Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems

SIAM Journal on Scientific Computing

Volumn 32, Issue 2, 2010, Pages 997-1019

  Dedé, Luca ^a  

^a POLITECNICO DI MILANO   (Italy)

Author keywords

A posteriori error estimation; Optimal control; Parametrized partial differential equations; Reduced basis method

Indexed keywords

ERROR ANALYSIS; OPTIMAL CONTROL SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS;

A-POSTERIORI ERROR ESTIMATES; A-POSTERIORI ERROR ESTIMATIONS; COMPUTATIONAL COSTS; COST SAVING; LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM; OPTIMAL CONTROL PROBLEM; OPTIMAL CONTROLS; PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS; PARAMETRIZED PARTIAL DIFFERENTIAL EQUATIONS; REDUCED BASIS METHODS;

QUADRATIC PROGRAMMING;

EID: 77950829866     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/090760453     Document Type: Article

References (42)

1. B. O. ALMROTH, P. STERN, AND F. A. BROGAN, Automatic choice of global shape functions in structural analysis, AIAA J., 16 (1978), pp. 525-528.
2. J. ATWELL AND B. KING, Proper orthogonal decomposition for reduced basis feedback controllers for parabolic equations, Math. Comput. Modelling, 33 (2001), pp. 1-19.
3. R. BECKER, H. KAPP, AND R. RANNACHER, Adaptive finite element methods for optimal control of partial differential equations: Basic concepts, SIAM J. Control Optim., 39 (2000), pp. 113-132.
4. R. BECKER AND R. RANNACHER, An optimal control approach to a posteriori error estimation in finite element methods, Acta Numer., 10 (2001), pp. 1-102.
5. C. CANUTO, T. ToNN, AND K. URBAN, A posteriori error analysis of the reduced basis method for nonaffine parametrized non-linear PDEs, SIAM J. Numer. Anal., 47 (2009), pp. 2001-2022.
6. L. DEDÉ, Adaptive and Reduced Basis Methods for Optimal Control Problems in Environmental Applications, Ph.D. thesis, Politecnico di Milano, Milano, Italy, 2008.
7. L. DEDÉ, reduced basis method for parametrized advection-reaction problems, J. Comput. Math., 28 (2010), pp. 122-148.
8. L. DEDÉ AND A. QuARTERONI, Optimal control and numerical adaptivity for advection-diffusion equations, M2AN Math. Model. Numer. Anal., 39 (20 05), pp. 1019-1040.
9. M. FAHL AND E. SACHS, Reduced order modelling approaches to PDE-constrained optimization based on proper orthogonal decomposition, in Lecture Notes in Computational Science and Engineering, Springer, Berlin, 2003, pp. 268-280.
10. W. R. GRAHAM, J. PERAIRE, AND K. Y. TANG, Optimal control of vortex shedding using low-order models. Part I: Open-loop model development, Internat. J. Numer. Methods Engrg., 44 (1999), pp. 945-972.
11. M.A. GREPL, Reduced-Basis Approximations and A Posteriori Error Estimation for Parabolic Partial Differential Equations, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA., 2005.
12. M. A. GREPL, N. C. NGUYEN, K. VEROY, A. T. PATERA, AND G. R. LIU, Certified rapid solution of partial differential equations for real-time parameter estimation and optimization,'m Real-time PDE-Constrained Optimization, L. Biegler, O. Ghattas, M. Heinkenschloss, D. Keyes, and B. van Bloemen Waanders, eds., SIAM, P hiladelphia, 2007, pp. 197-215.
13. M. A. GREPL AND A. T. PATERA, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations, M2AN Math. Model. Numer. Anal., 39 (2005), pp. 157-181.
14. R. GRIESSE, Parametric sensitivity analysis in optimal control of a reaction diffusion system. Part I: Solution differentiability, Numer. Funct. Anal. Optim., 25 (2004), pp. 93-117.
15. R. GRIESSE AND B. VEXLER, Numerical sensitivity analysis for the quantity of interest in PDE-constrained optimization, SIAM J. Sci. Comput ., 29 (2007), pp. 22-48.
16. M. D. GUNZBURGER, Adjoint equation-based methods for control problems in incompressible, viscous flows, Flow Turbul. Combust., 65 (2000), pp. 249-272.
17. M. D. GUNZBURGER, Perspectives in Flow Control and Optimization, Advances in Design and Control, SIAM, Philadelphia, 2003.
18. M. HINZE AND S. VOLKWEIN, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition, Comput. Optim. Appl., 39 (2008), pp. 319-345.
19. K. ITO AND S. S. RAVINDRAN, A reduced basis method for control problems governed by PDEs, in International Series of Numerical Mathematics, Birkhäuser, Basel, 1998, pp. 153-168.
