Optimal control and numerical adaptivity for advection–diffusion equations

Luca Dede'; Alfio Quarteroni

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

  • Volume: 39, Issue: 5, page 1019-1040
  • ISSN: 0764-583X

Abstract

top
We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection–diffusion equation, based on a stabilization method applied to the lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting the error on the cost functional into the sum of an iteration error plus a discretization error. Once the former is reduced below a given threshold (and therefore the computed solution is “near” the optimal solution), the adaptive strategy is operated on the discretization error. To prove the effectiveness of the proposed methods, we report some numerical tests, referring to problems in which the control term is the source term of the advection–diffusion equation.

How to cite

top

Dede', Luca, and Quarteroni, Alfio. "Optimal control and numerical adaptivity for advection–diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 39.5 (2005): 1019-1040. <http://eudml.org/doc/245174>.

@article{Dede2005,
abstract = {We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection–diffusion equation, based on a stabilization method applied to the lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting the error on the cost functional into the sum of an iteration error plus a discretization error. Once the former is reduced below a given threshold (and therefore the computed solution is “near” the optimal solution), the adaptive strategy is operated on the discretization error. To prove the effectiveness of the proposed methods, we report some numerical tests, referring to problems in which the control term is the source term of the advection–diffusion equation.},
author = {Dede', Luca, Quarteroni, Alfio},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {optimal control problems; partial differential equations; finite element approximation; stabilized lagrangian; numerical adaptivity; advection–diffusion equations; stabilized Lagrangian; advection-diffusion equations.},
language = {eng},
number = {5},
pages = {1019-1040},
publisher = {EDP-Sciences},
title = {Optimal control and numerical adaptivity for advection–diffusion equations},
url = {http://eudml.org/doc/245174},
volume = {39},
year = {2005},
}

TY - JOUR
AU - Dede', Luca
AU - Quarteroni, Alfio
TI - Optimal control and numerical adaptivity for advection–diffusion equations
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2005
PB - EDP-Sciences
VL - 39
IS - 5
SP - 1019
EP - 1040
AB - We propose a general approach for the numerical approximation of optimal control problems governed by a linear advection–diffusion equation, based on a stabilization method applied to the lagrangian functional, rather than stabilizing the state and adjoint equations separately. This approach yields a coherently stabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order terms are needless. Our a posteriori estimates stems from splitting the error on the cost functional into the sum of an iteration error plus a discretization error. Once the former is reduced below a given threshold (and therefore the computed solution is “near” the optimal solution), the adaptive strategy is operated on the discretization error. To prove the effectiveness of the proposed methods, we report some numerical tests, referring to problems in which the control term is the source term of the advection–diffusion equation.
LA - eng
KW - optimal control problems; partial differential equations; finite element approximation; stabilized lagrangian; numerical adaptivity; advection–diffusion equations; stabilized Lagrangian; advection-diffusion equations.
UR - http://eudml.org/doc/245174
ER -

References

top
  1. [1] V.I. Agoshkov, Optimal Control Methods and Adjoint Equations in Mathematical Physics Problems. Institute of Numerical Mathematics, Russian Academy of Science, Moscow (2003). Zbl1236.49002
  2. [2] A.K. Aziz, J.W. Wingate and M.J. Balas, Control Theory of Systems Governed by Partial Differential Equations. Academic Press, New York (1971). Zbl0409.00018MR428151
  3. [3] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1–102. Zbl1105.65349
  4. [4] 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) 113–132. Zbl0967.65080
  5. [5] M. Braack and A. Ern, A posteriori control of modelling errors and Discretization errors. SIAM Multiscale Model. Simul. 1 (2003) 221–238. Zbl1050.65100
  6. [6] G. Finzi, G. Pirovano and M. Volta, Gestione della Qualità dell’aria. Modelli di Simulazione e Previsione. Mc Graw-Hill, Milano (2001). 
  7. [7] L. Formaggia, S. Micheletti and S. Perotto, Anisotropic mesh adaptation in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math. 51 (2004) 511–533. Zbl1107.65098
  8. [8] A.N. Kolmogorov and S.V. Fomin, Elements of Theory of Functions and Functional Analysis. V.M. Tikhomirov, Nauka, Moscow (1989). Zbl0672.46001MR1025126
  9. [9] R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distribuited elliptic optimal control problems. SIAM J. Control Optim. 41 (2001) 1321–1349. Zbl1034.49031
  10. [10] J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag, New York (1971). Zbl0203.09001MR271512
  11. [11] W. Liu and N. Yan, A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math. 120 (2000) 159–173. Zbl0963.65072
  12. [12] W. Liu and N. Yan, A Posteriori error estimates for distribuited convex optimal control problems. Adv. Comput. Math. 15 (2001) 285–309. Zbl1008.49024
  13. [13] B. Mohammadi and O. Pironneau, Applied Shape Optimization for Fluids. Clarendon Press, Oxford (2001). Zbl0970.76003MR1835648
  14. [14] M. Picasso, Anisotropic a posteriori error estimates for an optimal control problem governed by the heat equation. Int. J. Numer. Method PDE (2004), submitted. Zbl1111.65060
  15. [15] O. Pironneau and E. Polak, Consistent approximation and approximate functions and gradients in optimal control. SIAM J. Control Optim. 41 (2002) 487–510. Zbl1011.49020
  16. [16] A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations. Springer-Verlag, Berlin and Heidelberg (1994). Zbl0803.65088MR1299729
  17. [17] J. Sokolowski and J.P. Zolesio, Introduction to Shape Optimization (Shape Sensitivity Analysis). Springer-Verlag, New York (1991). Zbl0761.73003MR1215733
  18. [18] R.B. Stull, An Introduction to Boundary Layer Meteorology. Kluver Academic Publishers, Dordrecht (1988). Zbl0752.76001
  19. [19] F.P. Vasiliev, Methods for Solving the Extremum Problems. Nauka, Moscow (1981). 
  20. [20] D.A. Venditti and D.L. Darmofal, Grid adaption for functional outputs: application to two-dimensional inviscid flows. J. Comput. Phys. 176 (2002) 40–69. Zbl1120.76342
  21. [21] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Teubner (1996). Zbl0853.65108

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.