this technique can potentially be unreliable when analyzing non-normal model outputs, and it does not inform analysts about specific values of the model input or output that may be decision-relevant. We demonstrate three emerging methods that build on variance-based global sensitivity analysis and that can provide new insights on uncertainty in typical LCA applications that present non-normal output distributions, trade-offs between environmental impacts, and interactions between model inputs. Methods To identify influential model inputs, trade-offs, and decision-relevant interactions, we implement techniques for distribution-based global sensitivity analysis (PAWN technique), spectral clustering, and scenario discovery (patient rule induction method: PRIM). We choose these techniques because they are applicable with generic Monte Carlo sampling and common LCA software. We compare these techniques with variance-based Sobol indices, using a previously published LCA case study of geothermal heating networks. We assess eight environmental impacts under uncertainty for three design alternatives, spanning different geothermal production temperatures and heating network configurations. Results In the application case on geothermal heating networks, PAWN distribution-based sensitivity indices generally identify influential model parameters consistently with Sobol indices. However, some discrepancies highlight the potentially misleading interpretation of Sobol indices on the non-normal distributions obtained in our analysis, where variance may not meaningfully describe uncertainty. Spectral clustering highlights groups of model results that present different trade-offs between environmental impacts. Compared to second-order Sobol interaction indices, PRIM then provides more precise information regarding the combinations of input values associated with these different groups of calculated impacts. PAWN indices, spectral clustering, and PRIM have a computational advantage because they yield stable results at relatively small sample sizes (n = 12,000), unlike Sobol indices (n = 100,000 for second-order indices). Conclusions We recommend adding these new techniques to global sensitivity analysis in LCA as they give more precise as well as additional insights on uncertainty regardless of the distribution of the model outputs. PAWN distribution-based global sensitivity analysis provides a computationally efficient assessment of input sensitivities as compared to variance-based global sensitivity analysis. The combination of clustering and scenario discovery enables analysts to precisely identify combinations of input parameters or uncertainties associated with different outcomes of environmental impacts.

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Mestre, Luis, David Allison, Keisuke Ejima, Nianjun Liu, and Greg Pavela. "Determining an Appropriate Statistical Model when Assessing the Association between Obesity and Mortality." Current Developments in Nutrition 6, Supplement_1 (June 2022): 38. http://dx.doi.org/10.1093/cdn/nzac047.038.

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Abstract Objectives To determine a statistical model appropriate to assess the association between Obesity and mortality using epidemiological data. We hypothesized that the model with the lowest AIC, BIC, and lowest Sobol's Indices is the appropriate model, compared to the others, to assess the association between obesity and mortality. Methods The datasets used were the National Child Development Study, the Health Retirement Study, the National Health Interview Survey, and the National Health and Nutrition Examination Survey. The statistical models compared were the Cox PH Model and the AFT Model. For the AFT Model, we used the Weibull Distribution, Lognormal Distribution, and Exponential Distribution. Body mass index (BMI) was used as a proxy for obesity, the exposure of interest. Mortality, the event of interest, was calculated using age and a death indicator of the participants. Covariates were sex, socioeconomic status (SES), education, smoking status, drinking status. The interactions were between BMI and SES, BMI and Sex, BMI and Smoking Status. Results The results of the NCDS dataset indicated that the Cox PH Model has an AIC of 11528.81, BIC of 11611.81 without interactions, an AIC of 11502.04, and BIC of 11640.36 with interaction. The Cox PH model has the highest AIC and BIC regardless of the inclusion of interactions compared to the AFT Models. The AFT model with a Weibull distribution had the lowest AIC and BIC (without interaction: 8367.61 and 8521.30, respectively and with interaction: 8346.29 and 8592.19, respectively). The Sobol's Indices for the models, without interaction, were close to zero for all covariates except education, where all models were close to one. However, with interactions, the Sobol's Indices of the education covariate decreases for all AFT Models though the Cox PH Model remains the highest and without change. Conclusions Given our sample and specified model, the AFT model with a Weibull distribution appeared to fit the data better relative to the Cox PH Model or the other AFT Models. The Cox PH model is more sensitive to interactions than the AFT Models. Funding Sources National Institute of Aging (NIA): 3R01AG057703-02S1

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Srivastava, Ankur, Arun K. Subramaniyan, and Liping Wang. "Analytical global sensitivity analysis with Gaussian processes." Artificial Intelligence for Engineering Design, Analysis and Manufacturing 31, no. 3 (August 2017): 235–50. http://dx.doi.org/10.1017/s0890060417000142.

