Abstract
Inverse modeling involves repeated evaluations of forward models, which can be computationally prohibitive for large numerical models. To reduce the overall computational burden of these simulations, we study the use of reduced order models (ROMs) as numerical surrogates. These ROMs usually involve using solutions to high-fidelity models at different sample points within the parameter space to construct an approximate solution at any point within the parameter space. This paper examines an input–output relational approach based on Gaussian process regression (GPR). We show that these ROMs are more accurate than the linear lookup tables with the same number of high-fidelity simulations. We describe an adaptive sampling procedure that automatically selects optimal sample points and demonstrate the use of GPR to a smooth response surface and a response surface with abrupt changes. We also describe how GPR can be used to construct ROMs for models with heterogeneous material properties. Finally, we demonstrate how the use of a GPR-based ROM in two many-query applications—uncertainty quantification and global sensitivity analysis—significantly reduces the total computational effort.
Similar content being viewed by others
References
Schmit, L.A., Farshi, B.: Some approximation concepts for structural synthesis. AIAA J. 12, 692–699 (1974)
Barthelemy, J.F.M., Haftka, R.T.: Approximation concepts for optimum structural design—a review. Struct. Multidisc. Optim. 5, 129–144 (1993)
Simpson, T., Peplinski, J., Koch, P., Allen, J.: Metamodels for computer-based engineering design: survey and recommendations. Eng. Comput. 17, 129–150 (2001)
Lucia, D.J., Beran, P.S., Silva, W.A.: Reduced-order modeling: new approaches for computational physics. Progr. Aero. Sci. 40, 51–117 (2004)
Saridakis, K.M., Dentsoras, A.J.: Soft computing in engineering design—a review. Adv. Eng. Informat. 22, 202–221 (2008)
Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Progr. Aero. Sci. 45, 50–79 (2009)
Razavi, S., Tolson, B.A., Burn, D.H.: Review of surrogate modeling in water resources. Water Resour. Res. 48, W07401 (2012)
Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. MIT Press, Cambridge (2006)
Marrel, A., Iooss, B., Van Dorpe, F., Volkova, E.: An efficient methodology for modeling complex computer codes with Gaussian processes. Comput. Stat. Data Anal. 52, 4731–4744 (2008)
Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Statist. Sci. 4, 409–435 (1989)
Kleijnen, J.P.C.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192, 707–716 (2009)
Prudhomme, C., Rovas, D.V., Veroy, K., Machiels, L., Maday, Y., Patera, A.T., Turinici, G.: Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluids Eng. 124, 70–80 (2002)
Cardoso, M.A., Durlofsky, L.J., Sarma, P.: Development and application of reduced-order modeling procedures for subsurface flow simulation. Int. J. Numer. Meth. Engng. 77, 1322–1350 (2009)
Lieberman, C., Willcox, K., Ghattas, O.: Parameter and state model reduction for large-scale statistical inverse problems. SIAM J. Sci. Comput. 32, 2523 (2010)
Finsterle, S., Doughty, C., Kowalsky, M.B., Moridis, G.J., Pan, L., Xu, T., Zhang, Y., Pruess, K.: Advanced vadose zone simulations using TOUGH. Vadose Zone J. 7, 601 (2008)
Cybenko, G.: Approximation by superpositions of a sigmoidal function. Math. Control Signals Syst. (MCSS) 5, 455–455 (1992)
Zadeh, L.A.: Fuzzy logic. Comput. 21, 83–93 (1988)
MacKay, D.J.C.: Information-based objective functions for active data selection. Neural Comput. 4, 590–604 (1992)
Razavi, S., Tolson, B.A., Burn, D.H.: Numerical assessment of metamodelling strategies in computationally intensive optimization. Environ. Model Softw. 34, 67–86 (2012)
Iooss, B., Boussouf, L., Feuillard, V., Marrel, A.: Numerical studies of the metamodel fitting and validation processes. Int. J. Advance. Syst. Meas. 3, 11–21 (2010)
Finsterle, S.: iTOUGH2 User’s Guide, pp. 1–137 (2007)
Marrel, A., Iooss, B., Laurent, B., Roustant, O.: Calculations of Sobol indices for the Gaussian process metamodel. Reliab. Eng. Syst. Saf. 94, 742–751 (2009)
