Adaptive Model Reduction for Sensitivity Analysis
Résumé
The bad accuracy of a simplified model can compromise the sensitivity analysis. We propose to build accurate simplified models for sensitivity analysis thanks to an adaptive model reduction method: the A Priori Hyper Reduction (APHR) method. The adaptivity allows to guaranty both the quality of the state estimation and a small number of shape functions involved in the reduced order model (ROM). This approach is very convenient for time dependent problems described by the finite element method. In case of non-linear problems, a reduction of integration point number named Hyper Reduction improves the efficiency of the simplified computations. This method can be thought of as an adaptive Snapshot POD. The ROM adaptations are based on iterative finite element computations. The initial guess of such iterative computations is obtained thanks to the current ROM. So the proposed method can also be interpreted as a convergence acceleration method based on model reduction. A non linear thermal example illustrates the capability of the APHR method. A numerical sensitivity analysis is solved to validate the efficiency of the adaptive strategy. The ROMs we proposed are time independent but slightly parameter dependent. The effect of the values of the parameters at the beginning of an inverse problem treatment can be forgotten, while the ROM is efficient for values of parameters in the vicinity of the optimal parameters.
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