Distributed Hierarchical SVD in the Hierarchical Tucker Format
This work addresses the need for scalable parallel computation with high-dimensional tensors, which is crucial for applications like parametric PDEs.
The paper presents parallel algorithms for tensor operations in the Hierarchical Tucker format, achieving a parallel runtime that grows logarithmically with tensor dimension. Numerical experiments on a parameter-dependent diffusion equation demonstrate the scalability.
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.