Skip to main content

Certified reduced basis method in geosciences

Addressing the challenge of high-dimensional problems


One of the biggest challenges in Computational Geosciences is finding ways of efficiently simulating high-dimensional problems. In this paper, we demonstrate how the RB method can be gainfully exploited to solve problems in the Geosciences. The reduced basis method constructs low-dimensional approximations to (high-dimensional) solutions of parametrized partial differential equations. In contrast to other widely used geoscientific reduction techniques, the reduced basis method reduces the Galerkin approximation space, and not the physical space and is consequently much less restrictive. Another advantage of the method is that for the problems considered in this paper, the method provides a bound to the error in the reduced order approximation, thus permitting an objective evaluation of the approximation quality. Using a geothermal conduction problem, we demonstrate that depending on the model, we obtain a maximum speed-up of three orders of magnitude with an approximation error that is very small in comparison with typical measurement errors. This significant reduction of the cost of the forward simulation allows performing uncertainty quantification, inversions, and parameter studies for larger and more complex models than currently possible.

This is a preview of subscription content, access via your institution.


  1. Abdulle, A., Budác, O.: A Petrov–Galerkin reduced basis approximation of the Stokes equation in parameterized geometries. Compt. Rend. Math. 353(7), 641–645 (2015)

    Article  Google Scholar 

  2. Adams, B.M., Ebeida, M., Eldred, M., Geraci, G., Jakeman, J., Maupin, K., Monschke, J., Swiler, L., Stephens, J., Vigil, D., Wildey, T., Bohnhoff, W., Dalbey, K., Eddy, J., Frye, J., Hooper, R., Hu, K., Hough, P., Khalil, M., Ridgway, E., Rushdi, A.: Dakota, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: version 6.6 user’s manual. Sandia National Laboratories, Tech. Rep SAND2014-4633 (2017)

  3. Balay, S., Abhyankar, S., Adams, M.F., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., May, D.A., McInnes, L.C., Rupp, K., Smith, B.F., Zampini, S., Zhang, H., Zhang, H.: PETSc Web page. (2017)

  4. Ballarin, F., Sartori, A., Rozza, G.: RBniCS-reduced order modelling in FEniCS. (2017)

  5. Brazell, O., Messenger, S., Abusalbi, N., Fjerstad, P.: Multicore Evaluation and Performance Analysis of the Eclipse and Intersect Reservoir Simulation Codes. In: Oil and Gas High Performance Computing Workshop (2010)

  6. Cacace, M., Blöcher, G.: Meshit - a software for three dimensional volumetric meshing of complex faulted reservoirs. Environ. Earth Sci. 74(6), 5191–5209 (2015)

    Article  Google Scholar 

  7. Clapp, R.G., Fu, H., Lindtjorn, O.: Selecting the right hardware for reverse time migration. Lead. edge 29(1), 48–58 (2010)

    Article  Google Scholar 

  8. Curtis, A., Lomax, A.: Prior information, sampling distributions, and the curse of dimensionality. Geophysics 66(2), 372–378 (2001)

    Article  Google Scholar 

  9. Floris, F., Bush, M., Cuypers, M., Roggero, F., Syversveen, A.R.: Methods for quantifying the uncertainty of production forecasts: a comparative study. Pet. Geosci. 7(S), S87–S96 (2001)

    Article  Google Scholar 

  10. Fu, H., Clapp, R.G., Lindtjorn, O.: Revisiting Convolution and Fft on Parallel Computation Platforms. In: SEG Technical Program Expanded Abstracts 2010, pp. 3071–3075. Society of Exploration Geophysicists (2010)

  11. Geuzaine, C., Remacle, J.F.: Gmsh: a 3-D finite element mesh generator with built-in pre-and post-processing facilities. Int. J. Numer. Methods Eng. 79(11), 1309–1331 (2009)

    Article  Google Scholar 

  12. Ghasemi, M., Gildin, E.: Model order reduction in porous median flow simulation using quadratic bilinear formulation. Comput. Geosci. 20(3), 723–735 (2016)

    Article  Google Scholar 

  13. Grepl, M.A., Patera, A.T.: A posteriori error bounds for reduced basis approximations of parametrized parabolic partial differential equations. ESAIM: Math. Modell. Numer. Anal. 39(1), 157–181 (2005)

    Article  Google Scholar 

  14. Hecht, F., Pironneau, O., Le Hyaric, A., Ohtsuka, K.: Freefem++. Numerical Mathematics and Scientific Computation. Laboratoire JL Lions, Université Pierre et Marie Curie. (2007)

  15. Herrmann, F.J., Friedlander, M.P., Yilmaz, O.: Fighting the curse of dimensionality: compressive sensing in exploration seismology. IEEE Signal Process. Mag. 29(3), 88–100 (2012)

    Article  Google Scholar 

  16. Hesthaven, J.S., Rozza, G., Stamm, B., et al.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer Briefs in Mathematics, Springer (2016)

  17. Huynh, D., Nguyen, N., Rozza, G., Patera, A.: Documentation for rbMIT software: I. Reduced basis (RB) for dummies. Massachusetts Institute of Technology, 2007-2010.

