# Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies

@article{Hampton2015CompressiveSO, title={Compressive sampling of polynomial chaos expansions: Convergence analysis and sampling strategies}, author={Jerrad Hampton and Alireza Doostan}, journal={J. Comput. Phys.}, year={2015}, volume={280}, pages={363-386} }

## Figures and Tables from this paper

## 167 Citations

Coherence motivated sampling and convergence analysis of least squares polynomial Chaos regression

- Mathematics
- 2015

Sparse polynomial chaos expansions - Benchmark of compressive sensing solvers and experimental design techniques

- Computer Science
- 2019

This work proposes a general modular framework for adaptive sparse PCE computations, in which most of the methods put forward in the literature can be fit, and collects and explains the available methods and analyse their behavior on various analytical and numerical examples.

A Generalized Sampling and Preconditioning Scheme for Sparse Approximation of Polynomial Chaos Expansions

- Computer ScienceSIAM J. Sci. Comput.
- 2017

An algorithm for recovering sparse orthogonal polynomial expansions via collocation that solves a preconditioned $\ell^1$-minimization problem and presents theoretical analysis and numerical results that show the method is superior to standard Monte Carlo methods in many situations of interest.

Sparse polynomial chaos expansions via compressed sensing and D-optimal design

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2018

Polynomial chaos expansions for dependent random variables

- Computer ScienceComputer Methods in Applied Mechanics and Engineering
- 2019

Sparse Polynomial Chaos Expansions: Literature Survey and Benchmark

- Computer ScienceSIAM/ASA J. Uncertain. Quantification
- 2021

It is found that the choice of sparse regression solver and sampling scheme for the computation of a sparse PCE surrogate can make a significant difference, of up to several orders of magnitude in the resulting mean-square error.

A near-optimal sampling strategy for sparse recovery of polynomial chaos expansions

- Computer ScienceJ. Comput. Phys.
- 2018

Sparse approximation of data-driven Polynomial Chaos expansions: an induced sampling approach

- Computer ScienceCommunications in Mathematical Research
- 2020

The capacity of the proposed induced sampling algorithm via sparse representation with limited data on test functions, and on a Kirchoff plating bending problem with random Young's modulus, is demonstrated.

## References

SHOWING 1-10 OF 58 REFERENCES

Selection of polynomial chaos bases via Bayesian model uncertainty methods with applications to sparse approximation of PDEs with stochastic inputs

- Computer Science, MathematicsJ. Comput. Phys.
- 2014

Adaptive sparse polynomial chaos expansion based on least angle regression

- Computer ScienceJ. Comput. Phys.
- 2011

A weighted l1-minimization approach for sparse polynomial chaos expansions

- Computer Science, MathematicsJ. Comput. Phys.
- 2014

Sparse Legendre expansions via l1-minimization

- Mathematics, Computer ScienceJ. Approx. Theory
- 2012

Monte Carlo Sampling Methods Using Markov Chains and Their Applications

- Mathematics
- 1970

SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and…

Reweighted ℓ1ℓ1 minimization method for stochastic elliptic differential equations

- Mathematics, Computer ScienceJ. Comput. Phys.
- 2013

DIMENSIONALITY REDUCTION FOR COMPLEX MODELS VIA BAYESIAN COMPRESSIVE SENSING

- Computer Science
- 2014

This work implements a PC-based surrogate model construction that “learns” and retains only the most relevant basis terms of the PC expansion, using sparse Bayesian learning, which dramatically reduces the dimensionality of the problem, making it more amenable to further analysis such as sensitivity or calibration studies.

Numerical Methods for Stochastic Computations: A Spectral Method Approach

- Computer Science
- 2010

This book describes the class of numerical methods based on generalized polynomial chaos (gPC), an extension of the classical spectral methods of high-dimensional random spaces designed to simulate complex systems subject to random inputs.

Postmaneuver Collision Probability Estimation Using Sparse Polynomial Chaos Expansions

- Computer Science
- 2015

Results demonstrate that these polynomial chaos-based methods provide a Monte Carlo-like estimate of the collision probability, including a potential collision with debris in low Earth orbit.

STOCHASTIC COLLOCATION ALGORITHMS USING 𝓁 1 -MINIMIZATION

- Computer Science
- 2012

The analysis suggests that using the Chebyshev measure to precondition the ‘1-minimization, which has been shown to be numerically advantageous in one dimension in the literature, may in fact become less efficient in high dimensions.