Hombal, V., Mahadevan, S.: Bias minimization in Gaussian process surrogate modeling for uncertainty quantification. Int. J. Uncert. Quantific. 1, 321–349 (2011)
Rohmer, J., Foerster, E.: Global sensitivity analysis of large-scale numerical landslide models based on Gaussian-process meta-modeling. Comput. Geosci. 37, 917–927 (2011)
Conti, S., O’Hagan, A.: Bayesian emulation of complex multi-output and dynamic computer models. J. Stat. Plann. Infer. 140, 640–651 (2009)
Alvarez, M.A.: Kernels for vector-valued functions: a review. Foundations Trend. Mach. Learn. 4, 195–266 (2012)
Higdon, D., Gattiker, J., Williams, B., Rightley, M.: Computer model calibration using high-dimensional output. J. Am. Stat. Assoc. 103, 570–583 (2008)
Lawrence, N.D.: Gaussian process latent variable models for visualisation of high dimensional data. Adv. Neural. Inform. Process. Syst. 16, 329–336 (2004)
Bayarri, M.J., Berger, J.O., Cafeo, J., Garcia-Donato, G., Liu, F., Palomo, J., Parthasarathy, R.J., Paulo, R., Sacks, J., Walsh, D.: Computer model validation with functional output. Ann. Stat. 35, 1874–1906 (2007)
Drignei, D., Forest, C.E., Nychka, D.: Parameter estimation for computationally intensive nonlinear regression with an application to climate modeling. Ann. Appl. Stat. 2, 1217–1230 (2008)
Marrel, A., Iooss, B., Jullien, M., Laurent, B., Volkova, E.: Global sensitivity analysis for models with spatially dependent outputs. Environmetrics 22, 383–397 (2010)
Wang, J., Fleet, D., Hertzmann, A.: Gaussian process dynamical models. Adv. Neural. Inform. Process. Syst. 18, 1441 (2006)
Neal, R.M.: Bayesian learning for neural networks. Springer, New York (1996)
Rasmussen, C.E., Nickisch, H.: Gaussian processes for machine learning (GPML) toolbox. J. Mach. Learn. Res. 11, 3011–3015 (2010)
Carr, J.C., Beatson, R.K., Cherrie, J.B., Mitchell, T.J., Fright, W.R., McCallum, B.C., Evans, T.R.: Reconstruction and representation of 3D objects with radial basis functions. Proceedings of ACM SIGGRAPH (2001)
Bichon, B.J., Eldred, M.S., Swile, L.P., Mahadevan, S., Mcfarland, J.M.: Efficient global reliability analysis for nonlinear implicit performance functions. AIAA J. 46, 2459–2468 (2008)
Bui-Thanh, T., Ghattas, O., Higdon, D.: Adaptive Hessian-based non-stationary Gaussian process response surface method for probability density approximation with application to Bayesian solution of large-scale inverse problems. SIAM J. Sci. Comput. 34, A2837—A2871 (2012)
Gramacy, R.B., Lee, H.: Adaptive design and analysis of supercomputer experiments. Technometrics 51(2), 130–142 (2009)
Janusevskis, J., Le, R.R., Ginsbourger, D.: Parallel expected improvements for global optimization: summary, bounds and speed-up. OMD2 deliverable Nmbr. 2.1.1-B (2011)
Finsterle, S., Pruess, K.: Solving the estimation-identification problem in two-phase flow modeling. Water Resour. Res. 31, 913–924 (1995)
Pruess, K., Moridis, G., Oldenburg, C.: TOUGH2 user’s guide, version 2.0 (1999)
Paciorek, C., Schervish, M.: Nonstationary covariance functions for Gaussian process regression. Adv. Neural. Inform. Process. Syst. 16, 273–280 (2004)
Homma, T., Saltelli, A.: Importance measures in global sensitivity analysis of nonlinear models. Reliab. Eng. Syst. Saf. 52, 1–17 (1996)
Saltelli, A., Ratto, M., Andres, T., Campolongo, F., Cariboni, J., Gatelli, D., Saisana, M., Tarantola, S.: Global sensitivity analysis: the primer. Wiley, Chichester (2008)
Archer, G., Saltelli, A., Sobol, I.M.: Sensitivity measures, ANOVA-like techniques and the use of bootstrap. J. Stat. Comput. Simulat. 58, 99–120 (1997)
Rozza, G., Huynh, D.B.P., Patera, A.T.: Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch. Computat. Methods Eng. 15, 229–275 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pau, G.S.H., Zhang, Y. & Finsterle, S. Reduced order models for many-query subsurface flow applications. Comput Geosci 17, 705–721 (2013). https://doi.org/10.1007/s10596-013-9349-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-013-9349-z