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AbstractMethods for efficient variance-based global sensitivity analysis of complex high-dimensional problems are presented and compared. Variance decomposition methods rank inputs according to Sobol indices that can be computationally expensive to evaluate. Main and interaction effect Sobol indices can be computed analytically in the Kennedy and O'Hagan framework with Gaussian processes. These methods use the high-dimensional model representation concept for variance decomposition that presents a unique model representation when inputs are uncorrelated. However, when the inputs are correlated, multiple model representations may be possible leading to ambiguous sensitivity ranking with Sobol indices. In this work, we present the effect of input correlation on sensitivity analysis and discuss the methods presented by Li and Rabitz in the context of Kennedy and O'Hagan's framework with Gaussian processes. Results are demonstrated on simulated and real problems for correlated and uncorrelated inputs and demonstrate the utility of variance decomposition methods for sensitivity analysis.

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Hart, Joseph, and Pierre A. Gremaud. "ROBUSTNESS OF THE SOBOL' INDICES TO DISTRIBUTIONAL UNCERTAINTY." International Journal for Uncertainty Quantification 9, no. 5 (2019): 453–69. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.2019030553.

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Hart, J. L., A. Alexanderian, and P. A. Gremaud. "Efficient Computation of Sobol' Indices for Stochastic Models." SIAM Journal on Scientific Computing 39, no. 4 (January 2017): A1514—A1530. http://dx.doi.org/10.1137/16m106193x.

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Dimov, I., and R. Georgieva. "Monte Carlo algorithms for evaluating Sobol’ sensitivity indices." Mathematics and Computers in Simulation 81, no. 3 (November 2010): 506–14. http://dx.doi.org/10.1016/j.matcom.2009.09.005.

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Le Guyadec, Mathias, Laurent Gerbaud, Emmanuel Vinot, and Benoit Delinchant. "Sensitivity analysis using Sobol indices for the thermal modelling of an electrical machine for sizing by optimization." COMPEL - The international journal for computation and mathematics in electrical and electronic engineering 38, no. 3 (May 7, 2019): 965–76. http://dx.doi.org/10.1108/compel-09-2018-0360.

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Purpose The thermal modelling of an electrical machine is difficult because the thermal behavior depends on its geometry, the used materials and its manufacturing process. In the paper, such a thermal model is used during the sizing process by optimization of a hybrid electric vehicle (HEV). This paper aims to deal with the sensitivities of thermal parameters on temperatures inside the electrical machine to allow the assessment of the influence of thermal parameters that are hard to assess. Design/methodology/approach A sensitivity analysis by Sobol indices is used to assess the sensitivities of the thermal parameters on electrical machine temperatures. As the optimization process needs fast computations, a lumped parameter thermal network (LPTN) is proposed for the thermal modelling of the machine, because of its fastness. This is also useful for the Sobol method that needs too many calls to this thermal model. This model is also used in a global model of a hybrid vehicle. Findings The difficulty is the thermal modelling of the machine on the validity domain of the sizing problem. The Sobol indices allow to find where a modelling effort has to be carried out. Research limitations/implications The Sobol indices have a significant value according to the number of calls of the model and their type (first-order, total, etc.). Therefore, the quality of the thermal sensitivity analysis is a compromise between computation times and modelling accuracy. Practical implications Thermal modelling of an electrical machine in a sizing process by optimization. Originality/value The use of Sobol indices for the sensitivity analysis of the thermal parameters of an electrical machine.

