Abstract
The main focus of this paper is on an a-posteriori analysis for different model-order strategies applied to optimal control problems governed by linear parabolic partial differential equations. Based on a perturbation method it is deduced how far the suboptimal control, computed on the basis of the reduced-order model, is from the (unknown) exact one. For the model-order reduction, $\mathcal H_{2,\alpha}$-norm optimal model reduction (H2), balanced truncation (BT), and proper orthogonal decomposition (POD) are studied. The proposed approach is based on semi-discretization of the underlying dynamics for the state and the adjoint equations as a large scale linear time-invariant (LTI) system. This system is reduced to a lower-dimensional one using Galerkin (POD) or Petrov-Galerkin (H2, BT) projection. The size of the reduced-order system is iteratively increased until the error in the optimal control, computed with the a-posteriori error estimator, satisfies a given accuracy. The method is illustrated with numerical tests.
Highlights
Model reduction is a powerful tool widely used for solving partial differential equations (PDEs) or large-scale ordinary differential equations (ODEs) where the latter may arise from semi-discretization of PDEs in space
Methods such as Balanced Truncation (BT) and H2 compute the reduced system on basis of an approximation of the operator mapping from inputs to outputs. This approach is applicable for linear time-invariant (LTI) large-scale ODE systems which can be obtained by semi-discretization of a PDE
The focus is on projection based model reduction, in particular Proper Orthogonal Decomposition (POD), H2,α-norm reduction (H2) and Balanced Truncation (BT) to solve the underlying dynamics efficiently
Summary
Model reduction is a powerful tool widely used for solving partial differential equations (PDEs) or large-scale ordinary differential equations (ODEs) where the latter may arise from semi-discretization of PDEs in space. Methods such as BT and H2 compute the reduced system on basis of an approximation of the operator mapping from inputs (i.e. usually timedependent parameters to control the system) to outputs (i.e. observations of the system) This approach is applicable for linear time-invariant (LTI) large-scale ODE systems which can be obtained by semi-discretization of a PDE. To estimate the error in the optimal control a-posteriori, we will apply a technique called perturbation method which was first used by Dontchev et al [8] and Malanowski, Buskens and Maurer [23] for optimal control of ODEs. The method has later been adapted to PDEs [3, 33, 34], where in the latter two references the error of the solution of the reduced problem obtained by POD and/or reduced-basis model reduction was estimated. We combine our a-posteriori error analysis with the following update strategy (compare [1, 28]): Assuming that the main computational effort in determining the POD basis is the solution of the full state and adjoint PDE, one may update the POD basis for each r since these solutions are necessary for the error estimator in each step
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