Control variate Monte Carlo estimators based on sparse polynomial chaos expansions
Article Full Text (PDF)


control variate
Monte Carlo
Bayesian polynomial chaos
Sobol' sensitivity indices

How to Cite

Duan, H. ., & Okten, G. (2023). Control variate Monte Carlo estimators based on sparse polynomial chaos expansions. Socio-Environmental Systems Modelling, 5, 18568.


We introduce two control variate Monte Carlo estimators where the control is based on the truncated sparse polynomial chaos expansion of the function in hand. We use the control variate estimators to estimate the lower and upper Sobol' indices in some applications, and compare them numerically with some of the best Monte Carlo estimators in the literature. The results suggest that in computationally expensive problems where a low-order polynomial chaos expansion is not an accurate approximation of the model but highly correlated with it, the control variate estimators are either the best or among the best in terms of efficiency.

Article Full Text (PDF)


Azzini, I., Mara, T., & Rosati, R. (2020). Monte Carlo estimators of first-and total-orders Sobol’ indices. arXiv preprint arXiv:2006.08232.

Babacan, S. D., Molina, R., & Katsaggelos, A. K. (2009). Bayesian compressive sensing using Laplace priors. IEEE Transactions on Image Processing, 19(1):53–63.

Chakraborty, R. & Vemuri, B. C. (2019). Statistics on the stiefel manifold: theory and applications. The Annals of Statistics, 47(1):415–438.

Edelman, A., Arias, T. A., & Smith, S. T. (1998). The geometry of algorithms with orthogonality constraints. SIAM Journal on Matrix Analysis and Applications, 20(2): 303–353.

Elsawah, S., Filatova, T., Jakeman, A. J., Kettner, A. J., Zellner, M. L., Athanasiadis, I. N., Hamilton, S. H., Axtell, R. L., Brown, D. G., Gilligan, J. M., et al. (2020). Eight grand challenges in socio-environmental systems modeling. Socio-Environmental Systems Modelling, 2: 16226.

Fox, J. & Ökten, G. (2021). Polynomial chaos as a control variate method. SIAM Journal on Scientific Computing, 43(3): A2268–A2294.

Ishigami, T. & Homma, T. (1990). An importance quantification technique in uncertainty analysis for computer models. In [1990] Proceedings. First International Symposium on Uncertainty Modeling and Analysis, pages 398–403. IEEE.

Janon, A., Klein, T., Lagnoux, A., Nodet, M., & Prieur, C. (2014). Asymptotic normality and efficiency of two Sobol’ index estimators. ESAIM: Probability and Statistics, 18: 342–364.

Jansen, M. J. (1999). Analysis of variance designs for model output. Computer Physics Communications, 117(1-2):35–43.

Kucherenko, S., Delpuech, B., Iooss, B., & Tarantola, S. (2015). Application of the control variate technique to estimation of total sensitivity indices. Reliability Engineering & System Safety, 134: 251–259.

Lemieux, C. & Owen, A. B. (2002). Quasi-regression and the relative importance of the ANOVA components of a function. In Monte Carlo and Quasi-Monte Carlo Methods 2000. Springer, pp. 331–344.

Lüthen, N., Marelli, S., & Sudret, B. (2021). Sparse polynomial chaos expansions: Literature survey and benchmark. SIAM/ASA Journal on Uncertainty Quantification, 9(2): 593–649.

Marelli, S. & Sudret, B. (2014). Uqlab: A framework for uncertainty quantification in Matlab. In Vulnerability, uncertainty, and risk: quantification, mitigation, and management, pp. 2554–2563.

Matoušek, J. (1998). On the L2-discrepancy for anchored boxes. Journal of Complexity, 14(4): 527–556.

Morris, M. D. (1991). Factorial sampling plans for preliminary computational experiments. Technometrics, 33(2): 161–174. Owen, A. B. (2013). Better estimation of small Sobol’ sensitivity indices. ACM Transactions on Modeling and Computer Simulation (TOMACS), 23(2): 1–17.

Puy, A., Becker, W., Piano, S. L., & Saltelli, A. (2022). A comprehensive comparison of total-order estimators for global sensitivity analysis. International Journal for Uncertainty Quantification, 12(2): 1-18.

Saltelli, A., Ratto, M., Tarantola, S., & Campolongo, F. (2006). Sensitivity analysis practices: Strategies for model-based inference. Reliability Engineering & System Safety, 91(10-11): 1109–1125.

Sobol’, I. M. (1993). Sensitivity analysis for non-linear mathematical models. Mathematical Modelling and Computational Experiment, 1: 407–414.

Sudret, B. (2008). Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 93(7): 964–979.

Zimmermann, R. (2017). A matrix-algebraic algorithm for the Riemannian logarithm on the Stiefel manifold under the canonical metric. SIAM Journal on Matrix Analysis and Applications, 38(2): 322–342.

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Copyright (c) 2023 Hui Duan, Giray Okten