  We investigate mathematically a nonlinear approximation type approach
recently introduced in [A. Ammar et al., J. Non-Newtonian Fluid Mech., 2006] to
solve high dimensional partial differential equations. We show the link between
the approach and the greedy algorithms of approximation theory studied e.g. in
[R.A. DeVore and V.N. Temlyakov, Adv. Comput. Math., 1996]. On the prototypical
case of the Poisson equation, we show that a variational version of the
approach, based on minimization of energies, converges. On the other hand, we
show various theoretical and numerical difficulties arising with the non
variational version of the approach, consisting of simply solving the first
order optimality equations of the problem. Several unsolved issues are
indicated in order to motivate further research.
