Skip to main content
SpringerLink
Log in

Distributed quasi-Newton derivative-free optimization method for optimization problems with multiple local optima

  • Original Paper
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

The distributed Gauss-Newton (DGN) optimization method performs quite efficiently and robustly for history-matching problems with multiple best matches. However, this method is not applicable for generic optimization problems, e.g., life-cycle production optimization or well location optimization. This paper introduces a generalized form of the objective functions F(x, y(x)) = f(x) with both explicit variables x and implicit variables (or simulated responses), y(x). The split in explicit and implicit variables is such that partial derivatives of F(x, y) with respect to both x and y can be computed analytically. An ensemble of quasi-Newton optimization threads is distributed among multiple high-performance-computing (HPC) cluster nodes. The simulation results generated from one optimization thread are shared with others by updating a common set of training data points, which records simulated responses of all simulation jobs. The sensitivity matrix at the current best solution of each optimization thread is approximated by the linear-interpolation method. The gradient of the objective function is then analytically computed using its partial derivatives with respect to x and y and the estimated sensitivities of y with respect to x. The Hessian is updated using the quasi-Newton formulation. A new search point for each distributed optimization thread is generated by solving a quasi-Newton trust-region subproblem (TRS) for the next iteration. The proposed distributed quasi-Newton (DQN) method is first validated on a synthetic history matching problem and its performance is found to be comparable with the DGN optimizer. Then, the DQN method is tested on a variety of optimization problems. For all test problems, the DQN method can find multiple optima of the objective function with reasonably small numbers of iterations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer, New York City (1999)

    Book  Google Scholar 

  2. Jansen, J.D.: Adjoint-based optimization of multi-phase flow through porous media—a review. Comput. Fluids. 46(1), 40–51 (2011)

    Article  Google Scholar 

  3. Li, R., Reynolds, A.C., Oliver, D.S.: History matching of three-phase flow production data. SPE J. 8(4), 328–340 (2003)

    Article  Google Scholar 

  4. Sarma, P., Durlofsky, L., Aziz, K.: Implementation of Adjoint Solution for Optimal Control of Smart Wells. Paper SPE 92864 presented at the SPE Reservoir Simulation Symposium held in the Woodlands, Texas, USA, 31 Jan.-2 Feb. (2005)

  5. Volkov, O., Voskov, D.: Effect of time stepping strategy on adjoint-based production optimization. Comput. Geosci. 20(3), 707–722 (2016)

    Article  Google Scholar 

  6. Nelder, J.A., Mead, R.: A simplex method for function minimization. Comput. J. 7, 308–313 (1965)

    Article  Google Scholar 

  7. Audet, C., Dennis Jr., J.E.: Mesh adaptive direct search algorithms for constrained optimization. SIAM J. Opt. 17, 188–217 (2006)

    Article  Google Scholar 

  8. Hooke, R., Jeeves, T.A.: Direct search solution of numerical and Statical problems. J. Assoc. Comput. Mach. 8, 212–229 (1961)

    Article  Google Scholar 

  9. Gao, G., Vink, J.C., Chen, C., Alpak, F.O., du, K.: A parallelized and hybrid data-integration algorithm for history matching of geologically complex reservoirs. SPE J. 21(6), 2155–2174 (2016)

    Article  Google Scholar 

  10. Wild, S.M.: Derivative Free Optimization Algorithms for Computationally Expensive Functions. Ph.D Thesis, Cornell University (2009)

  11. Powell, M.J.D.: Least Frobenius norm updating of quadratic models that satisfy interpolation conditions. Math. Program. 100(1), 183–215 (2004)

    Article  Google Scholar 

  12. Zhao, H., Li, G., Reynolds, A.C., Yao, J.: Large-scale history matching with quadratic interpolation models. Comput. Geosci. 17, 117–138 (2012)

    Article  Google Scholar 

  13. Chen, Y., Oliver, D.S., Zhang, D.: Efficient ensemble-based closed-loop production optimization. SPE J. 14(4), 634–645 (2009)

    Article  Google Scholar 

  14. Chen, Y., Oliver, D.S.: Ensemble-based closed-loop optimization applied to Brugge field. SPE Reserv. Eval. Eng. 13(1), 56–71 (2010)

    Article  Google Scholar 

  15. Do, S.T., Reynolds, A.C.: Theoretical connections between optimization algorithms based on an approximation gradient. Comput. Geosci. 17(6), 959–973 (2013)

    Article  Google Scholar 

  16. Fonseca, R.R., Chen, B., Jansen, J.D., Reynolds, A.C.: A stochastic simplex approximate gradient (StoSAG) for optimization under uncertainty. Int. J. Numer. Methods Eng. 109, 1756–1776 (2017)

    Article  Google Scholar 

  17. Gao, G., Vink, J.C., Chen, C., el Khamra, Y., Tarrahi, M.: Distributed gauss-Newton optimization method for history matching problems with multiple best matches. Comput. Geosci. 21(5–6), 1325–1342 (2017)

