Premium
Distributed hierarchical SVD in the Hierarchical Tucker format
Author(s) -
Grasedyck Lars,
Löbbert Christian
Publication year - 2018
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2174
Subject(s) - multigrid method , dimension (graph theory) , tensor (intrinsic definition) , computer science , conjugate gradient method , tucker decomposition , hierarchical database model , mathematics , theoretical computer science , algorithm , partial differential equation , tensor decomposition , geometry , combinatorics , data mining , mathematical analysis
Summary 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 that 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 A X = B directly in the Hierarchical Tucker format using iterative methods such as 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.