  This paper is concerned with the construction, analysis and realization of a
numerical method to approximate the solution of high dimensional elliptic
partial differential equations. We propose a new combination of an Adaptive
Wavelet Galerkin Method (AWGM) and the well known Hierarchical Tensor (HT)
format. The arising HT-AWGM is adaptive both in the wavelet representation of
the low dimensional factors and in the tensor rank of the HT representation.
The point of departure is an adaptive wavelet method for the HT format using
approximate Richardson iterations from [1] and an AWGM method as described in
[13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncated
preconditioned conjugate gradient (PCG) algorithm from [33] in combination with
a tensor-based preconditioner from [3]. Our analysis starts by showing
convergence of the truncated conjugate gradient method. The next step is to add
routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed
concerning convergence and complexity. We show that the performance of the
scheme asymptotically depends only on the desired tolerance with convergence
rates depending on the Besov regularity of low dimensional quantities and the
low rank tensor structure of the solution. The complexity in the ranks is
algebraic with powers of four stemming from the complexity of the tensor
truncation. Numerical experiments show the quantitative performance.
