Mixed Finite Elements for spatial regression with PDE penalization

20/2013

Mixed Finite Elements for spatial regression with PDE penalization

Wednesday 1st May 2013

Azzimonti, L.; Nobile, F.; Sangalli, L.M.; Secchi, P.

We study a class of models at the interface between statistics and numerical analysis. Specifically, we consider non-parametric regression models for the estimation of spatial fields from pointwise and noisy observations, that account for problem specific prior information, described in terms of a PDE governing the phenomenon under study. The prior information is incorporated in the model via a roughness term using a penalized regression framework. We prove the well-posedness of the estimation problem and we resort to a mixed equal order Finite Element method for its discretization. We prove the well posedness and the optimal convergence rate of the proposed discretization method. Finally the smoothing technique is extended to the case of areal data, particularly interesting in many applications. Keywords: mixed Finite Element method, fourth order problems, non-parametric regression, smoothing.

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Laura Azzimonti, Fabio Nobile, Laura M. Sangalli, Piercesare Secchi (2014), Mixed Finite Elements for spatial regression with PDE penalization, SIAM/ASA Journal on Uncertainty Quantification, Vol. 2, No. 1, pp. 305-335.