Constrained Proper Orthogonal Decomposition based on QR-factorization for aerodynamical shape optimization

While performing numerical simulations for design optimization, one of the major issues is reaching a good compromise between accuracy and computational effort. Simulation methods such as finite elements, finite volumes, etc. provide 'high fidelity' numerical solutions at a cost which may be prohibitive in optimization problems requiring frequent calls to the discretized equations' solver. The Proper Orthogonal Decomposition (POD) constitutes an economical and efficient option to decrease the cost of the solution; however, the truncation of the POD basis implies an error in the calculation of the global quantities used as objectives and optimization constraints, which in turn might bias the optimization results. Our idea is thus to improve the snapshot POD by means of a reduced basis constructed to provide an exact interpolation of the quantities of interest obtained by integration of the 'physical' fields. To this end, we reformulate the POD as a minimization problem where the desired properties are expressed as a set of constraints impacting the calculation of both the modes and the coefficients. The main contribution of this paper is to provide a detailed mathematical justification for our constrained variant of the POD, including a graphical interpretation of the proposed approach. The constrained POD method is then applied to the problem of representing the pressure field around a 2D wing, and is compared with the traditional POD.