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This paper is concerned with the construction, analysis and realization of anumerical method to approximate the solution of high dimensional ellipticpartial differential equations. We propose a new combination of an AdaptiveWavelet Galerkin Method (AWGM) and the well known Hierarchical Tensor (HT)format. The arising HT-AWGM is adaptive both in the wavelet representation ofthe 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 usingapproximate Richardson iterations from [1] and an AWGM method as described in[13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncatedpreconditioned conjugate gradient (PCG) algorithm from [33] in combination witha tensor-based preconditioner from [3]. Our analysis starts by showingconvergence of the truncated conjugate gradient method. The next step is to addroutines realizing the adaptive refinement. The resulting HT-AWGM is analyzedconcerning convergence and complexity. We show that the performance of thescheme asymptotically depends only on the desired tolerance with convergencerates depending on the Besov regularity of low dimensional quantities and thelow rank tensor structure of the solution. The complexity in the ranks isalgebraic with powers of four stemming from the complexity of the tensortruncation. Numerical experiments show the quantitative performance.

HT-AWGM: A Hierarchical Tucker-Adaptive Wavelet Galerkin Method for High Dimensional Elliptic Problems