In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space spanned by solutions of the...
We consider the efficient and reliable solution of linear-quadratic optimal control problems governed by parametrized parabolic partial differential equations. To this end, we employ the reduced basis method as a low-dimensional surrogate model to solve the optimal control problem and develop error estimation procedures that provide rigorous bounds for the error in the optimal control and the associated cost functional. We show that our approach can be applied to problems involving control constraints...
In this paper, we employ the reduced basis method as a surrogate model for the solution of linear-quadratic optimal control problems governed by parametrized elliptic partial differential equations. We present error estimation and dual procedures that provide rigorous bounds for the error in several quantities of interest: the optimal control, the cost functional, and general linear output functionals of the control, state, and adjoint variables. We show that, based on the assumption of affine parameter...
In this paper, we extend the reduced-basis methods and associated error estimators developed earlier for elliptic partial
differential equations to parabolic problems with affine parameter
dependence. The essential new ingredient is the presence of time in the
formulation and solution of the problem – we shall “simply” treat
time as an additional, albeit special, parameter. First, we introduce
the reduced-basis recipe – Galerkin projection onto a space
spanned by solutions...
In this paper, we extend the reduced-basis approximations developed earlier for elliptic and parabolic partial differential equations with parameter
dependence to problems involving (a) dependence on the
parameter, and (b) dependence on the field variable.
The method replaces the nonaffine and nonlinear terms with a coefficient function approximation which then permits an efficient offline-online computational
decomposition. We first review the coefficient function approximation procedure:...
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