A Julia libray for dealing with Hierarchical Tensors.
WORK IN PROGRESS
Code and interface are still strongly in flux!
- a type
HTensorNodewhich holds the nodes of the tree-structured HTensor - indexing and slicing (
getelts) - a type
HTensorEvaluatorfor zero-allocation evaluation of individual elements of the HTensor (getelt) - converting into a full multi-dimensional array (
array)
Most importantly, methods for converting a full multi-dimensional array (e.g. stored in memory
as a Julia Array, or computed on the fly as a ComputedArray, etc.)
into a Hierarchical Tensor. This conversion will be lossy (i.e. only approximate) but should be controllable
either via an a-priori error threshold, or via an a-posteriori error estimate.
Based on the algorithms described in:
L. Grasedyck,
Hierarchical Singular Value Decomposition of Tensors,
SIAM J. Matrix Anal. Appl. 31, 2029 (2010)
DOI: 10.1137/090764189
The original Hierarchical SVD needs access to all elements of the full tensor, which is prohibitive for high tensor orders. If the tensor has regularity, it can be sufficient to randomly (or not so randomly) sample from the full tensor. Initial ideas and results are presented in:
F. Otto, Y.-C. Chiang, and D. Peláez,
Accuracy of Potfit-based potential representations and its impact on the performance of (ML-)MCTDH.
Chem. Phys. (2017), in press
DOI: 10.1016/j.chemphys.2017.11.013 -- arXiv preprint
For the mathematical background on Hierarchical Tensors, see:
W. Hackbusch and S. Kühn,
A new scheme for the tensor representation.
J. Fourier Anal. Appl. 15, 706 (2009)
DOI: 10.1007/s00041-009-9094-9
Additional helpful literature:
W. Hackbusch,
Tensor Spaces and Numerical Tensor Calculus.
Springer Series in Computational Mathematics 42 (2012), Springer, Heidelberg
DOI: 10.1007/978-3-642-28027-6
L. Grasedyck, D. Kressner, and C. Tobler,
A literature survey of low-rank tensor approximation techniques.
GAMM-Mitteilungen, 36: 53-78 (2013)
DOI: 10.1002/gamm.201310004