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Local improvements to reduced-order approximations of PDE-constrained optimization problems

Akman, Tuğba
Optimal control problems (OCPs) governed partial differential equations (PDEs) arise in environmental control problems, optimal control of fluid flow, petroleum reservoir simulation, laser surface hardening of steel, parameter estimation and in many other applications. Although the OCPs governed by elliptic and parabolic problems are investigated theoretically and numerically in several papers, the studies concerning the optimal control of evolutionary diffusion-convection-reaction (DCR) equation and Burgers equation are quite rare. In this study, we consider the optimal control problem governed by the unsteady diffusion-convection-reaction equation and Burgers equation without control constraints. These problems gain importance, especially when the diffusive term is small. In such cases, the numerical solution exhibit interior/boundary layers and classical finite element method (FEM) is not efficient for derivation of an accurate numerical solution and methods requiring higher regularity of the solution might not be practical. Therefore, we solve these problems using variational time discretization method, which is a stable, superconvergent technique requiring less regularity when compared to the methods of the same order; and symmetric interior penalty Galerkin (SIPG) with upwinding in space, which flexes inter element continuity of the solution. We provide a priori error estimates for space-time discontinuous Galerkin method and present numerical findings. An accurate and stable numerical solution requires a fine grid/mesh, which increases the dimension of the discrete problem, so the computational time. In case of perturbations in the data, full-order model (FOM) is required to be solved for each new parameter in the data set. In case of optimization problems, FOM associated to the differential equations must be resolved after updating the control. Therefore, we use a model-order reduction (MOR) technique that eliminates the necessity of the solution of the FOM for each parameter and that enables us to solve the problem in a fast way. We use one of the most popular and successful MOR techniques, namely the proper orthogonal decomposition (POD) method. The idea behind the POD method is to derive a new basis spanning the space whose dimension is lower than the finite element space. Then, the FOM is projected onto the low-dimensonal space using the new optimal POD basis as we proceed in Galerkin projection. In addition, a priori error estimates associated to reduced-order model (ROM) based on space-time dG method are proven and numerical results are shown. The POD basis is computed using the snapshots of a particular problem which is interpreted by a mathematical model and data. Because there is a link between the data and the snapshots, some perturbation in the data may lead to larger changes in the snapshots depending on the problem at hand. This leads the nominal/baseline POD basis, which depends on the nominal/baseline parameters, not to approximate the perturbed problem accurately. In such cases, one has to solve the full problem for each parameter in the data set again and regenerate the POD basis. This approach is expensive especially for nonlinear problems or optimal control problems which requires the solution of a set of differential equations. Thus, POD sensitivities are used to enrich the low-dimensional subspace for a wider range of parameters and the quantity of interest is the diffusion term $\epsilon$, the convection term $\beta$ and the reaction term $r$. We generate two new bases, i.e. extrapolated POD (ExtPOD) and expanded POD (ExpPOD) and compare these bases in terms of advantages and discuss the main drawbacks of them.