×

zbMATH — the first resource for mathematics

Hybrid evolutionary algorithm with Hermite radial basis function interpolants for computationally expensive adjoint solvers. (English) Zbl 1147.49301
Summary: We present an evolutionary algorithm hybridized with a gradient-based optimization technique in the spirit of Lamarckian learning for efficient design optimization. In order to expedite gradient search, we employ local surrogate models that approximate the outputs of a computationally expensive Euler solver. Our focus is on the case when an adjoint Euler solver is available for efficiently computing the sensitivities of the outputs with respect to the design variables. We propose the idea of using Hermite interpolation to construct gradient-enhanced radial basis function networks that incorporate sensitivity data provided by the adjoint Euler solver. Further, we conduct local search using a trust-region framework that interleaves gradient-enhanced surrogate models with the computationally expensive adjoint Euler solver. This ensures that the present hybrid evolutionary algorithm inherits the convergence properties of the classical trust-region approach. We present numerical results for airfoil aerodynamic design optimization problems to show that the proposed algorithm converges to good designs on a limited computational budget.

MSC:
49K10 Optimality conditions for free problems in two or more independent variables
49K40 Sensitivity, stability, well-posedness
65D05 Numerical interpolation
68T05 Learning and adaptive systems in artificial intelligence
92D15 Problems related to evolution
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Alexandrov, N., Dennis, J.E., Lewis, R.M., Torczon, V.: A trust region framework for managing the use of approximation models in optimization. Struct. Optim. 15(1), 16–23 (1998)
[2] Bishop, C.: Neural Networks for Pattern Recognition. Oxford University Press, Oxford (1995) · Zbl 0868.68096
[3] Booker, A.J., Dennis, Jr J.E., Frank, P.D., Serafini, D.B., Torczon, V., Trosset, M.W.: A rigorous framework for optimization of expensive functions by surrogates. Struct. Optim. 17(1), 1–13 (1998)
[4] Burgreen, G.W., Baysal, O.: Three-dimensional aerodynamic shape optimization of wings using discrete sensitivity analysis. AIAA J. 34(9), 1761–1770 (1996) · Zbl 0909.76082
[5] El-Beltagy, M.A., Nair, P.B., Keane, A.J.: Metamodelling techniques for evolutionary optimization of computationally expensive problems: promises and limitations. In: Proceedings of the Genetic and Evolutionary Computation Conference, pp. 196–203, Morgan Kaufman, Los Altos (1999)
[6] Fasshauer, G.: Hermite interpolation with radial basis functions on spheres. Adv. Comput. Math. 10, 81–96 (1999) · Zbl 0918.41003
[7] Foster, I., Kesselman, C., Tuecke, S.: The anatomy of the grid: enabling scalable virtual organizations, Int. J. Supercomput. Appl. 15(3) (2001)
[8] Friedman, J.H.: Multivariate adaptive regression splines. Ann. Stat. 19, 1–141 (1991) · Zbl 0765.62064
[9] Giannakoglou, K.C.: Design of optimal aerodynamic shapes using stochastic optimization methods and computational intelligence. Prog. Aerosp. Sci. 38, 43–76 (2001)
[10] Giles, M.B., Pierce, N.A.: An introduction to the adjoint approach to design. Flow Turbul. Combust. 65, 393–415 (2000) · Zbl 0996.76023
[11] Hardy, R.L.: Theory and applications of the multiquadric-biharmonic method. Comput. Math. Appl. 19, 163–208 (1990) · Zbl 0692.65003
[12] Hicks, R.M., Henne, P.A.: Wing design by numerical optimization. J. Aircr. 15(7), 407–412 (1978)
[13] Ho, Q.T., Ong, Y.S., Cai, W.T.: Gridifying aerodynamic design problem using GridRPC. In: Second Grid and Cooperative Computing: Second International Workshop 2003, Part I, Shanghai, China. Lecture Notes in Computer Science, vol. 3032, pp. 83–90. Springer, Heidelberg (2004)
[14] Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988) · Zbl 0676.76055
[15] Jameson, A., Reuther, J.: Control theory based airfoil design using the Euler equations. AIAA 94-4272-CP (1994)
[16] Jameson, A., Vassberg, J.C.: Computational fluid dynamics for aerodynamic design: its current and future impact. AIAA 2001-0538 (January 2001)
