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Derivative-extended POD reduced-order modeling for parameter estimation. (English) Zbl 1285.65039

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.

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

Software:

NewtonLib
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