Noname manuscript No. (will be inserted by the editor) Proper Generalized Decomposition for Nonlinear Convex Problems in Tensor Banach Spaces

the date of receipt and acceptance should be inserted later Abstract Tensor-based methods are receiving a growing interest in scientific comput-ing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called Proper Generalized Decompositions methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space