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

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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 \((4 \times 8\times 8\times 64\times 64\times 64\times 64)\) 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.

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Data Availability

The data that support the findings of this study are not available due to change of affiliation of the author. The data used in the last section is obtained through a software that has not been made public at the moment of writing this article and consequently cannot be shared. Synthetic data used in Sect. 4.2 can be directly reproduced using the ipython notebook provided with pydecomp library. For other information, please contact the author.

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Acknowledgements

The author gratefully acknowledges using IHPC’s simulation platform for additive manufacturing to generate some data used in this paper. For further information on the simulation platform, readers can contact the author. The author would like to thank Pr. Mejdi Azaez for the support he provided in the writing of this paper and frequent discussions on the subject displayed here, in particular in the valorization of the numerical results. Many thanks to Diego Britez as well, who coded many of the functions in pydecomp as part of his masters’ internship at I2M. The author would like to thank former colleagues at I2M Bordeaux and in particular notus CFD dev team for providing data as well as Pr. T.K. Sengupta (IIT Kanpur) for providing high precision LDC simulation code and the many discussions we had.

Funding

Financial support was provided by Grant MOE2018-T2-1-05 in the context of the author’s research fellowship at NTU in the team of Pr. Wang Li-Lian. Financial support was provided by the Science and Engineering Research Council, A*STAR, Singapore (Grant No. A19E1a0097) in the context of the author’s new position as a Scientist at A*STAR, IHPC, Engineering Mechanics Department, Singapore.

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  1. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Lucas Lestandi

  2. Engineering Mechanics Department, IHPC, A*STAR, Singapore, Singapore

    Lucas Lestandi

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  1. Lucas Lestandi
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Correspondence to Lucas Lestandi.

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The main code used for this data is publicly available under CECILL license. https://git.notus-cfd.org/llestandi/python_decomposition_library

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Lestandi, L. Numerical Study of Low Rank Approximation Methods for Mechanics Data and Its Analysis. J Sci Comput 87, 14 (2021). https://doi.org/10.1007/s10915-021-01421-2

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