Optimal control of viscous Burgers equation via an adaptive nonmonotone Barzilai–Borwein gradient method

References

• M.A. Abdulwahhab, Optimal system and exact solutions for the generalized system of 2-dimensional Burgers equations with infinite Reynolds number, Commun. Nonlinear Sci. Numer. Simul. 20 (2015), pp. 98–112. doi: 10.1016/j.cnsns.2014.05.008
   Web of Science ®Google Scholar
• S. Anita, V. Arnautu, and V. Capasso, An Introduction to Optimal Control Problems in Life Sciences and Economics: From Mathematical Models to Numerical Simulation with MATLAB, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser, Boston, MA, 2011.
   Google Scholar
• J. Barzilai and J.M. Borwein, Two-point step size gradient methods, IMA J. Numer. Anal. 8 (1988), pp. 141–148. doi: 10.1093/imanum/8.1.141
   Web of Science ®Google Scholar
• L.T. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, Real-time PDE-constrained Optimization, SIAM, Philadelphia, 2007.
   Google Scholar
• F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, Springer, New York, 2011.
   Google Scholar
• G. Buttazzo and A. Frediani, Control and Optimization of Multiscale Process Systems, Springer Optimization and Its Applications, Springer-Verlag, New York, 2009.
   Google Scholar
• P. Christofides, A. Armaou, Y. Lou, and A. Varshney, Control and Optimization of Multiscale Process Systems, Control Engineering, Birkhäuser, Boston, MA, 2008.
   Google Scholar
• I. Da, A. Canvar, and ŞahinA., Taylor-Galerkin and Ttaylor-collocation methods for the numerical solutions of Burgers equation using B-splines, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 2696–2708. doi: 10.1016/j.cnsns.2010.10.009
   Web of Science ®Google Scholar
• H. Egger and H.W. Engl, Tikhonov regularization applied to the inverse problem of option pricing: Convergence analysis and rates, Inverse Probl. 21 (2005), pp. 1027–1045. doi: 10.1088/0266-5611/21/3/014
   Web of Science ®Google Scholar
• L. Ge, W. Liu, and D. Yang, Adaptive finite element approximation for a constrained optimal control problem via multi-meshes, J. Sci. Comput. 41 (2009), pp. 238–255. doi: 10.1007/s10915-009-9296-y
   Web of Science ®Google Scholar
• L. Grippo, F. Lampariello, and S. Lucidi, A nonmonotone line search technique for Newtons method, SIAM J. Numer. Anal. 23 (1986), pp. 707–716. doi: 10.1137/0723046
   Web of Science ®Google Scholar
• L. Grippo and M. Sciandrone, Nonmonotone globalization techniques for the Barzilai–Borwein gradient method, Comput. Optim. Appl. 23 (2002), pp. 143–169. doi: 10.1023/A:1020587701058
   Web of Science ®Google Scholar
• I. Griva, S. Nash, and A. Sofer, Linear and Nonlinear Optimization, 2nd ed., Society for Industrial and Applied Mathematics, SIAM, Philadelphia, PA, 2009.
   Google Scholar
• M. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms, Computer Science and Scientific Computing, Academic Press, San Diego, 1989.
   Google Scholar
• S. Hazra, Large-scale PDE-constrained Optimization in Applications, Lecture Notes in Applied and Computational Mechanics, Springer, Berlin, 2009.
   Google Scholar
• M. Heinkenschloss, Numerical Solution of Implicitly Constrained Optimization Problems, Technical Report, Department of Computational and Applied Mathematics, Rice University, Houston, TX, 2008.
   Google Scholar
• M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, Springer, Netherlands, 2008.
   Google Scholar
• M. Holmes, Introduction to Numerical Methods in Differential Equations, Texts in Applied Mathematics, Springer, New York, 2007.
   Google Scholar
• N. Hritonenko, N. Kato, and Y. Yatsenko, Optimal control of investments in old and new capital under improving technology, J. Optim. Theory Appl. 172 (2017), pp. 247–266. doi: 10.1007/s10957-016-1022-y
   Web of Science ®Google Scholar
• B. Karaszen and F. Ylmaz, Optimal boundary control of the unsteady Burgers equation with simultaneous space–time discretization, Optim. Control Appl. Methods 35 (2014), pp. 423–434. doi: 10.1002/oca.2079
   Web of Science ®Google Scholar
• K. Kunisch and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (1999), pp. 345–371. doi: 10.1023/A:1021732508059
   Web of Science ®Google Scholar
• J. Nocedal and S. Wright, Numerical Optimization, Springer Series in Operations Research and Financial Engineering, Springer, New York, 2006.
   Google Scholar
• H. Nosratipour, O.S. Fard, A.H. Borzabadi, and F. Sarani, Stable equilibrium configuration of two bar truss by an efficient nonmonotone global Barzilai–Borwein gradient method in a fuzzy environment, Afr. Mat. 28 (2017), pp. 333–356. doi: 10.1007/s13370-016-0451-y
   Google Scholar
• L. Qi, K. Teo, and X. Yang, Optimization and Control with Applications, Applied Optimization, Springer, Boston, 2006.
   Google Scholar
• M. Raydan, The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem, SIAM J. Optim. 7 (1997), pp. 26–33. doi: 10.1137/S1052623494266365
   Web of Science ®Google Scholar
• J. Reddy and D. Gartling, The Finite Element Method in Heat Transfer and Fluid Dynamics, 3rd ed., Computational Mechanics and Applied Analysis, CRC Press, Boca Raton, 2010.
   Google Scholar
• Z. Sabeh, M. Shamsi, and M. Dehghan, Distributed optimal control of the viscous Burgers equation via a Legendre pseudo-spectral approach, Math. Methods Appl. Sci. 39 (2016), pp. 3350–3360. doi: 10.1002/mma.3779
   Web of Science ®Google Scholar
• A. Shenoy, M. Heinkenschloss, and E.M. Cliff, Airfoil design by an all-at-once method, Int. J. Comput. Fluid Dyn. 11 (1998), pp. 3–25. doi: 10.1080/10618569808940863
   Web of Science ®Google Scholar
• R. Tavakoli and H. Zhang, A nonmonotone spectral projected gradient method for large-scale topology optimization problems, Numer. Algebra Control Optim. 2 (2012), pp. 395–412. doi: 10.3934/naco.2012.2.395
   Google Scholar
• S. Volkwein, Mesh-independence for an augmented Lagrangian-SQP method in Hilbert spaces, SIAM J. Control Optim. 38 (2000), pp. 767–785. doi: 10.1137/S0363012998334468
   Web of Science ®Google Scholar
• G. Yuan, Z. Meng, and Y. Li, A modified Hestenes and Stiefel conjugate gradient algorithm for large-scale nonsmooth minimizations and nonlinear equations, J. Optim. Theory Appl. 168 (2016), pp. 129–152. doi: 10.1007/s10957-015-0781-1
   Web of Science ®Google Scholar
• G. Yuan and Z. Wei, The Barzilai and Borwein gradient method with nonmonotone line search for nonsmooth convex optimization problems, Math. Model. Anal. 17 (2012), pp. 203–216. doi: 10.3846/13926292.2012.661375
   Web of Science ®Google Scholar
• G. Yuan and M. Zhang, A three-terms Polak–Rribiere–Polyak conjugate gradient algorithm for large-scale nonlinear equations, J. Comput. Appl. Math. 286 (2015), pp. 186–195. doi: 10.1016/j.cam.2015.03.014
   Web of Science ®Google Scholar