Abstract: When proper orthogonal decomposition (POD) or another projection based technique is used to generate reduced order models, the number of equations and unknowns is typically reduced dramatically. However, for nonlinear or parametrically varying problems, the cost of evaluating the reduced order models still depends on the size of the full order model and is still expensive. To overcome this bottleneck, Chaturantabut and Sorensen developed the Discrete Empirical Interpolation Method (DEIM), which generates reduced order models that typically can be evaluated at a cost that only depends on the size of the reduced order model and therefore generates a truly useful reduced order model.
We demonstrate why model reduction by POD alone is insufficient and outline the DEIM. We then extend the POD-DEIM method to finite element solutions of nonlinear partial differential equations (PDEs) and derive a-priori error estimate between the solution of the full and the reduced order problems. Further we apply POD-DEIM to shape optimization problems governed by PDEs and finally give an a-priori estimate of error between the full and the reduced order shape optimization problems.
Joint work with M. Heinkenschloss and D. C. Sorensen.
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