536 RWTH Publication No: 711159        2017       
TITLE Distributed Hierarchical SVD in the Hierarchical Tucker Format
AUTHORS Lars Grasedyck, Christian Löbbert
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 AX=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.
DOI 10.1002/nla.2174
PUBLICATION Numerical Linear Algebra with Applications
Special Issue: 7th Workshop on Matrix Equations and Tensor Techniques
Volume25, Issue 6, December 2018, e2174