A continuous analogue of the tensor-train decomposition
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...
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| Format: | Article |
| Language: | English |
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Elsevier BV
2020
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| Online Access: | https://hdl.handle.net/1721.1/124519 |
| _version_ | 1826202868464484352 |
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| author | Gorodetsky, Alex Karaman, Sertac Marzouk, Youssef M. |
| author2 | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics |
| author_facet | Massachusetts Institute of Technology. Department of Aeronautics and Astronautics Gorodetsky, Alex Karaman, Sertac Marzouk, Youssef M. |
| author_sort | Gorodetsky, Alex |
| collection | MIT |
| description | 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 |
| first_indexed | 2024-09-23T12:23:10Z |
| format | Article |
| id | mit-1721.1/124519 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T12:23:10Z |
| publishDate | 2020 |
| publisher | Elsevier BV |
| record_format | dspace |
| spelling | mit-1721.1/1245192022-09-28T07:54:28Z A continuous analogue of the tensor-train decomposition Gorodetsky, Alex Karaman, Sertac Marzouk, Youssef M. Massachusetts Institute of Technology. Department of Aeronautics and Astronautics 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 National Science Foundation (Grant IIS-1452019) US Department of Energy, Office of Advanced Scientific Computing Research (Award no. DE-SC0007099) 2020-04-07T21:09:37Z 2020-04-07T21:09:37Z 2018-12 2018-04 2019-10-29T15:10:59Z Article http://purl.org/eprint/type/JournalArticle 1879-2138 0045-7825 https://hdl.handle.net/1721.1/124519 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) en 10.1016/J.CMA.2018.12.015 Computer methods in applied mechanics and engineering Creative Commons Attribution-NonCommercial-NoDerivs License http://creativecommons.org/licenses/by-nc-nd/4.0/ application/pdf Elsevier BV arXiv |
| spellingShingle | Gorodetsky, Alex Karaman, Sertac Marzouk, Youssef M. A continuous analogue of the tensor-train decomposition |
| title | A continuous analogue of the tensor-train decomposition |
| title_full | A continuous analogue of the tensor-train decomposition |
| title_fullStr | A continuous analogue of the tensor-train decomposition |
| title_full_unstemmed | A continuous analogue of the tensor-train decomposition |
| title_short | A continuous analogue of the tensor-train decomposition |
| title_sort | continuous analogue of the tensor train decomposition |
| url | https://hdl.handle.net/1721.1/124519 |
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