1887
  1. Home
  2. Conferences
  3. Conference Proceedings
  4. ECMOR XVII
  5. Conference paper

Abstract

Summary

For highly nonlinear problems, the objective function f(x) may have multiple local optima and it is desired to locate all of them. Analytical or adjoint-based derivatives may not be available for most real optimization problems, especially, when responses of a system are predicted by numerical simulations. 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.

In this paper, we generalized the distribution techniques of the DGN optimization method and developed a new distributed quasi-Newton (DQN) optimization method that is applicable to generic optimization problems. It can handle generalized objective functions F(x,y(x))=f(x) with both explicit variables x and implicit variables, i.e., simulated responses, y(x). The partial derivatives of F(x,y) with respect to both x and y can be computed analytically, whereas the partial derivatives of y(x) with respect to x (sensitivity matrix) is estimated by applying the same efficient information sharing mechanism implemented in the DGN optimization method. An ensemble of quasi-Newton optimization tasks is distributed among multiple high-performance-computing (HPC) cluster nodes. The simulation results generated from one optimization task 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 task is approximated by either the linear-interpolation (LI) method or the support-vector-regression (SVR) method, using some or all training data points. 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 task is generated by solving a quasi-Newton trust-region subproblem for the next iteration.

The proposed 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 different optimization problems. For all test problems, the DQN method can find multiple optima of the objective function with reasonably small numbers of iterations (30 to 50). Compared to sequential model-based derivative-free optimization methods, the DQN method can reduce the computational cost, in terms of the number of simulations required for convergence, by a factor of 3 to 10.

© EAGE Publications BV
Loading

Article metrics loading...

/content/papers/10.3997/2214-4609.202035131
2020-09-14
2024-04-16

  • From This Site

    /content/papers/10.3997/2214-4609.202035131
    dcterms_title,dcterms_subject,pub_keyword
    -contentType:Journal -contentType:Contributor -contentType:Concept -contentType:Institution
    10
    5
Loading full text...

Full text loading...

