# Polynomial chaos representation of databases on manifolds

## Abstract

Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.

- Authors:

- Université Paris-Est, Laboratoire Modélisation et Simulation Multi-Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-La-Vallée, Cedex 2 (France)
- University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)

- Publication Date:

- OSTI Identifier:
- 22622278

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 335; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; CHAOS THEORY; CONCENTRATION RATIO; CONVERGENCE; DATASETS; DIFFUSION; DISTRIBUTION; EXPANSION; POLYNOMIALS; PROBABILITY; RANDOMNESS; SAMPLING; STATISTICS

### Citation Formats

```
Soize, C., E-mail: christian.soize@univ-paris-est.fr, and Ghanem, R., E-mail: ghanem@usc.edu.
```*Polynomial chaos representation of databases on manifolds*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.01.031.

```
Soize, C., E-mail: christian.soize@univ-paris-est.fr, & Ghanem, R., E-mail: ghanem@usc.edu.
```*Polynomial chaos representation of databases on manifolds*. United States. doi:10.1016/J.JCP.2017.01.031.

```
Soize, C., E-mail: christian.soize@univ-paris-est.fr, and Ghanem, R., E-mail: ghanem@usc.edu. Sat .
"Polynomial chaos representation of databases on manifolds". United States.
doi:10.1016/J.JCP.2017.01.031.
```

```
@article{osti_22622278,
```

title = {Polynomial chaos representation of databases on manifolds},

author = {Soize, C., E-mail: christian.soize@univ-paris-est.fr and Ghanem, R., E-mail: ghanem@usc.edu},

abstractNote = {Characterizing the polynomial chaos expansion (PCE) of a vector-valued random variable with probability distribution concentrated on a manifold is a relevant problem in data-driven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by data-driven queries.},

doi = {10.1016/J.JCP.2017.01.031},

journal = {Journal of Computational Physics},

number = ,

volume = 335,

place = {United States},

year = {Sat Apr 15 00:00:00 EDT 2017},

month = {Sat Apr 15 00:00:00 EDT 2017}

}