Comparison of the performance and reliability between improved sampling strategies for polynomial chaos expansion

TL;DR

This paper compares various advanced sampling strategies for polynomial chaos expansion to improve the efficiency and reliability of uncertainty quantification in complex models, demonstrating significant differences in convergence and success rates.

Contribution

It provides a comprehensive empirical comparison of space-filling and L1-optimal sampling schemes for polynomial chaos, highlighting their performance in diverse problem settings.

Findings

L1-optimal sampling shows higher stability in surrogate modeling.

Space-filling designs improve convergence in high-dimensional problems.

Sampling scheme effectiveness varies significantly across different test cases.

Abstract

As uncertainty and sensitivity analysis of complex models grows ever more important, the difficulty of their timely realizations highlights a need for more efficient numerical operations. Non-intrusive Polynomial Chaos methods are highly efficient and accurate methods of mapping input-output relationships to investigate complex models. There is substantial potential to increase the efficacy of the method regarding the selected sampling scheme. We examine state-of-the-art sampling schemes categorized in space-filling-optimal designs such as Latin Hypercube sampling and L1-optimal sampling and compare their empirical performance against standard random sampling. The analysis was performed in the context of L1 minimization using the least-angle regression algorithm to fit the GPCE regression models. Due to the random nature of the sampling schemes, we compared different sampling approaches using statistical stability measures and evaluated the success rates to construct a surrogate model with relative errors of $<0.1$\%, $<1$\%, and $<10$\%, respectively. The sampling schemes are thoroughly investigated by evaluating the y of surrogate models constructed for various distinct test cases, which represent different problem classes covering low, medium and high dimensional problems. Finally, the sampling schemes are tested on an application example to estimate the sensitivity of the self-impedance of a probe that is used to measure the impedance of biological tissues at different frequencies. We observed strong differences in the convergence properties of the methods between the analyzed test functions.