[17] Jin, R., Chen, W., Simpson, T.W.: Comparative studies of metamodeling techniques under multiple modeling criteria. Struct. Multidiscip. Optim. 23(1), 1–13 (2001)
[18] Jin, Y.: A comprehensive survey of fitness approximation in evolutionary computation. Soft Comput. J. 9(1), 3–12 (2005) · Zbl 05036836
[19] Jin, Y., Olhofer, M., Sendhoff, B.: A framework for evolutionary optimization with approximate fitness functions. IEEE Trans. Evol. Comput. 6(5), 481–494 (2002) · Zbl 05452037
[20] Keane, A.J., Nair, P.B.: Computational Approaches for Aerospace Design. Wiley, New York (2005). Chapter 4
[21] Lawrence, C.T., Tits, A.L.: A computionally efficient feasible sequential quadratic programming algorithm. SIAM J. Optim. 11(4), 1092–1118 (2001) · Zbl 1035.90105
[22] Liang, K.H., Yao, X., Newton, C.: Evolutionary search of approximated N-dimensional landscapes. Int. J. Knowl.-Based Intell. Eng. Syst. 4(3), 172–183 (2001)
[23] Lions, J.L.: Optimal Control Of Systems Governed by Partial Differential Equations. Springer, Berlin (1971). Translated by S.K. Mitter · Zbl 0203.09001
[24] Myers, R.H., Montgomery, D.C.: Response Surface Methodology: Process and Product Optimization Using Designed Experiments. Wiley, New York (1995) · Zbl 1161.62392
[25] Nanyang Campus Grid: http://ntu-cg.ntu.edu.sg/
[26] Narcowich, F.J., Ward, J.D.: Generalized Hermite interpolation via matrix-valued conditionally positive definite functions. Math. Comput. 63(208), 661–687 (1994) · Zbl 0806.41003
[27] Ng, H.K., Lim, D., Ong, Y.S., Lee, B.S., Freund, L., Parvez, S., Sendhoff, B.: A multi-cluster grid enabled evolution framework for aerodynamic airfoil design optimization. In: Wang, L.P., Chen, K., Ong, Y.S. (eds.) International Conference on Natural Computing. Lecture Notes in Computer Science, vol. 3611, pp. 1112–1121. Springer, New York (2005)
[28] Ong, Y.S., Keane, A.J.: Meta-Lamarckian learning in memetic algorithm. IEEE Trans. Evol. Comput. 8(2), 99–110 (2004) · Zbl 05452044
[29] Ong, Y.S., Nair, P.B., Keane, A.J.: Evolutionary optimization of computationally expensive problems via surrogate modeling. Am. Inst. Aeronaut. Astronaut. J. 41(4), 687–696 (2003)
[30] Ong, Y.S., Nair, P.B., Keane, A.J., Wong, K.W.: Surrogate-assisted evolutionary optimization frameworks for high-fidelity engineering design problems. In: Jin, Y. (ed.) Knowledge Incorporation in Evolutionary Computation, pp. 307–331. Studies in Fuzziness and Soft Computing Series. Springer, New York (2004)
[31] Ong, Y.S., Nair, P.B., Lum, K.Y.: Max-min surrogate-assisted evolutionary algorithm for robust aerodynamic design. IEEE Trans. Evol. Comput. 10(4), 392–404 (2006) · Zbl 05452149
[32] Reuther, J., Jameson, A., Alonso, J.J., Rimlinger, M.J., Saunders, D.: Constrained multipoint aerodynamic shape optimization using adjoint formulation and parallel computers. AIAA Paper 97-0103 (January 1997)
[33] Rodriguez, J.F., Renaud, J.E., Watson, L.T.: Convergence of trust region augmented Lagrangian methods using variable fidelity approximation data. Struct. Optim. 5(3–4), 141–156 (1998)
[34] Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989) · Zbl 0955.62619
[35] Serafini, D.B.: A framework for managing models in nonlinear optimization of computally expensive functions. Ph.D. Thesis, Rice University (1998)
[36] Simpson, T.W., Booker, A.J., Ghosh, D., Giunta, A.A., Koch, P.N., Yang, R.J.: Approximation methods in multidisciplinary analysis and optimization: a panel discussion, In: Proceedings of the Third ISSMO/AIAA Internet Conference on Approximations in Optimization, pp. 14–25 (2002)
[37] Song, W.B., Keane, A.J.: A study of shape parameterisation methods for airfoil optimisation. In: 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, pp. 2031–2038 (2004). AIAA 2004-4482
[38] Williams, C.K.I., Rasmussen, C.E.: Gaussian processes for regression. In: Touretsky, D.S., Mozer, M.C., Hasselmo, M.E. (eds.) Advances in Neural Information Processing Systems. MIT Press, Cambridge (1996)
[39] Zhongmin, W.: Hermite-Birkhoff interpolation of scattered data by radial basis functions. Approx. Theory Appl. 8, 1–10 (1992)
[40] Zhou, Z.Z., Ong, Y.S., Nair, P.B., Keane, A.J., Lum, K.Y.: Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans. Syst. Man Cybern. Part C 36(6), 814–823 (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.