Optimal control and numerical adaptivity for advection–diffusion equations

Luca Dede'; Alfio Quarteroni

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

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

Abstract

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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

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Dede', Luca, and Quarteroni, Alfio. "Optimal control and numerical adaptivity for advection–diffusion equations." ESAIM: Mathematical Modelling and Numerical Analysis 39.5 (2010): 1019-1040. <http://eudml.org/doc/194289>.

@article{Dede2010,
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},
keywords = {Optimal control problems; partial differential equations; finite element approximation; stabilized Lagrangian; numerical adaptivity; advection–diffusion equations.; optimal control problems; partial differential equations; stabilized Lagrangian; numerical adaptivity; advection-diffusion equations.},
language = {eng},
month = {3},
number = {5},
pages = {1019-1040},
publisher = {EDP Sciences},
title = {Optimal control and numerical adaptivity for advection–diffusion equations},
url = {http://eudml.org/doc/194289},
volume = {39},
year = {2010},
}

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
DA - 2010/3//
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.; optimal control problems; partial differential equations; stabilized Lagrangian; numerical adaptivity; advection-diffusion equations.
UR - http://eudml.org/doc/194289
ER -

References

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