We present a certified reduced basis (RB) framework for the efficient solution of PDE-constrained parametric optimization problems. We consider optimization problems (such as optimal control and optimal design) governed by elliptic PDEs and involving possibly non-convex cost functionals, assuming that the control functions are described in terms of a parameter vector. At each optimization step, the high-fidelity approximation of state and adjoint problems is replaced by a certified RB approximation, thus yielding a very efficient solution through an “optimize-then-reduce” approach. We develop a posteriori error estimates for the solutions of state and adjoint problems, the cost functional, its gradient and the optimal solution. We confirm our theoretical results in the case of optimal control/design problems dealing with potential and thermal flows.
If the inline PDF is not rendering correctly, you can download the PDF file here.
1. A. Borzì and V. Schulz Computational Optimization of Systems Governed by Partial Differential Equations. SIAM 2011.
2. M. Hinze R. Pinnau M. Ulbrich and S. Ulbrich Optimization with PDE Constraints. Springer 2009.
3. A. Quarteroni G. Rozza and A. Quaini Reduced basis methods for optimal control of advection- diffusion problem in Advances in Numerical Mathematics W. Fitzgibbon R. Hoppe J. Periaux O. Pironneau and Y. Vassilevski Editors pp. 193–216 2007.
4. L. Dedè Reduced basis method and a posteriori error estimation for parametrized linear-quadratic optimal control problems SIAM Journal on Scientific Computing vol. 32 no. 2 pp. 997–1019 2010.
5. L. Dedè Reduced basis method and error estimation for parametrized optimal control problems with control constraints Journal of Scientific Computing vol. 50 no. 2 pp. 287–305 2012.
6. T. Tonn K. Urban and S. Volkwein Comparison of the reduced basis and pod a posteriori error estimators for an elliptic linear-quadratic optimal control problem Mathematical and Computer Modelling of Dynamical Systems vol. 17 no. 4 pp. 355–369 2011.
7. M. A. Dihlmann and B. Haasdonk Certified PDE-constrained parameter optimization using reduced basis surrogate models for evolution problems Computational Optimization and Applications vol. 60 no. 3 pp. 753–787 2015.
8. M. Dihlmann and B. Haasdonk Certified nonlinear parameter optimization with reduced basis surrogate models Proceedings in Applied Mathematics & Mechanics vol. 13 no. 1 pp. 3–6 2013.
9. Y. Zhang L. Feng S. Li and P. Benner Accelerating PDE constrained optimization by the reduced basis method: application to batch chromatography International Journal for Numerical Methods in Engineering vol. 104 no. 11 pp. 983–1007 2015.
10. K. Kunisch and S. Volkwein Proper orthogonal decomposition for optimality systems ESAIM: Mathematical Modelling and Numerical Analysis vol. 42 no. 1 pp. 1–23 2008.
11. M. Kahlbacher and S. Volkwein Galerkin proper orthogonal decomposition methods for parameter dependent elliptic systems Disc. Math.: Diff. Incl. Control and Optim. vol. 27 pp. 95–117 2007.
12. E. W. Sachs and S. Volkwein Pod-Galerkin approximations in pde-constrained optimization GAMM- Mitteilungen vol. 33 no. 2 pp. 194–208 2010.
13. M. Kahlbacher and S. Volkwein POD a-posteriori error based inexact SQP method for bilinear elliptic optimal control problems ESAIM: Mathematical Modelling and Numerical Analysis vol. 46 no. 02 pp. 491–511 2012.
14. O. Lass Reduced Order Modeling and Parameter Identification for Coupled Nonlinear PDE Systems. PhD thesis University of Konstanz 2014.
15. D. Amsallem M. J. Zahr Y. Choi and C. Farhat Design optimization using hyper-reduced order models Struct. Multidisc. Optim. 2014.
16. M. Gubisch and S. Volkwein Pod a-posteriori error analysis for optimal control problems with mixed control-state constraints Computational Optimization and Applications vol. 58 no. 3 pp. 619–644 2014.
17. M. Gubisch and S. Volkwein Proper orthogonal decomposition for linear-quadratic optimal control in Model Reduction and Approximation: Theory and Algorithms (P. Benner A. Cohen M. Ohlberger and K. Willcox eds.) pp. 5–66 SIAM 2017.
18. F. Negri G. Rozza A. Manzoni and A. Quarteroni Reduced basis method for parametrized elliptic optimal control problems SIAM Journal on Scientific Computing vol. 35 no. 5 pp. A2316–A2340 2013.
19. F. Negri A. Manzoni and G. Rozza Reduced basis approximation of parametrized optimal flow control problems for the Stokes equations Computers & Mathematics with Applications vol. 69 no. 4 pp. 319–336 2015.
20. M. Kärcher and M. Grepl A certified reduced basis method for parametrized elliptic optimal control problems ESAIM: Control Optimisation and Calculus of Variations vol. 20 no. 2 pp. 416–441 2014.
21. M. Kärcher Z. Tokoutsi M. Grepl and K. Veroy Certified reduced basis methods for parametrized elliptic optimal control problems with distributed controls Journal of Scientific Computing vol. 75 no. 1 pp. 276–307 2018.
22. P. Benner E. Sachs and S. Volkwein Model order reduction for PDE constrained optimization in Trends in PDE Constrained Optimization (G. Leugering P. Benner S. Engell A. Griewank H. Harbrecht M. Hinze R. Rannacher and S. Ulbrich eds.) pp. 303–326 Springer International Publishing 2014.
23. A. Borzì and G. von Winckel A pod framework to determine robust controls in pde optimization Computing and Visualization in Science vol. 14 no. 3 pp. 91–103 2011.
