In this article, we analyse three related preconditioned steepest descent algorithms, which are partially popular in Hartree-Fock and Kohn-Sham theory as well as invariant subspace computations, from the viewpoint of minimization of the corresponding functionals, constrained by orthogonality conditions. We exploit the geometry of the of the admissible manifold, i.e. the invariance with respect to unitary transformations, to reformulate the problem on the Grassmann manifold as the admissible set. We then prove asymptotical linear convergence of the algorithms under the condition that the Hessian of the corresponding Lagrangian is elliptic on the tangent space of the Grassmann manifold at the minimizer.
Recently, the format of TT tensors
\cite{hackbuschHT,osele1,tyrtosele2,tyrtosele3} has turned out to be
a promising new format for the approximation of solutions of high
dimensional problems. In this paper, we prove some new results for
the TT representation of a tensor $U \in \R^{n_1\times \ldots\times
n_d}$ and for the manifold of tensors of TT-rank $\underline{r}$.\As a first result, we prove that the TT (or compression) ranks $r_i$
of a tensor $U$ are unique and equal to the respective separation
ranks of $U$ if the components of the TT decomposition are required to
fulfil a certain maximal rank condition. We then show that the set
$\mathcal{T}$ of TT tensors of fixed rank $\underline{r}$ forms an embedded
manifold in $\R^{n^d}$, therefore preserving the essential theoretical
properties of the Tucker format, but often showing an improved scaling
behaviour. Extending a similar approach for matrices \cite{conte_lub},
we introduce certain gauge conditions to obtain a unique
representation of the tangent space $\cT_U\mathcal{T}$ of $\mathcal{T}$
and deduce a
local parametrization of the TT manifold. The parametrisation of
$\cT_{U}\mathcal{T}$ is often crucial for an algorithmic treatment of
high-dimensional time-dependent PDEs and minimisation problems
\cite{lubuch_blau}. We conclude with remarks on those applications and
present some numerical examples.