Low-Rank Tensor Approximations versus Polynomial Chaos Expansions for Meta-Modeling in High-Dimensional Spaces

Meta-models developed with low-rank tensor approximations are investigated for propagating uncertainty through computational models with high-dimensional input. Of interest are meta-models based on polynomial functions, because of the combination of simplicity and versatility they offer. The popular approach of polynomial chaos expansions faces the curse of dimensionality, meaning the exponential growth of the size of the candidate basis with the input dimension. By exploiting the tensor-product form of the polynomial basis, low-rank approximations drastically decrease the number of unknown coefficients, which therein grows only linearly with the input dimension. The construction of such approximations relies on the sequential updating of the polynomial coefficients along separate dimensions, which involves minimization problems of only small size. However, the specification of stopping criteria in the sequential updating of the coefficients and the selection of optimal rank and polynomial degrees remain open questions. In this paper, first, we shed light on the aforementioned issues through extensive numerical investigations. In the sequel, the newly-emerged meta-modeling approach is confronted with state-of-art methods of polynomial chaos expansions. The considered applications involve models of varying dimensionality, i.e. the deflections of two simple engineering structures subjected to static loads and the temperature in stationary heat conduction with spatially varying thermal conductivity. It is found that the comparative accuracy of the two approaches in terms of the generalization error depends on both the application and the size of the experimental design. Nevertheless, low-rank approximations are found superior to polynomial chaos expansions in predicting extreme values of model responses in cases when the two types of meta-models demonstrate similar generalization errors.