Local Improvements to Reduced-Order Approximations of Optimal Control Problems Governed by Diffusion-Convection-Reaction Equation

We consider the optimal control problem governed by diffusion convection reaction equation without control constraints. The proper orthogonal decomposition(POD) method is used to reduce the dimension of the problem. The POD method may be lack of accuracy if the POD basis depending on a set of parameters is used to approximate the problem depending on a different set of parameters. We are interested in the perturbation of diffusion term. To increase the accuracy and robustness of the basis, we compute three bases additional to the baseline POD. The first two of them use the sensitivity information to extrapolate and expand the POD basis. The other one is based on the subspace angle interpolation method. We compare these different bases in terms of accuracy and complexity and investigate the advantages and main drawbacks of them.Comment: 19 pages, 5 figures, 2 table

Adaptative reduced order model to control non linear partial differential equations

In classical adjoint based optimal control of unsteady dynamical systems, requirements of CPU time and storage memory are known to be very important. To overcome this issue, model order reduction techiques operating by the construction of a separated representation of the solution are considered. A spatial basis must be calcu- lated for each variation in control parameters, followed by a Galerkin projection of the equations’s residuals on this basis, that results in a low dimentional system of ordinary differential equations. These steps need to be carried out in every iteration of the control algorithm. The most popular reduced order model method is the Proper Orthogonal De- composition (POD). It is used here for the construction of reduced bases. The interest in this communication is turned to the adaptation of these bases respectivly to control parameter variations. Two adaptation approaches are considered. The first one uses a powerfull interpolation method based on calculus of geodesic paths on the Grassmann manifold. This approach needs a precomputed set of bases associated to a distribution of opetating points, that are calculated using POD method. The second approach uses the Proper Generalized Decomposition (PGD) considered here as a correction method. This method consists in enriching a basis by reducing the error of the approximated solution

Model Order Reduction

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This three-volume handbook covers methods as well as applications. This third volume focuses on applications in engineering, biomedical engineering, computational physics and computer science