Recent tensor formats for solving high-dimensional PDEs

Hierarchical Tucker tensor format introduced by Hackbusch et al. including Tensor Trains (TT) (Tyrtyshnikov) have been introduced recently. These representations offer stable and robust approximation of high order tensors and multi-variate functions by a low order cost. For many high dimensional problems, including many body quantum mechanics, uncertainty quantification and Fokker-Planck equation, this approach has a certain potential to circumvent from the curse of dimensionality.

In case $\mathcal{V}=\bigotimes_{i=1}^d\mathcal{V}_i$ complexity is proportional to $d$ and polynomial in some multilinear ranks. The approximation properties w.r.t. to these ranks are depending on bilinear approximation rates and corresponding trace class norms. Despite fundamental problems in multilinear approximation, under certain conditions optimal convergence rates could be shown. In case $\mathcal{V}=\bigotimes_{i=1}^d\mathbb{C}^2 $, these formats are equivalent to tree tensor networks states and matrix product states (MPS) introduced for the treatment of quantum spin systems. (Under the assumption of moderate ranks, i.e. low entanglement, this approximation enables quantum computing without quantum computers.)

For numerical computations, we consider the solution of quadratic optimization problems constraint by the restriction to discrete tensors of bounded multilinear ranks $\mathbf{r} $. The underlying admissible set is no longer convex, but it is an algebraic variety. Outside singular points, it is an analytic (open) Riemannian manifold of such tensors. We consider optimization on this manifold and corresponding gradient schemes. The loss of convexity is the reason of essential difficulties for proving convergence. However some convergence results could be established by applying Lojasiewicz inequality and related techniques.