20. K. ITO AND S. S. RAVINDRAN, A reduced-order method for simulation and control of fluid flow, J. Comput. Phys., 143 (1998), pp. 403-425.
21. K. ITO AND S. S. RAVINDRAN, Reduced basis methods for optimal control of unsteady viscous flows, Int. J. Comput. Fluid Dyn., 15 (2001), pp. 97-113.
22. K. KUNISCH AND S. VOLKWEIN, Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics, SIAM J. Numer. Anal., 40 (2002), pp. 492-515.
23. K. KUNISCH AND S. VOLKWEIN, Proper orthogonal decomposition for optimality systems,M2AN Math. Model. Numer. Anal., 42 (2008), pp. 1-23.
24. H. MAURER, First and second order sufficient optimality conditions in mathematical programming and optimal control, Math. Programming Stud., 14 (1981), pp. 163-177.
25. D. MEIDNER AND B. VEXLER, Adaptive space-time finite element methods for parabolic optimization problems, SIAM J. Control Optim., 46 (2007), pp. 116-142.
26. N. C. NGUYEN, Reduced Basis Approximation and A Posteriori Error Bounds for Nonaffine and Non-linear Partial Differential Equations: Application to Inverse Analysis,Ph.D. thesis, Singapore-MIT Alliance, National University of Singapore, Singapore, 2005.
27. N. C. NGUYEN, G. ROZZA, D. B. P. HUYNH, AND A. T. PATERA, Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs; Application to realtime Bayesian parameter estimation, in Computational Methods for Large Scale Inverse Problems and Uncertainty Quantification, L. Biegler, G. Biros, O. Ghattas, M. Heinken- schloss, D. Keyes, B. Mallick, L. Tenorio, B. van Bloemen Waanders, and K. Willcox, eds., John Wiley & Sons, London, UK, 2009.
28. N. C. NGUYEN, G. ROZZA, AND A. T. PATERA, Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers equation, Calcolo, 46 (2009), pp. 157-185.
29. N. C. NGUYEN, K. VEROY, AND A. T. PATERA, Certified real-time solution of parametrized partial differential equations, in Handbook of Materials Modeling, S. Yip, ed., Springer, New York, 2005, pp. 1523-1558.
30. J. NOCEDAL AND S. J. WRIGHT, Numerical Optimization, Springer, New York, 2006.
31. A. K. NOOR AND J. M. PETERS, Reduced basis technique for non-linear analysis of structures, AIAA J., 19 (1980), pp. 455-462.
32. A. T. PATERA AND G. ROZZA, reduced basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006-2007, to appear in (tentative rubric) MIT Pappalardo Grad uate Monographs in Mechanical Engineering, http://augustine.mit.edu/.
33. T. A. PORSCHING, Estimation of the error in the reduced basis method solution of non-linear equations, Math. Comp., 45 (1985), pp. 487-496.
34. C. PRUD'HOMME, D. ROVAS, K. VEROY, Y. MADAY, A. T. PATERA, AND G. TURINICI, Reliable real-time solution of parametrized partial differential equations: Reduced-basis outpu bound methods, J. Fluids Engrg., 124 (2002), pp. 70-80.
35. A. QUARTERONI, A. QUAINI, AND G. ROZZA, Reduced basis methods for advection-diffusion optimal control problems, in Advances in Numerical Mat hematics, W. Fitzgibbon, R. Hoppe, J. Periaux, O. Pironneau, and Y.Vassilevski, eds., Russian Academy of Science, Moscow, and University of Houston, Houston, 2006, pp. 193-216.
36. A. QUARTERONI, G. ROZZA, L. DEDÈ, AND A. QUAINI, Numerical approximation of a control problem for advection-diffusion processes, in System Modeling and Optimization, Springer, New York, 2006, pp. 261-273.
37. A. QUARTERONI AND A. VALLI, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1994.
38. S. S. RAVINDRAN, A reduced-order approach for optimal control of fluids using proper orthogonal decomposition, Internat. J. Numer. Methods Flu ids, 34 (2000), pp. 425-448.
39. S. SEN, K. VEROY, D. B. P. HUYNH, S. DEPARIS, N. C. NGUYEN, AND A. T. PATERA, "Natural norm" a posteriori error estimators for reduced basis approximations, J. Comput. Phys., 217 (2006), pp. 37-62.
40. F. TRÖLTZSCH AND S. VOLKWEIN, POD a-posteriori error estimates for linear-quadratic optimal control problems, Comput. Optim. Appl., 44 (2009), pp. 83-115.
41. B. VEXLER AND W. WOLLNER, Adaptive finite elements for elliptic optimization problems with control constraints, SIAM J. Control Optim., 47 (2008), pp. 509-534.
42. rbMIT©MIT Software, Massachusetts Institute of Technology, Cambridge, MA, http://augustine.mit.edu/, 2007.