A non-linear manifold reduction based on ridge regression
Résumé
The present work deals with non-linear manifold reduction. A database of snapshots is represented by introducing an encoder-decoder pair. In the spirit of quadratic manifold reduction, we consider a linear encoder. The reconstruction of the snapshots (decoder) is then performed by using a dof-wise regression based on ridge functions. The non-linear function for each of the ridge regressions is not fixed a priori. Instead, it is constructively computed by using P 3-Hermite finite elements. In order to promote sparsity and enhance the decoder performances (both in terms of memory burden and number of elementary operations needed to apply it) we define a greedy algorithm which progressively adapt the dof-wise regression until a prescribed accuracy is fulfilled. A theoretical analysis is presented as well as several numerical experiments in order to assess the performances of the proposed reduced representation.
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