Schmidt, Andreas; Potschka, Andreas; Körkel, Stefan; Bock, Hans Georg Derivative-extended POD reduced-order modeling for parameter estimation. (English) Zbl 1285.65039 SIAM J. Sci. Comput. 35, No. 6, A2696-A2717 (2013). Summary: We consider model reduction via proper orthogonal decomposition (POD) and its application to parameter estimation problems constrained by parabolic partial differential equations. We use a ‘first discretize then optimize’ approach to solve the parameter estimation problem and show that the use of derivative information in the reduced-order model is important. We include directional derivatives directly in the POD snapshot matrix and show that, equivalently to the stationary case, this extension yields a more robust model with respect to changes in the parameters. Moreover, we propose an algorithm that uses derivative-extended POD models together with a Gauss-Newton method. We give an a posteriori error estimate that indicates how far a suboptimal solution obtained with the reduced problem deviates from the solution of the high-dimensional problem. Finally, we present numerical examples that showcase the efficiency of the proposed approach. Cited in 4 Documents MSC: 65K10 Numerical optimization and variational techniques 49J20 Existence theories for optimal control problems involving partial differential equations 49M15 Newton-type methods 35K58 Semilinear parabolic equations Keywords:proper orthogonal decomposition; parameter estimation; error estimates; semilinear parabolic equation; model reduction; first discretize then optimize; algorithm; Gauss-Newton method; numerical example Software:NewtonLib PDFBibTeX XMLCite \textit{A. Schmidt} et al., SIAM J. Sci. Comput. 35, No. 6, A2696--A2717 (2013; Zbl 1285.65039) Full Text: DOI