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

• 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
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-021-01421-2

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.