We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

Quasi-best approximation in optimization with PDE constraints / F. Gaspoz, C. Kreuzer, A. Veeser, W. Wollner. - In: INVERSE PROBLEMS. - ISSN 0266-5611. - 36:1(2020 Jan). [10.1088/1361-6420/ab47f3]

Quasi-best approximation in optimization with PDE constraints

A. Veeser
Penultimo
;
2020

Abstract

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.
a priori error estimates; PDE constrained optimization; quasi-best approximation
Settore MAT/08 - Analisi Numerica
   Numerical analysis for full and reduced order methods for the efficient and accurate solution of complex systems held by partial differential equations
   MINISTERO DELL'ISTRUZIONE E DEL MERITO
   201752HKH8_006
gen-2020
dic-2019
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/807875
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