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Xie, Guangheng, Dachun Yang, and Wen Yuan. "Pointwise Characterizations of Even Order Sobolev Spaces via Derivatives of Ball Averages." Canadian Mathematical Bulletin 62, no. 3 (October 15, 2018): 681–99. http://dx.doi.org/10.4153/s000843951800005x.

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AbstractLet $\ell \in \mathbb{N}$ and $p\in (1,\infty ]$. In this article, the authors establish several equivalent characterizations of Sobolev spaces $W^{2\ell +2,p}(\mathbb{R}^{n})$ in terms of derivatives of ball averages. The novelty in the results of this article is that these equivalent characterizations reveal some new connections between the smoothness indices of Sobolev spaces and the derivatives on the radius of ball averages and also that, to obtain the corresponding results for higher order Sobolev spaces, the authors first establish the combinatorial equality: for any $\ell \in \mathbb{N}$ and $k\in \{0,\ldots ,\ell -1\}$, $\sum _{j=0}^{2\ell }(-1)^{j}\binom{2\ell }{j}|\ell -j|^{2k}=0$.

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Kalra, Tarandeep S., Alfredo Aretxabaleta, Pranay Seshadri, Neil K. Ganju, and Alexis Beudin. "Sensitivity analysis of a coupled hydrodynamic-vegetation model using the effectively subsampled quadratures method (ESQM v5.2)." Geoscientific Model Development 10, no. 12 (December 8, 2017): 4511–23. http://dx.doi.org/10.5194/gmd-10-4511-2017.

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Abstract. Coastal hydrodynamics can be greatly affected by the presence of submerged aquatic vegetation. The effect of vegetation has been incorporated into the Coupled Ocean–Atmosphere–Wave–Sediment Transport (COAWST) modeling system. The vegetation implementation includes the plant-induced three-dimensional drag, in-canopy wave-induced streaming, and the production of turbulent kinetic energy by the presence of vegetation. In this study, we evaluate the sensitivity of the flow and wave dynamics to vegetation parameters using Sobol' indices and a least squares polynomial approach referred to as the Effective Quadratures method. This method reduces the number of simulations needed for evaluating Sobol' indices and provides a robust, practical, and efficient approach for the parameter sensitivity analysis. The evaluation of Sobol' indices shows that kinetic energy, turbulent kinetic energy, and water level changes are affected by plant stem density, height, and, to a lesser degree, diameter. Wave dissipation is mostly dependent on the variation in plant stem density. Performing sensitivity analyses for the vegetation module in COAWST provides guidance to optimize efforts and reduce exploration of parameter space for future observational and modeling work.

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Guo, Jianbin, Shaohua Du, Yao Wang, and Shengkui Zeng. "Time-Dependent Global Sensitivity Analysis for Long-Term Degeneracy Model Using Polynomial Chaos." Advances in Mechanical Engineering 6 (January 1, 2014): 719825. http://dx.doi.org/10.1155/2014/719825.

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Global sensitivity is used to quantify the influence of uncertain model inputs on the output variability of static models in general. However, very few approaches can be applied for the sensitivity analysis of long-term degeneracy models, as far as time-dependent reliability is concerned. The reason is that the static sensitivity may not reflect the completed sensitivity during the entire life circle. This paper presents time-dependent global sensitivity analysis for long-term degeneracy models based on polynomial chaos expansion (PCE). Sobol’ indices are employed as the time-dependent global sensitivity since they provide accurate information on the selected uncertain inputs. In order to compute Sobol’ indices more efficiently, this paper proposes a moving least squares (MLS) method to obtain the time-dependent PCE coefficients with acceptable simulation effort. Then Sobol’ indices can be calculated analytically as a postprocessing of the time-dependent PCE coefficients with almost no additional cost. A test case is used to show how to conduct the proposed method, then this approach is applied to an engineering case, and the time-dependent global sensitivity is obtained for the long-term degeneracy mechanism model.

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Puy, Arnald, Samuele Lo Piano, and Andrea Saltelli. "Is VARS more intuitive and efficient than Sobol’ indices?" Environmental Modelling & Software 137 (March 2021): 104960. http://dx.doi.org/10.1016/j.envsoft.2021.104960.

