Article
Full entry |
PDF
(0.8 MB)
Feedback
PDF
(0.8 MB)
Feedback
Keywords:
optimal control; parabolic partial differential equations; backward Euler method; nonmonotone LBFGS method
optimal control; parabolic partial differential equations; backward Euler method; nonmonotone LBFGS method
Summary:
In this paper a nonmonotone limited memory BFGS (NLBFGS) method is applied for approximately solving optimal control problems (OCPs) governed by one-dimensional parabolic partial differential equations. A discretized optimal control problem is obtained by using piecewise linear finite element and well-known backward Euler methods. Afterwards, regarding the implicit function theorem, the optimal control problem is transformed into an unconstrained nonlinear optimization problem (UNOP). Finally the obtained UNOP is solved by utilizing the NLBFGS method. In comparison to other existing methods, the NLBFGS method shows a significant improvement especially for nonlinear and ill-posed control problems.
References:
[1] Albrecher, H., Runggaldier, W. J., Schachermayer, W.: Advanced Financial Modelling. Radon series on computational and applied mathematics, Walter de Gruyter, 2009. DOI 10.1515/9783110213140 | MR 2605639
[2] Amini, K., Ahookhosh, M., Nosratipour, H.: An inexact line search approach using modified nonmonotone strategy for unconstrained optimization. Numer. Algor. 66 (2014), 49-78. DOI 10.1007/s11075-013-9723-x | MR 3197357
[3] Aniţa, S., Arnautu, V., Capasso, V.: An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB. Birkhäuser, Boston 2011. MR 2761466
[4] Bazaraa, M. S., Sherali, H. D., Shetty, C. M.: Nonlinear Programming: Theory and Algorithms. Wiley, New York 2006. DOI 10.1002/0471787779 | MR 2218478 | Zbl 1140.90040
[5] Borzi, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations. SIAM, 2012. DOI 10.1137/1.9781611972054 | MR 2895881
[6] Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York 2012. DOI 10.1007/978-1-4612-3172-1 | MR 1115205
[7] Cantrell, S., Cosner, C., Ruan, S.: Spatial Ecology. CRC Mathematical and Computational Biology, CRC Press 2009. DOI 10.1201/9781420059861 | MR 2664165
[8] Chang, R. Y., Yang, S. Y.: Solution of two point boundary value problems by generalized orthogonal polynomials and application to optimal control of lumped and distributed parameter systems. International Journal of Control 43 (1986), 1785-1802. DOI 10.1080/00207178608933572 | MR 0838620
[9] Christofides, P., Armaou, A., Lou, Y., Varshney, A.: Control and Optimization of Multiscale Process Systems, Control Engineering. Birkhäuser, Boston 2008. DOI 10.1007/978-0-8176-4793-3 | MR 2489170
[10] Klerk, E. De, Roos, C., Terlaky, T.: Nonlinear Optimization. University Of Waterloo, Waterloo 2005.
[11] Griva, I., Nash, S. G., Sofer, A.: Linear and Nonlinear Optimization. SIAM, Philadelphia 2009. DOI 10.1137/1.9780898717730 | MR 2472514
[12] Haslinger, J., Neittaanmäki, P.: Finite Element Approximation for Optimal Shape, Material and Topology Design. Wiley, 1996. MR 1419500
[13] Heinkenschloss, M.: Numerical Solution of Implicitly Constrained Optimization Problems. CAAM Technical Report TR08-05, Rice University (2008).
[14] Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Springer, Netherlands 2008. MR 2516528
[15] Horng, I. R., Chou, J. H.: Application of shifted Chebyshev series to the optimal control of linear distributed-parameter systems. Int. J. Control 42 (1985), 233-241. DOI 10.1080/00207178508933359 | MR 0802185
[16] Hu, W. W.: Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems. Ph.D. Thesis, Department of Mathematics, Virginia Tech. 2012.
