# Least Squares Polynomial Chaos Expansion: A Review of Sampling Strategies

## TL;DR

This paper reviews various sampling strategies for least squares polynomial chaos expansion, introduces a hybrid method, and compares their performance across multiple numerical examples to guide practitioners.

## Contribution

It presents a comprehensive review of sampling methods for PCE, introduces a novel hybrid sampling technique, and provides empirical comparisons to inform method selection.

## Key findings

- Alphabetic-coherence-optimal sampling outperforms others in high-order and low oversampling scenarios.
- Hybrid sampling method combines advantages of ODE and coherence-optimal techniques.
- Performance varies significantly depending on problem complexity and sampling strategy.

## Abstract

As non-institutive polynomial chaos expansion (PCE) techniques have gained growing popularity among researchers, we here provide a comprehensive review of major sampling strategies for the least squares based PCE. Traditional sampling methods, such as Monte Carlo, Latin hypercube, quasi-Monte Carlo, optimal design of experiments (ODE), Gaussian quadratures, as well as more recent techniques, such as coherence-optimal and randomized quadratures are discussed. We also propose a hybrid sampling method, dubbed alphabetic-coherence-optimal, that employs the so-called alphabetic optimality criteria used in the context of ODE in conjunction with coherence-optimal samples. A comparison between the empirical performance of the selected sampling methods applied to three numerical examples, including high-order PCE's, high-dimensional problems, and low oversampling ratios, is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand. We observed that the alphabetic-coherence-optimal technique outperforms other sampling methods, specially when high-order ODE are employed and/or the oversampling ratio is low.