Numerical Study of Low Rank Approximation Methods for Mechanics Data and Its Analysis

This paper proposes a comparison of the numerical aspect and efficiency of several low rank approximation techniques for multidimensional data, namely CPD, HOSVD, TT-SVD, RPOD, QTT-SVD and HT. This approach is different from the numerous papers that compare the theoretical aspects of these methods o...

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Published in	Journal of scientific computing Vol. 87; no. 1
Main Author	Lestandi, Lucas
Format	Journal Article
Language	English
Published	New York Springer US 01.04.2021 Springer Verlag
Subjects	Algorithms Computational Mathematics and Numerical Analysis Computer Science Data Analysis, Statistics and Probability Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Mechanics Modeling and Simulation Physics Theoretical Tensor decomposition Hierarchical SVD HOSVD POD Canonical decomposition 15A69 Tensor train HT 78M34 QTT RPOD 35Q35 15A21 ST-HOSVD Low rank approximation Canonical Decomposition Tensor Train SVD Mathematics Subject Classification 35Q35 tensor decomposition
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Abstract	This paper proposes a comparison of the numerical aspect and efficiency of several low rank approximation techniques for multidimensional data, namely CPD, HOSVD, TT-SVD, RPOD, QTT-SVD and HT. This approach is different from the numerous papers that compare the theoretical aspects of these methods or propose efficient implementation of a single technique. Here, after a brief presentation of the studied methods, they are tested in practical conditions in order to draw hindsight at which one should be preferred. Synthetic data provides sufficient evidence for dismissing CPD, T-HOSVD and RPOD. Then, three examples from mechanics provide data for realistic application of TT-SVD and ST-HOSVD. The obtained low rank approximation provides different levels of compression and accuracy depending on how separable the data is. In all cases, the data layout has significant influence on the analysis of modes and computing time while remaining similarly efficient at compressing information. Both methods provide satisfactory compression, from 0.1% to 20% of the original size within a few percent error in L 2 norm. ST-HOSVD provides an orthonormal basis while TT-SVD doesn’t. QTT is performing well only when one dimension is very large. A final experiment is applied to an order 7 tensor with ( 4 × 8 × 8 × 64 × 64 × 64 × 64 ) entries (32 GB) from complex multi-physics experiment. In that case, only HT provides actual compression (50%) due to the low separability of this data. However, it is better suited for higher order d . Finally, these numerical tests have been performed with pydecomp , an open source python library developed by the author.
AbstractList	This paper proposes a comparison of the numerical aspect and efficiency of several low rank approximation techniques for multidimensional data, namely CPD, HOSVD, TT-SVD, RPOD, QTT-SVD and HT. This approach is different from the numerous papers that compare the theoretical aspects of these methods or propose efficient implementation of a single technique. Here, after a brief presentation of the studied methods, they are tested in practical conditions in order to draw hindsight at which one should be preferred. Synthetic data provides sufficient evidence for dismissing CPD, T-HOSVD and RPOD. Then, three examples from mechanics provide data for realistic application of TT-SVD and ST-HOSVD. The obtained low rank approximation provides different levels of compression and accuracy depending on how separable the data is. In all cases, the data layout has significant influence on the analysis of modes and computing time while remaining similarly efficient at compressing information. Both methods provide satisfactory compression, from 0.1% to 20% of the original size within a few percent error in L 2 norm. ST-HOSVD provides an orthonormal basis while TT-SVD doesn’t. QTT is performing well only when one dimension is very large. A final experiment is applied to an order 7 tensor with ( 4 × 8 × 8 × 64 × 64 × 64 × 64 ) entries (32 GB) from complex multi-physics experiment. In that case, only HT provides