References

  1. [1]Audet, C. and Dennis,Jr.J.E.. Mesh Adaptive Direct search Algorithms for Constrained Optimization, SIAM J. Opt., 17: 188–217, 2006.
     [Google Scholar]
  2. [2]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.
     [Google Scholar]
  3. [3]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 August2016.
     [Google Scholar]
  4. [4]Chen, C, et al.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.
     [Google Scholar]
  5. [5]Chen, Y., Oliver, D.S., and Zhang, D.Efficient Ensemble-based Closed-loop Production Optimization. SPE J., 14(4): 634–645, 2009.
     [Google Scholar]
  6. [6]Chen, Y. and Oliver, D.S.Ensemble-based Closed-loop Optimization Applied to Brugge Field. SPE Reservoir Engineering & Evaluation, 13(1): 56–71, 2010.
     [Google Scholar]
  7. [7]Ding, Y., Lushi, E. and Li, Q.Investigation of Quasi-Newton methods for Unconstrained Optimization, (http://people.math.sfu.ca/∼elushi/project_833.pdf), Simon Fraser University, Canada, 2004.
     [Google Scholar]
  8. [8]Do, S.T. and Reynolds, A.C.Theoretical Connections between Optimization Algorithms Based on an Approximation Gradient. Computational Geosciences, 17(6): 959–973, 2013.
     [Google Scholar]
  9. [9]Erway, J.B., Gill, P.E. and Griffin, J.D.Iterative Methods for Finding a Trust-Region Step. SIAM J. Optim., 20(2): 1110–1131, 2009.
     [Google Scholar]
  10. [10]Fonseca, et al.Robust Ensemble-based Multi-objective Optimization. Procedding of the 14th European Conference on the Mathematics of Oil Recovery (ECMOR XIV), held in Catania, Italy, 8–11 Sept., 2014.
     [Google Scholar]
  11. [11]Fonseca, et al.Ensemble-based Multi-objective Optimization of On-off Control Devices Under Geological Uncertainly. SPE Reservoir Engineering & Evaluation, 18(4): 554–563, 2015.
     [Google Scholar]
  12. [12]Fonseca, R.R., Chen, B., Jansen, J.D., Reynolds, A.C.A Stochastic Simplex Approximate Gradient (StoSAG) for Optimization Under Uncertainty. Int. J. Numer. Meth. Engng, 109: 1756–1776, 2017.
     [Google Scholar]
  13. [13]Gao, G. et al.A Parallelized and Hybrid Data-Integration Algorithm for History Matching of Geologically Complex Reservoirs. SPE J., 21(6): 2155–2174, 2016.
     [Google Scholar]
  14. [14]Gao, G., et al.Distributed Gauss-Newton Optimization Method for History Matching Problems with Multiple Best Matches. Computational Geosciences, 21(5–6): 1325–1342, 2017a.
     [Google Scholar]
  15. [15]Gao, G. et al.A Gauss-Newton Trust Region Solver for Large Scale History Matching Problems. SPE J, 22(6), December 2017b (DOI: http://dx.doi.org/10.2118/182602-PA).
     [Google Scholar]
  16. [16]Gao, et al.Performance Enhancement of Gauss-Newton Trust Region Solver for Distributed Gauss-Newton Optimization methods.Computational Geosciences (published online), June 2019a.
     [Google Scholar]
  17. [17]Gao, et al.Gaussian Mixture Model Fitting Method for Uncertainly Quantification by Conditioning to Production Data.Computational Geosciences (published online), June 2019b.
     [Google Scholar]
  18. [18]Gao, et al.Integration of Support Vector Regression with Distributed Gauss-Newton Optimization Method and Its Applications to the Uncertainly Assessment of Unconventional Assets. SPE Reservoir Evaluation & Engineering. 21(4): 1007–1026, 2018(a).
     [Google Scholar]
  19. [19]Gao, 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(b).
     [Google Scholar]
  20. [20]Gould, N.I.M., et al.Solving the Trust-Region Subproblem Using the Lanczos Method. SIAM J. Optim., 9(2): 504–525, 1999.
     [Google Scholar]
  21. [21]Gould, N.I.M., Robinson, D. and Thorne, H.S.On Solving Trust-Region and Other Regularized Subproblems in Optimization, Math. Prog. Comp., 2(1): 21–57, 2010.
     [Google Scholar]
  22. [22]Hooke, R. and Jeeves, T.A.Direct Search solution of Numerical and Statical Problems, J. Assoc. Comput. Machinery, 8: 212–229, 1961.
     [Google Scholar]
  23. [23]Jansen, J.D.Adjoint-based Optimization of Multi-phase Flow through Porous Media—A Review. Computers & Fluids, 46(1): 40–51, 2011.
     [Google Scholar]
  24. [24]Li, R., Reynolds, A.C, and Oliver, D.S.History Matching of Three-Phase Flow Production Data. SPE J., 8(4): 328–340, 2003.
     [Google Scholar]
  25. [25]More, J.J. and Sorensen, D.C.Computing a Trust Region Step, SIAM Journal, on Scientific and Statistical Computing, 4(3): 553–572, 1983.
     [Google Scholar]
  26. [26]Nelder, J.A. and Mead, R.A Simplex Method for Function Minimization, The Computer Journal, 7:308–313, 1965.
     [Google Scholar]
  27. [27]Nocedal, J. and Wright, S.J.Numerical Optimization. Springer, New York City, 1999.
     [Google Scholar]
  28. [28]Oliver, D.S.On Conditional Simulation to Inaccurate Data. Math. Geology, 28: 811–817, 1996.
     [Google Scholar]
  29. [29]Oliver, D.S. and Chen, Y.Recent Progress on Reservoir History Matching: a Review. Computational Geoscience, 15(1): 185–211,2011.
     [Google Scholar]
  30. [30]Oliver, D.S., Reynolds, A.C, and Liu. Inverse Theory for Petroleum Reservoir Characterization and History Matching. Cambridge University Press, 2008.
     [Google Scholar]
  31. [31]Powell, M.J.D.Least Frobenius Norm Updating of Quadratic Models That satisfy Interpolation Conditions, Mathematical Programming, 100(1): 183–215, 2004.
     [Google Scholar]
  32. [32]Rafiee, J. and Reynolds, A.CA 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.
     [Google Scholar]
  33. [33]Rojas, M., Santos, S.A. and Sorensen, D.C.Algorithm 873: LSTRS: MATLAB Software for Large-Scale Trust-Region Subproblems and Regularization. ACM Trans. Math. Software. 34(2): 1–28, 2008.
     [Google Scholar]
  34. [34]Sarma, P., Durlofsky, L., and 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.
     [Google Scholar]
  35. [35]Tarantola, A.Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM. 2005.
     [Google Scholar]
  36. [36]Volkov, O. and Voskov, D.Effect of time stepping strategy on adjoint-based production optimization. Computational Geosciences, 20(3): 707–722, 2016.
     [Google Scholar]
  37. [37]Wild, S.M.Derivative Free Optimization Algorithms for Computationally Expensive Functions. Ph.D Thesis, Cornell University, 2009.
     [Google Scholar]
  38. [38]Zhao, H., Li, G., Reynolds, A.C., Yao, J.Large-scale History Matching with Quadratic Interpolation Models. Computational Geosciences, 17: 117–138, 2012.
     [Google Scholar]
http://instance.metastore.ingenta.com/content/papers/10.3997/2214-4609.202035131
Loading
Distributed Quasi-Newton Derivative-Free Optimization Method for Optimization Problems with Multiple Local Optima
Conference Proceedings 2020, 1 (2020); https://doi.org/10.3997/2214-4609.202035131
/content/papers/10.3997/2214-4609.202035131
/content/papers/10.3997/2214-4609.202035131
Loading

Data & Media loading...

Most Cited This Month Most Cited RSS feed

This is a required field
Please enter a valid email address
Approval was a Success
Invalid data
An Error Occurred
Approval was partially successful, following selected items could not be processed due to error