  A class of mixed-order \emph{PDE}-constraint regularizer for image processing
problem is proposed, generalizing the standard first order total variation
$(TV)$. A semi-supervised (bilevel) training scheme, which provides a
simultaneous optimization with respect to parameters and the new class of
regularizers, is studied. Also, A finite approximation method, which used to
solve the global optimization solutions of such training scheme, is introduced
and analyzed.

Analysis of PDEs