actual compression (50%) due to the low separability of this data. However, it is better suited for higher order d . Finally, these numerical tests have been performed with pydecomp , an open source python library developed by the author. This paper proposes a comparison of the numerical aspect and efficiency of several low rank approximation techniques for multidimensional data, namely CPD, HOSVD, TT-SVD, RPOD, QTT-SVD and HT. This approach is different from the numerous papers that compare the theoretical aspects of these methods or propose efficient implementation of a single technique. Here, after a brief presentation of the studied methods, they are tested in practical conditions in order to draw hindsight at which one should be preferred. Synthetic data provides sufficient evidence for dismissing CPD, T-HOSVD and RPOD. Then, three examples from mechanics provide data for realistic application of TT-SVD and ST-HOSVD. The obtained low rank approximation provides different levels of compression and accuracy depending on how separable the data is. In all cases, the data layout has significant influence on the analysis of modes and computing time while remaining similarly efficient at compressing information. Both methods provide satisfactory compression, from 0.1% to 20% of the original size within a few percent error in L2 norm. ST-HOSVD provides an orthonormal basis while TT-SVD doesn’t. QTT is performing well only when one dimension is very large. A final experiment is applied to an order 7 tensor with (4 × 8 × 8 × 64 × 64 × 64 × 64) entries (32GB) from complex multi-physics experiment. In that case, only HT provides actual compression (50%) due to the low separability of this data. However, it is better suited for higher order d. Finally, these numerical tests have been performed with pydecomp , an open source python library developed by the author.
ArticleNumber	14
Author	Lestandi, Lucas
Author_xml	– sequence: 1 givenname: Lucas orcidid: 0000-0001-8457-1131 surname: Lestandi fullname: Lestandi, Lucas email: Lucas_Lestandi@ihpc.a-star.edu.sg organization: School of Physical and Mathematical Sciences, Nanyang Technological University, Engineering Mechanics Department, IHPC, ASTAR
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Cites_doi	10.1007/s00365-011-9131-1 10.1007/978-1-4684-9464-8 10.1007/978-3-642-28027-6 10.1016/j.compfluid.2018.01.038 10.1090/qam/910462 10.1002/gamm.201310004 10.1016/j.cma.2018.12.015 10.1016/j.laa.2011.08.010 10.1002/sapm19287139 10.1007/978-3-030-17012-7_5 10.1016/B978-0-12-493240-1.50017-X 10.1007/BF02310791 10.1007/BF02288367 10.1137/090757861 10.1016/j.compfluid.2018.01.035 10.1007/s00365-012-9175-x 10.1137/110836067 10.2478/cmam-2011-0016 10.1016/j.jcp.2013.02.028 10.1137/100799010 10.1109/SC.2012.58 10.1137/090752286 10.1007/978-3-319-11259-6_21-1 10.1145/2538688 10.1137/090764189 10.1137/15M1036919 10.1137/07070111X 10.1016/j.jcp.2020.109704 10.1063/1.857881 10.1037/h0071325 10.1007/s001620050119 10.1137/140960980 10.1186/s40323-018-0119-2 10.1137/S0895479896305696 10.1007/978-3-7091-1794-1 10.1109/IPDPS.2016.67 10.1007/BF02289464 10.1186/s40323-015-0038-4 10.1063/1.869686 10.1016/j.jcp.2019.108973 10.1016/j.actamat.2017.11.033 10.1137/16M1082214 10.1006/jcph.1998.5943 10.1002/nme.5252
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Keywords	Tensor decomposition Hierarchical SVD HOSVD POD Canonical decomposition 15A69 Tensor train HT 78M34 QTT RPOD 35Q35 15A21 ST-HOSVD Low rank approximation Canonical Decomposition Tensor Train SVD Mathematics Subject Classification 35Q35 tensor decomposition
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Snippet	This paper proposes a comparison of the numerical aspect and efficiency of several low rank approximation techniques for multidimensional data, namely CPD,...
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SubjectTerms	Algorithms Computational Mathematics and Numerical Analysis Computer Science Data Analysis, Statistics and Probability Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Mechanics Modeling and Simulation Physics Theoretical
Title	Numerical Study of Low Rank Approximation Methods for Mechanics Data and Its Analysis
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