24. A. Manzoni A. Quarteroni and G. Rozza Shape optimization for viscous flows by reduced basis methods and free-form deformation International Journal for Numerical Methods in Fluids vol. 70 no. 5 pp. 646–670 2012.
25. T. Lassila and G. Rozza Parametric free-form shape design with pde models and reduced basis method Computer Methods in Applied Mechanics and Engineering vol. 199 no. 23–24 pp. 1583–1592 2010.
26. F. Negri A. Manzoni and D. Amsallem Efficient model reduction of parametrized systems by matrix discrete empirical interpolation Journal of Computational Physics vol. 303 pp. 431–454 2015.
27. M. Grepl N. Nguyen K. Veroy A. Patera and G. Liu Certified rapid solution of partial differential equations for real-time parameter estimation and optimization in Real-time PDE-Constrained Optimization (L. Biegler O. Ghattas M. Heinkenschloss D. Keyes and B. Van Bloemen Waanders eds.) pp. 197–215 SIAM 2007.
28. M. Bambach M. Heinkenschloss and M. Herty A method for model identification and parameter estimation Inverse Problems vol. 29 no. 2 2013.
29. T. Lassila A. Manzoni A. Quarteroni and G. Rozza A reduced computational and geometrical framework for inverse problems in haemodynamics International Journal for Numerical Methods in Biomedical Engineering vol. 29 no. 7 pp. 741–776 2013.
30. C. Lieberman K. Willcox and O. Ghattas Parameter and state model reduction for large-scale statistical inverse problems SIAM Journal on Scientific Computing vol. 32 no. 5 pp. 2523–2542 2010.
31. A. Manzoni S. Pagani and T. Lassila Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models SIAM/ASA Journal on Uncertainty Quantification vol. 4 no. 1 pp. 380–412 2016.
32. M. Zahr K. Carlberg and D. Kouri An efficient globally convergent method for optimization under uncertainty using adaptive model reduction and sparse grids arXiv preprint arXiv:1811.00177 2018.
33. A. Quarteroni A. Manzoni and F. Negri Reduced Basis Methods for Partial Differential Equations. An Introduction. Springer 2016.
34. E. Zeidler Nonlinear Functional Analysis and its Applications vol. I: Fixed-Point Theorems. Springer-Verlag 1985.
35. J. Nocedal and S. J. Wright Numerical Optimization. Springer New York 2006.
36. E. Arian M. Fahl and E. Sachs Trust-region proper orthogonal decomposition for flow control tech. rep. Institute for Computer Applications in Science and Engineering NASA Langley Research Center Hampton VA 2000. Technical report DTIC Document.
37. Y. Yue and K. Meerbergen Accelerating pde-constrained optimization by model order reduction with error control SIAM Journal on Optimization vol. 23 pp. 1344–1370 2013.
38. M. J. Zahr and C. Farhat Progressive construction of a parametric reduced-order model for PDE- constrained optimization International Journal for Numerical Methods in Engineering vol. 102 no. 5 pp. 1111–1135 2015.
39. C. Kelley Iterative Methods for Optimization. SIAM 1999.
40. G. Rozza D. Huynh and A. Patera Reduced basis approximation and a posteriori error estimation for a nely parametrized elliptic coercive partial differential equations Archives of Computational Methods in Engineering vol. 15 pp. 229–275 2008.
41. A. Manzoni and F. Negri Heuristic strategies for the approximation of stability factors in quadratically nonlinear parametrized PDEs Advances in Computational Mathematics vol. 41 no. 5 pp. 1255–1288 2015.
42. R. Becker and R. Rannacher An optimal control approach to a posteriori error estimation in finite element methods Acta Numerica vol. 10 pp. 1–102 2001.
43. R. Becker H. Kapp and R. Rannacher Adaptive finite element methods for optimal control of partial differential equations: Basic concept SIAM Journal on Control and Optimization vol. 39 no. 1 pp. 113–132 2000.
44. F. Brezzi J. Rappaz and P. A. Raviart Finite dimensional approximation of nonlinear problems Num. Math. vol. 38 no. 1 pp. 1–30 1982.
45. G. Caloz and J. Rappaz Numerical analysis for nonlinear and bifurcation problems in Handbook of Numerical Analysis Vol. V (P. Ciarlet and J. Lions eds.) pp. 487–637 Elsevier Science B.V. 1997.
46. G. Rozza and A. Manzoni Model order reduction by geometrical parametrization for shape optimization in computational fluid dynamics in Proceedings of the ECCOMAS CFD 2010 V European Conference on Computational Fluid Dynamics 2010.
47. H. Antil M. Heinkenschloss and D. C. Sorensen Application of the discrete empirical interpolation method to reduced order modeling of nonlinear and parametric systems in Reduced Order Methods for Modeling and Computational Reduction (A. Quarteroni and G. Rozza eds.) pp. 101–136 Springer 2014.
48. B. Mohammadi and O. Pironneau Applied Shape Optimization for Fluids. Oxford University Press 2001.
49. A. Manzoni F. Salmoiraghi and L. Heltai Reduced basis isogeometric methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils Computer Methods in Applied Mechanics and Engineering vol. 284 pp. 1147 – 1180 2015.
50. G. Rozza T. Lassila and A. Manzoni Reduced basis approximation for shape optimization in thermal flows with a parametrized polynomial geometric map in Spectral and High Order Methods for Partial Differential Equations. Selected papers from the ICOSAHOM ’09 conference June 22-26 Trondheim Norway (J. S. Hesthaven and E. Rønquist eds.) pp. 307–315 Springer 2011.