[17] Ji, Y., Li, Y., Zhang, K., Zhan, X.: A new nonmonotone trust-region method of conic model for solving unconstrained optimization. J. Comput. Appl. Math. 233 (2010), 1746-1754. DOI 10.1016/j.cam.2009.09.011 | MR 2564012
[18] Kunisch, K., Volkwein, S.: Control of the burgers equation by a reduced-order approach using proper orthogonal decomposition. J. Optim. Theory Appl. 102 (1999), 345-371. DOI 10.1023/a:1021732508059 | MR 1706822
[19] Lions, J. L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, 1971. DOI 10.1007/978-3-642-65024-6 | MR 0271512
[20] Liu, D., Nocedal, J.: On the limited memory BFGS method for large-scale optimization. Math. Program. 45 (1989), 503-528. DOI 10.1007/bf01589116 | MR 1038245
[21] Merino, P.: Finite element error estimates for an optimal control problem governed by the Burgers equation. Comput. Optim. Appl. 63 (2016), 793-824. DOI 10.1007/s10589-015-9790-0 | MR 3465457
[22] Meyer, C., Philip, P., Tröltzsch, F.: Optimal control of a semilinear PDE with nonlocal radiation interface conditions. SIAM J. Control Optim. 45 (2006), 699-721. DOI 10.1137/040617753 | MR 2246096
[23] Noack, B. R., Morzynski, M., Tadmor, G.: Reduced-Order Modelling for Flow Control. Springer, Vienna 2011. DOI 10.1007/978-3-7091-0758-4
[24] Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35 (1980) 773-782. DOI 10.1090/s0025-5718-1980-0572855-7 | MR 0572855 | Zbl 0464.65037
[25] Nocedal, J., Wright, S.: Numerical Optimization. Springer, New York 2006. DOI 10.1007/b98874 | MR 2244940 | Zbl 1104.65059
[26] Nosratipour, H., Borzabadi, A. H., Fard, O. S.: Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai-Borwein gradient method. Int. J. Comput. Math. 95 (2018) 1858-1873. DOI 10.1080/00207160.2017.1343472 | MR 3817213
[27] Nosratipour, H., Borzabadi, A. H., Fard, O. S.: On the nonmonotonicity degree of nonmonotone line searches. Calcolo 54 (2017) 1217-1242. DOI 10.1007/s10092-017-0226-3 | MR 3735813
[28] Nosratipour, H., Fard, O. S., Borzabadi, A. H.: An adaptive nonmonotone global Barzilai-Borwein gradient method for unconstrained optimization. Optimization 66 (2017) 641-655. DOI 10.1080/02331934.2017.1287702 | MR 3610318
[29] Rad, J. A., Kazem, S., Parand, K.: Optimal control of a parabolic distributed parameter system via radial basis functions. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2559-2567. DOI 10.1016/j.cnsns.2013.01.007 | MR 3168052
[30] Razzaghi, M., Arabshahi, A.: Optimal control of linear distributed-parameter systems via polynomial series. Int. J. Systems Sci. 20 (1989), 1141-1148. DOI 10.1080/00207728908910200 | MR 0988803
[31] Sabeh, Z., Shamsi, M., Dehghan, M.: Distributed optimal control of the viscous Burgers equation via a Legendre pseudo-spectral approach. Math. Methods Appl. Sci. 39 (2016), 3350-3360. DOI 10.1002/mma.3779 | MR 3521259
[32] Strang, G., Fix, G.: An Analysis of the Finite Element Method. Wellesley-Cambridge Press, 2008. MR 2743037 | Zbl 0356.65096
[33] Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications. Graduate studies in mathematics, American Mathematical Society, 2010. DOI 10.1090/gsm/112 | MR 2583281
[34] Tröltzsch, F., Volkwein, S.: The SQP method for control constrained optimal control of the Burgers equation. ESAIM: COCV 6 (2001), 649-674. DOI 10.1051/cocv:2001127 | MR 1872392
[35] Wang, F. S., Jian, J. B.: A new nonmonotone linesearch SQP algorithm for unconstrained minimax problem. Numer. Funct. Anal. Optim. 35 (2014), 487-508. DOI 10.1080/01630563.2013.873454 | MR 3177067
[36] Yılmaz, F., Karasözen, B.: Solving distributed optimal control problems for the unsteady Burgers equation in COMSOL multiphysics. J. Comput. Appl. Math. 235 (2011), 4839-4850. DOI 10.1016/j.cam.2011.01.002 | MR 2805724
[37] Zhang, H., Hager, W. W.: A nonmonotone line search technique and its application to unconstrained optimization. SIAM J. Optim. 14 (2004), 1043-1056. DOI 10.1137/s1052623403428208 | MR 2112963
Search
Partner of
