Advertisement

Journal of Scientific Computing

, Volume 79, Issue 2, pp 1335–1359 | Cite as

Sensitivity-Driven Adaptive Construction of Reduced-space Surrogates

  • Manav Vohra
  • Alen Alexanderian
  • Cosmin Safta
  • Sankaran MahadevanEmail author
Article

Abstract

Surrogate modeling has become a critical component of scientific computing in situations involving expensive model evaluations. However, training a surrogate model can be remarkably challenging and even computationally prohibitive in the case of intensive simulations and large-dimensional systems. We develop a systematic approach for surrogate model construction in reduced input parameter spaces. A sparse set of model evaluations in the original input space is used to approximate derivative based global sensitivity measures (DGSMs) for individual uncertain inputs of the model. An iterative screening procedure is developed that exploits DGSM estimates in order to identify the unimportant inputs. The screening procedure forms an integral part of an overall framework for adaptive construction of a surrogate in the reduced space. The framework is tested for computational efficiency through an initial implementation in simple test cases such as the classic Borehole function, and a semilinear elliptic PDE with a random source function. The framework is then deployed for a realistic application from chemical kinetics, where we study the ignition delay in an \(\hbox {H}_2{/}\hbox {O}_2\) reaction mechanism with 19 and 33 uncertain rate-controlling parameters. It is observed that significant computational gains can be attained by constructing accurate low-dimensional surrogates using the proposed framework.

Keywords

Global sensitivity analysis Polynomial chaos Parameter screening Surrogate modeling 

Notes

Acknowledgements

M. Vohra and S. Mahadevan gratefully acknowledge funding support from the National Science Foundation (Grant No. 1404823, CDSE Program). C. Safta was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, as part of the Computational Chemical Sciences Program. The research of A. Alexanderian was partially supported by the National Science Foundation through the Grant DMS-1745654. M. Vohra would also like to sincerely thank Dr. Xun Huan at Sandia National Labs for his guidance pertaining to the usage of TChem for the chemical kinetics application in this work. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The views expressed in the article do not necessarily represent the views of the U.S. Department Of Energy or the United States Government.

