Proper orthogonal decomposition (POD) combined with hierarchical tensor approximation (HTA) in the context of uncertain parameters

TL;DR

The paper addresses efficient uncertainty quantification for high-dimensional parameter spaces in nonlinear structural mechanics by combining POD and low-rank tensor approaches. It develops adaptive POD (APOD) to improve accuracy for geometrically nonlinear Neo-Hookean deformation and introduces Hierarchical Tucker Approximation (HTA) to create compact surrogates over a tensorized parameter grid. Key findings show that APOD with a reduced number of modes can outperform standard POD, and HTA can achieve very low surrogate errors (full HTA ~10^{-4}, HTA of (A)POD < 0.15%), enabling rapid Monte Carlo estimates of mean and variance when coupled with snapshot-enrichment strategies. The combination of APOD and HTA demonstrates a scalable, multi-level surrogate framework for nonlinear, uncertain-parameter problems with practical impact on design and reliability assessments.

Abstract

The evaluation of robustness and reliability of realistic structures in the presence of uncertainty involves costly numerical simulations with a very high number of evaluations. This motivates model order reduction techniques like the proper orthogonal decomposition. When only a few quantities are of interest an approximative mapping from the high-dimensional parameter space onto each quantity of interest is sufficient. Appropriate methods for this task are for instance the polynomial chaos expansion or low-rank tensor approximations. In this work we focus on a non-linear neo-hookean deformation problem with the maximal deformation as our quantity of interest. POD and adaptive POD models of this problem are constructed and compared with respect to approximation quality and construction cost. Additionally, the adapative proper orthogonal decomposition (APOD) is introduced and compared to the regular POD. Building upon that, several hierarchical Tucker approximations (HTAs) are constructed from the reduced and unreduced models. A simple Monte Carlo method in combination with HTA and (A)POD is used to estimate the mean and variance of our quantity of interest. Furthermore, the HTA of the unreduced model is employed to find feasible snapshots for (A)POD.

Figures (21)

• Figure 1: Cube under compression
• Figure 2: Comparison of the accuracy of POD and APOD for a variation of the Young's modulus $E$. The number appending "POD" or "APOD" represents the number of used modes.
• Figure 3: Comparison of the absolute $L2$ error
• Figure 4: Comparison of the absolute QoI error
• Figure 5: Comparison of the accuracy of POD and APOD for uncertain parameters: $l_2=1.00$, $E=200$ and $10$ modes