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Chastaing, G., F. Gamboa, and C. Prieur. "Generalized Sobol sensitivity indices for dependent variables: numerical methods." Journal of Statistical Computation and Simulation 85, no. 7 (September 25, 2014): 1306–33. http://dx.doi.org/10.1080/00949655.2014.960415.

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Kucherenko, S., O. V. Klymenko, and N. Shah. "Sobol' indices for problems defined in non-rectangular domains." Reliability Engineering & System Safety 167 (November 2017): 218–31. http://dx.doi.org/10.1016/j.ress.2017.06.001.

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Marrel, Amandine, Bertrand Iooss, Béatrice Laurent, and Olivier Roustant. "Calculations of Sobol indices for the Gaussian process metamodel." Reliability Engineering & System Safety 94, no. 3 (March 2009): 742–51. http://dx.doi.org/10.1016/j.ress.2008.07.008.

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Grabisch, Michel, and Christophe Labreuche. "A note on the Sobol' indices and interactive criteria." Fuzzy Sets and Systems 315 (May 2017): 99–108. http://dx.doi.org/10.1016/j.fss.2016.05.009.

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Hart, Joseph L., and Pierre A. Gremaud. "Robustness of the Sobol' Indices to Marginal Distribution Uncertainty." SIAM/ASA Journal on Uncertainty Quantification 7, no. 4 (January 2019): 1224–44. http://dx.doi.org/10.1137/18m123387x.

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Benoumechiara, Nazih, and Kevin Elie-Dit-Cosaque. "Shapley effects for sensitivity analysis with dependent inputs: bootstrap and kriging-based algorithms." ESAIM: Proceedings and Surveys 65 (2019): 266–93. http://dx.doi.org/10.1051/proc/201965266.

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In global sensitivity analysis, the well-known Sobol’ sensitivity indices aim to quantify how the variance in the output of a mathematical model can be apportioned to the different variances of its input random variables. These indices are based on the functional variance decomposition and their interpretation becomes difficult in the presence of statistical dependence between the inputs. However, as there are dependencies in many application studies, this drawback enhances the development of interpretable sensitivity indices. Recently, the Shapley values that were developed in the field of cooperative games theory have been connected to global sensitivity analysis and present good properties in the presence of dependencies. Nevertheless, the available estimation methods do not always provide confidence intervals and require a large number of model evaluations. In this paper, a bootstrap resampling is implemented in existing algorithms to assess confidence intervals. We also propose to consider a metamodel in substitution of a costly numerical model. The estimation error from the Monte-Carlo sampling is combined with the metamodel error in order to have confidence intervals on the Shapley effects. Furthermore, we compare the Shapley effects with existing extensions of the Sobol’ indices in different examples of dependent random variables.

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Noi, Takahiro. "Duality of Variable Exponent Triebel-Lizorkin and Besov Spaces." Journal of Function Spaces and Applications 2012 (2012): 1–19. http://dx.doi.org/10.1155/2012/361807.

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We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive.

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Hart, Joseph L., and P. A. Gremaud. "AN APPROXIMATION THEORETIC PERSPECTIVE OF SOBOL' INDICES WITH DEPENDENT VARIABLES." International Journal for Uncertainty Quantification 8, no. 6 (2018): 483–93. http://dx.doi.org/10.1615/int.j.uncertaintyquantification.2018026498.

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Panin, Ivan. "Risk of estimators for Sobol’ sensitivity indices based on metamodels." Electronic Journal of Statistics 15, no. 1 (2021): 235–81. http://dx.doi.org/10.1214/20-ejs1793.

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Klein, Thierry, Nicolas Peteilh, and Paul Rochet. "Test Comparison for Sobol Indices over Nested Sets of Variables." SIAM/ASA Journal on Uncertainty Quantification 10, no. 4 (December 5, 2022): 1586–600. http://dx.doi.org/10.1137/21m1457370.

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