  18. Cacace, M., Jacquey, A.: Flexible parallel implicit modelling of coupled thermal–hydraulic–mechanical processes in fractured rocks. Solid Earth 8(5), 921–941 (2017)

    Article  Google Scholar 

  19. Jacquey, A., Cacace, M., Blöcher, G., Milsch, H., Deon, F.: From localized to homogeneous deformation of porous rocks–insights from laboratory experiments and numerical modelling. In: Geo-Proc 2017 Book of Abstracts: 6th International Conference on Coupled THMC Processes in Geosystems, pp. 50

  20. Jülich Supercomputing Centre: JURECA: General-purpose supercomputer at Jülich Supercomputing Centre. Journal of large-scale research facilities 2(A62). DOI (2016)

  21. Jung, N.: Error estimation for parametric model order reduction and its application VDI Verlag (2012)

  22. Kärcher, M., Tokoutsi, Z., Grepl, M.A., Veroy, K.: Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls. J. Sci. Comput. 75(1), 276–307 (2018)

    Article  Google Scholar 

  23. Kirk, B.S., Peterson, J.W., Stogner, R.H., Carey, G.F.: Libmesh: a c++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3–4), 237–254 (2006)

    Article  Google Scholar 

  24. Knezevic, D.J., Peterson, J.W.: A high-performance parallel implementation of the certified reduced basis method. Comput. Methods Appl. Mech. Eng. 200(13–16), 1455–1466 (2011).

    Article  Google Scholar 

  25. van Leeuwen, P.J.: Nonlinear data assimilation in geosciences: an extremely efficient particle filter. Q. J. Roy. Meteorol. Soc. 136(653), 1991–1999 (2010)

    Article  Google Scholar 

  26. Martini, I., Rozza, G., Haasdonk, B.: Reduced basis approximation and a-posteriori error estimation for the coupled Stokes-Darcy system. Adv. Comput. Math. 41(5), 1131–1157 (2015)

    Article  Google Scholar 

  27. Midttømme, K., Roaldset, E., Aagaard, P.: Thermal conductivity claystones and mudstones of selected from England. Clay Miner. 33(1), 131–145 (1998)

    Article  Google Scholar 

  28. Milk, R., Rave, S., Schindler, F.: Pymorgeneric algorithms and interfaces for model order reduction. SIAM J. Sci. Comput. 38(5), S194–S216 (2016)

    Article  Google Scholar 

  29. Poulet, T., Veveakis, M.: A viscoplastic approach for pore collapse in saturated soft rocks using redback: an open-source parallel simulator for rock mechanics with dissipative feedbacks. Comput. Geotech. 74, 211–221 (2016)

    Article  Google Scholar 

  30. Poulet, T., Veveakis, M., Paesold, M., Regenauer-Lieb, K.: REDBACK: an Open-Source Highly Scalable Simulation Tool for Rock Mechanics with Dissipative Feedbacks. In: AGU Fall Meeting Abstracts (2014)

  31. Prud’homme, 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(1), 70–80 (2002)

    Article  Google Scholar 

  32. Prud’homme, C., Chabannes, V., Doyeux, V., Ismail, M., Samake, A., Pena, G.: Feel++: a Computational Framework for Galerkin Methods and Advanced Numerical Methods. In: ESAIM: Proceedings, vol. 38, pp. 429–5455. EDP Sciences (2012)

  33. Quarteroni, A., Manzoni, A., Negri, F.: Reduced basis methods for partial differential equations: an introduction. UNITEXT Springer International Publishing (2015)

  34. Rizzo, C.B., de Barros, F.P., Perotto, S., Oldani, L., Guadagnini, A.: Adaptive POD model reduction for solute transport in heterogeneous porous media. Comput. Geosci. 22(1), 297–308 (2018)

    Article  Google Scholar 

  35. Rousset, M.A., Huang, C.K., Klie, H., Durlofsky, L.J.: Reduced order modeling for thermal recovery processes. Comput. Geosci. 18(3–4), 401–415 (2014)

    Article  Google Scholar 

  36. Tonks, M.R., Gaston, D., Millett, P.C., Andrs, D., Talbot, P.: An object-oriented finite element framework for multiphysics phase field simulations. Comput. Mater. Sci. 51(1), 20–29 (2012)

    Article  Google Scholar 

  37. University of Stuttgart, University of Münster, University of Ulm, Sandia Lawrence Livermore National Research Laboratory:

  38. de la Varga, M., Schaaf, A., Wellmann, F.: Gempy 1.0: open-source stochastic geological modeling and inversion. Geoscientific Model Development (2019)

  39. Veroy, K., Patera, A.: Certified real-time solution of the parametrized steady incompressible navier–stokes equations: rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Methods Fluids 47(8–9), 773–788 (2005)

    Article  Google Scholar 

  40. Veroy, K., Prud’homme, C., Rovas, D.V., Patera, A.T.: A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In: Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, vol. 3847, pp. 23–26, Orlando, FL (2003)

  41. Volkwein, S.: Model reduction using proper orthogonal decomposition. Lecture Notes, Institute of Mathematics and Scientific Computing, University of Graz. see (2011)

  42. Wellmann, J.F., Regenauer-Lieb, K.: Uncertainties have a meaning: information entropy as a quality measure for 3-D geological models. Tectonophysics 526, 207–216 (2012)

    Article  Google Scholar 

  43. Wellmann, J.F., Reid, L.B.: Basin-scale geothermal model calibration: experience from the Perth Basin, Australia. Energy Procedia 59, 382–389 (2014)

    Article  Google Scholar 

Download references


We would like to thank two anonymous reviewers for helping to improve this paper through their useful remarks and comments. We also gratefully acknowledge the computing time granted through JARA-HPC on the supercomputer JURECA at Forschungszentrum Jülich.


This study was financially supported by the DFG through DFG Project GSC111.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Denise Degen.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Degen, D., Veroy, K. & Wellmann, F. Certified reduced basis method in geosciences. Comput Geosci 24, 241–259 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Model order reduction
  • Reduced basis method
  • Finite element method
  • Geothermal conduction
  • MOOSE framework