References

  1. 1.
    Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Courier Corporation, North Chelmsford (2003)zbMATHGoogle Scholar
  3. 3.
    Le Maître, O., Knio, O.M.: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  4. 4.
    Friedman, J.H.: Fast MARS. Technical Report 110, Laboratory for Computational Statistics, Department of Statistics, Stanford University (1993)Google Scholar
  5. 5.
    Rasmussen, C.E.: Gaussian Processes in Machine Learning. Advanced Lectures on Machine Learning, pp. 63–71. Springer, Berlin (2004)zbMATHGoogle Scholar
  6. 6.
    Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer, Berlin (2012)Google Scholar
  7. 7.
    Funahashi, K.: On the approximate realization of continuous mappings by neural networks. Neural Netw. 2(3), 183–192 (1989)CrossRefGoogle Scholar
  8. 8.
    Specht, D.F.: Probabilistic neural networks. Neural Netw. 3(1), 109–118 (1990)CrossRefGoogle Scholar
  9. 9.
    Sobol’, I.M.: Sensitivity estimates for nonlinear mathematical models. Math. Model. Comput. Exp. 1, 407–414 (1993)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Alexanderian, A., Winokur, J., Sraj, I., Srinivasan, A., Iskandarani, M., Thacker, W.C., Knio, O.M.: Global sensitivity analysis in an ocean general circulation model: a sparse spectral projection approach. Comput. Geosci. 16(3), 757–778 (2012)CrossRefGoogle Scholar
  11. 11.
    Li, G., Iskandarani, M., Le Hénaff, M., Winokur, J., Le Maître, O.P., Knio, O.M.: Quantifying initial and wind forcing uncertainties in the Gulf of Mexico. Comput. Geosci. 20(5), 1133–1153 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Namhata, A., Oladyshkin, S., Dilmore, R.M., Zhang, L., Nakles, D.V.: Probabilistic assessment of above zone pressure predictions at a geologic carbon storage site. Sci. Rep. 6, 39536 (2016)CrossRefGoogle Scholar
  13. 13.
    Deman, G., Konakli, K., Sudret, B., Kerrou, J., Perrochet, P., Benabderrahmane, H.: Using sparse polynomial chaos expansions for the global sensitivity analysis of groundwater lifetime expectancy in a multi-layered hydrogeological model. Reliab. Eng. Syst. Saf. 147, 156–169 (2016)CrossRefGoogle Scholar
  14. 14.
    Saad, B., Alexanderian, A., Prudhomme, S., Knio, O.M.: Probabilistic modeling and global sensitivity analysis for \(co\_2\) storage in geological formations: a spectral approach. Appl. Math. Model. 53, 584–601 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Degasperi, A., Gilmore, S.: Sensitivity Analysis of Stochastic Models of Bistable Biochemical Reactions. Formal Methods for Computational Systems Biology, pp. 1–20. Springer, Berlin (2008)Google Scholar
  16. 16.
    Navarro Jimenez, M., Le Maître, O.P., Knio, O.M.: Global sensitivity analysis in stochastic simulators of uncertain reaction networks. J. Chem. Phys. 145(24), 244106 (2016)CrossRefGoogle Scholar
  17. 17.
    Vohra, M., Winokur, J., Overdeep, K.R., Marcello, P., Weihs, T.P., Knio, O.M.: Development of a reduced model of formation reactions in zr–al nanolaminates. J. Appl. Phys. 116(23), 233501 (2014)CrossRefGoogle Scholar
  18. 18.
    Sobol’, I.M., Kucherenko, S.: Derivative based global sensitivity measures and their link with global sensitivity indices. Math. Comput. Simul. 79(10), 3009–3017 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sobol, I.M., Kucherenko, S.: Derivative based global sensitivity measures. Procedia Soc. Behav. Sci. 2(6), 7745–7746 (2010)CrossRefGoogle Scholar
  20. 20.
    Lamboni, M., Iooss, B., Popelin, A.L., Gamboa, F.: Derivative-based global sensitivity measures: general links with sobol’ indices and numerical tests. Math. Comput. Simul. 87, 45–54 (2013)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kucherenko, S., Rodriguez-Fernandez, M., Pantelides, C., Shah, N.: Monte Carlo evaluation of derivative-based global sensitivity measures. Reliab. Eng. Syst. Saf. 94(7), 1135–1148 (2009)CrossRefGoogle Scholar
  22. 22.
    Kucherenko, S., Iooss, B.: Derivative-Based Global Sensitivity Measures. Springer, Berlin (2017)CrossRefGoogle Scholar
  23. 23.
    Kiparissides, A., Kucherenko, S.S., Mantalaris, A., Pistikopoulos, E.N.: Global sensitivity analysis challenges in biological systems modeling. Ind. Eng. Chem. Res. 48(15), 7168–7180 (2009)CrossRefGoogle Scholar
  24. 24.
    Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Gunzburger, M.D.: Perspectives in Flow Control and Optimization, vol. 5. SIAM, Philadelphia (2003)zbMATHGoogle Scholar
  26. 26.
    Borzì, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations, vol. 8. SIAM, Philadelphia (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Alexanderian, A., Petra, N., Stadler, G., Ghattas, O.: Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations. SIAM J. Uncertain. Quantif. 5, 1166–1192 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sobol’, I.M.: Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul. 55(1), 271–280 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Efron, B., Hastie, T., Johnstone, I., Tibshirani, R., et al.: Least angle regression. Ann. Stat. 32(2), 407–499 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B Methodol. 58, 267–288 (1996)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Blatman, G., Sudret, B.: Sparse polynomial chaos expansions and adaptive stochastic finite elements using a regression approach. C. R. Méc. 336(6), 518–523 (2008)CrossRefzbMATHGoogle Scholar
  32. 32.
    Marelli, S., Sudret, B.: UQLab: a framework for uncertainty quantification in MATLAB. In: Proceedings of the 2nd International Conference on Vulnerability, Risk Analysis and Management (ICVRAM 2014), ASCE, Reston, VA (2014)Google Scholar
  33. 33.
    Blatman, G.: Adaptive sparse polynomial chaos expansions for uncertainty propagation and sensitivity analysis. Ph.D. thesis, Clermont-Ferrand 2, (2009)Google Scholar
  34. 34.
    Griewank, A., Walther, A.: Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, vol. 105. SIAM, Philadelphia (2008)CrossRefzbMATHGoogle Scholar
  35. 35.
    Morris, M.D., Mitchell, T.J., Ylvisaker, D.: Bayesian design and analysis of computer experiments: use of derivatives in surface prediction. Technometrics 35(3), 243–255 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Yetter, R.A., Dryer, F.L., Rabitz, H.: A comprehensive reaction mechanism for carbon monoxide/hydrogen/oxygen kinetics. Combust. Sci. Technol. 79(1–3), 97–128 (1991)CrossRefGoogle Scholar
  37. 37.
    Das, L.M.: Hydrogen–oxygen reaction mechanism and its implication to hydrogen engine combustion. Int. J. Hydrog. Energy 21(8), 703–715 (1996)CrossRefGoogle Scholar
  38. 38.
    Loges, B., Boddien, A., Junge, H., Beller, M.: Controlled generation of hydrogen from formic acid amine adducts at room temperature and application in \(\text{ H }_2/\text{ O }_2\) fuel cells. Angew. Chem. Int. Ed. 47, 3962–3965 (2008)CrossRefGoogle Scholar
  39. 39.
    Cosnier, S., Gross, A.J., Le Goff, A., Holzinger, M.: Recent advances on enzymatic glucose/oxygen and hydrogen/oxygen biofuel cells: achievements and limitations. J. Power Sources 325, 252–263 (2016)CrossRefGoogle Scholar
  40. 40.
    Safta, C., Najm, H.N., Knio, O.M.: Tchem—a software toolkit for the analysis of complex kinetic models. Sandia Report, SAND2011-3282 (2011)Google Scholar
  41. 41.
    Crestaux, T., Le Maitre, O.P., Martinez, J.-M.: Polynomial chaos expansion for sensitivity analysis. Reliab. Eng. Syst. Saf. 94(7), 1161–1172 (2009). Special Issue on Sensitivity AnalysisCrossRefGoogle Scholar
  42. 42.
    Blatman, G., Sudret, B.: Efficient computation of global sensitivity indices using sparse polynomial chaos expansions. Reliab. Eng. Syst. Saf. 95(11), 1216–1229 (2010)CrossRefGoogle Scholar
  43. 43.
    Sudret, B.: Global sensitivity analysis using polynomial chaos expansions. Reliab. Eng. Syst. Saf. 93(7), 964–979 (2008)CrossRefGoogle Scholar
  44. 44.
    Borgonovo, E., Iooss, B.: Moment-Independent and Reliability-based Importance Measures. Springer, Berlin (2017)CrossRefGoogle Scholar
  45. 45.
    Iooss, B., Lemaître, P.: A Review on Global Sensitivity Analysis Methods. Uncertainty Management in Simulation-Optimization of Complex Systems, pp. 101–122. Springer, Berlin (2015)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringVanderbilt UniversityNashvilleUSA
  2. 2.Department of MathematicsNorth Carolina State UniversityRaleighUSA
  3. 3.Sandia National LaboratoriesLivermoreUSA

Personalised recommendations