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

TL;DR

This paper combines proper orthogonal decomposition with hierarchical tensor approximation to efficiently evaluate the robustness of structures under uncertainty, focusing on a non-linear deformation problem and introducing adaptive POD methods.

Contribution

It introduces adaptive POD and hierarchical Tucker approximations for non-linear problems, improving approximation quality and reducing computational cost.

Findings

Adaptive POD outperforms regular POD in accuracy and efficiency.

Hierarchical Tucker approximations enable efficient uncertainty quantification.

HTA combined with (A)POD provides reliable estimates of mean and variance.

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.