    Article  Google Scholar 

  18. Chen, C., Gao, G., Li, R., Cao, R., Chen, T., Vink, J.C., Gelderblom, P.: Global-search distributed-gauss-Newton optimization method and its integration with the randomized-maximum-likelihood method for uncertainty quantification of reservoir performance. SPE J. 23(5), 1496–1517 (2018)

    Article  Google Scholar 

  19. Oliver, D.S.: On conditional simulation to inaccurate data. Math. Geol. 28, 811–817 (1996)

    Article  Google Scholar 

  20. Guo, et al.: Integration of Support Vector Regression with Distributed Gauss-Newton Optimization Method and Its Applications to the Uncertainty Assessment of Unconventional Assets. SPE Reserv. Eval. Eng. 21(4), 1007–1026 (2018)

    Article  Google Scholar 

  21. Guo, et al.: Enhancing the Performance of the Distributed Gauss-Newton Optimization Method by Reducing the Effect of Numerical Noise and Truncation Error with Support-Vector-Regression. SPE J. 23(6), 2428–2443 (2018)

    Article  Google Scholar 

  22. Rafiee, J., Reynolds, A.C.: A Two-Level MCMC Based on the Distributed Gauss-Newton Method for Uncertainty Quantification. The 16th European Conference on the Mathematics of Oil Recovery, Barcelona, Spain, 3–6 September (2018)

  23. Gao, G., et al.: A gauss-Newton trust region solver for large scale history matching problems. SPE J. 22(6), (2017). https://doi.org/10.2118/182602-PA

  24. Gao, et al.: Performance Enhancement of Gauss-Newton Trust Region Solver for Distributed Gauss-Newton Optimization methods. Comput. Geosci. 24(2), 837–852 (2020)

    Article  Google Scholar 

  25. Ding, Y., Lushi, E., Li, Q.: Investigation of Quasi-Newton methods for Unconstrained Optimization. (http://people.math.sfu.ca/~elushi/project_833.pdf), Simon Fraser University, Canada (2004)

  26. Oliver, D.S., Chen, Y.: Recent Progress on reservoir history matching: a review. Comput. Geosci. 15(1), 185–211 (2011)

    Article  Google Scholar 

  27. Oliver, D.S., Reynolds, A.C., Liu, N.: Inverse theory for petroleum reservoir characterization and history matching. Cambridge University Press. (2008). https://doi.org/10.1017/CBO9780511535642

  28. Tarantola, A.: Inverse problem theory and methods for model parameter estimation. SIAM. (2005). https://doi.org/10.1137/1.9780898717921

  29. Chen, C., et al.: EUR Assessment of Unconventional Assets Using Parallelized History Matching Workflow Together with RML Method. Paper URTeC-2429986 presented at the Unconventional Resources Technology Conference held in San Antonio, Texas, USA, 1–3 August (2016)

  30. Gould, N.I.M., Robinson, D., Thorne, H.S.: On solving trust-region and other regularized subproblems in optimization. Math. Program. Comput. 2(1), 21–57 (2010)

    Article  Google Scholar 

  31. More, J.J., Sorensen, D.C.: Computing a trust region step. SIAM J. Sci. Stat. Comput. 4(3), 553–572 (1983)

    Article  Google Scholar 

  32. Erway, J.B., Gill, P.E., Griffin, J.D.: Iterative methods for finding a trust-region step. SIAM J. Optim. 20(2), 1110–1131 (2009)

    Article  Google Scholar 

  33. Gould, N.I.M., Lucidi, S., Roma, M., Toint, P.L.: Solving the trust-region subproblem using the Lanczos method. SIAM J. Optim. 9(2), 504–525 (1999)

    Article  Google Scholar 

  34. Rojas, M., Santos, S.A., Sorensen, D.C.: Algorithm 873: LSTRS: MATLAB software for large-scale trust-region subproblems and regularization. ACM Trans. Math. Softw. 34(2), 1–28 (2008)

    Article  Google Scholar 

  35. Gao, et al.: Gaussian Mixture Model Fitting Method for Uncertainty Quantification by Conditioning to Production Data. Comput. Geosci. 24(2), 663–681 (2020)

    Article  Google Scholar 

  36. Chen, C., et al.: Assisted History Matching Using Three Derivative-Free Optimization Algorithms. Paper SPE-154112-MS presented at the SPE Europec/EAGE Annual Conference held in Copenhagen, Denmark, 4–7 June (2012)

Download references

Author information

Authors and Affiliations

  1. Shell Global Solutions (US) Inc., 3333 HWY 6 S, Houston, TX, 77082, USA

    Guohua Gao, Yixuan Wang & Fredrik J.F.E. Saaf

  2. Shell Global Solutions International B.V., Kessler Park 1, 2288, GS, Rijswijk, The Netherlands

    Jeroen C. Vink & Terence J. Wells

Authors
  1. Guohua Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yixuan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jeroen C. Vink
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Terence J. Wells
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Fredrik J.F.E. Saaf
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guohua Gao.

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

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gao, G., Wang, Y., Vink, J.C. et al. Distributed quasi-Newton derivative-free optimization method for optimization problems with multiple local optima. Comput Geosci 26, 847–863 (2022). https://doi.org/10.1007/s10596-021-10101-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-021-10101-x

Keywords

Search

Navigation