# Sparse Polynomial Chaos Expansions via Compressed Sensing and D-optimal Design

arXiv: https://arxiv.org/abs/1712.10131 · HTML: https://mlsanity.com/paper/1712.10131/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2017.

## Built on by (most cited first, 10 of 10)

- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Manifold learning-based polynomial chaos expansions for high-dimensional surrogate models](https://mlsanity.com/paper/2107.09814/) (2021)
- [On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples](https://mlsanity.com/paper/2203.13908/) (2022)
- [Deep Adaptive Arbitrary Polynomial Chaos Expansion: A Mini-data-driven Semi-supervised Method for Uncertainty Quantification](https://mlsanity.com/paper/2107.10428/) (2021)
- [Polynomial Chaos Expansions on Principal Geodesic Grassmannian Submanifolds for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/2401.16683/) (2024)
- [Active Learning-based Domain Adaptive Localized Polynomial Chaos Expansion](https://mlsanity.com/paper/2301.13635/) (2023)
- [On Fractional Moment Estimation from Polynomial Chaos Expansion](https://mlsanity.com/paper/2403.01948/) (2024)
- [Consistency regularization-based Deep Polynomial Chaos Neural Network Method for Reliability Analysis](https://mlsanity.com/paper/2203.15655/) (2022)
- [Uncertainty-aware multi-fidelity surrogate modeling with noisy data](https://mlsanity.com/paper/2401.06447/) (2024)
- [Conformal prediction for full and sparse polynomial chaos expansions](https://mlsanity.com/paper/2601.16636/) (2026)

## Abstract (arXiv)

In the field of uncertainty quantification, sparse polynomial chaos (PC) expansions are commonly used by researchers for a variety of purposes, such as surrogate modeling. Ideas from compressed sensing may be employed to exploit this sparsity in order to reduce computational costs. A class of greedy compressed sensing algorithms use least squares minimization to approximate PC coefficients. This least squares problem lends itself to the theory of optimal design of experiments (ODE). Our work focuses on selecting an experimental design that improves the accuracy of sparse PC approximations for a fixed computational budget. We propose DSP, a novel sequential design, greedy algorithm for sparse PC approximation. The algorithm sequentially augments an experimental design according to a set of the basis polynomials deemed important by the magnitude of their coefficients, at each iteration. Our algorithm incorporates topics from ODE to estimate the PC coefficients. A variety of numerical simulations are performed on three physical models and manufactured sparse PC expansions to provide a comparative study between our proposed algorithm and other non-adaptive methods. Further, we examine the importance of sampling by comparing different strategies in terms of their ability to generate a candidate pool from which an optimal experimental design is chosen. It is demonstrated that the most accurate PC coefficient approximations, with the least variability, are produced with our design-adaptive greedy algorithm and the use of a studied importance sampling strategy. We provide theoretical and numerical results which show that using an optimal sampling strategy for the candidate pool is key, both in terms of accuracy in the approximation, but also in terms of constructing an optimal design.