  We consider tensors in the Hierarchical Tucker format and suppose the tensor
data to be distributed among several compute nodes. We assume the compute nodes
to be in a one-to-one correspondence with the nodes of the Hierarchical Tucker
format such that connected nodes can communicate with each other. An
appropriate tree structure in the Hierarchical Tucker format then allows for
the parallelization of basic arithmetic operations between tensors with a
parallel runtime which grows like $\log(d)$, where $d$ is the tensor dimension.
We introduce parallel algorithms for several tensor operations, some of which
can be applied to solve linear equations $\mathcal{A}X=B$ directly in the
Hierarchical Tucker format using iterative methods like conjugate gradients or
multigrid. We present weak scaling studies, which provide evidence that the
runtime of our algorithms indeed grows like $\log(d)$. Furthermore, we present
numerical experiments in which we apply our algorithms to solve a
parameter-dependent diffusion equation in the Hierarchical Tucker format by
means of a multigrid algorithm.
