Distributed Hierarchical SVD in the Hierarchical Tucker Format

TL;DR

This paper develops parallel algorithms for tensor operations in the Hierarchical Tucker format, enabling efficient distributed computations with logarithmic runtime growth in tensor dimension, and demonstrates their effectiveness through numerical experiments.

Contribution

It introduces parallel algorithms for tensor operations in the Hierarchical Tucker format, allowing scalable distributed computations and solving linear equations directly in this format.

Findings

Runtime grows like log(d) in experiments

Algorithms effectively solve linear equations in Hierarchical Tucker format

Numerical experiments validate scalability and efficiency

Abstract

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.