A continuous analogue of the tensor-train decomposition
Name
1510.09088.pdf
Description
Accepted version
Size
538.79 KB
Format
Adobe PDF
Checksum (MD5)
61e7d2b7120fd40320432e4302ce10ca
Author(s)
Gorodetsky, Alex
Karaman, Sertac
Marzouk, Youssef M.
Date Issued
December 2018
Journal
Computer methods in applied mechanics and engineering
Publisher
Elsevier BV
Citation
Gorodetsky, Alex, Sertac Karaman, and Youssef M. Marzouk, "A continuous analogue of the tensor-train decomposition." Computer methods in applied mechanics and engineering 347 (2018): p. 59-84 doi 10.1016/J.CMA.2018.12.015 ©2018 Author(s)
Version
Author's final manuscript
Abstract
We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of “continuous” matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities. ©2018
MIT Department
Massachusetts Institute of Technology. Department of Aeronautics and Astronautics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/J.CMA.2018.12.015
