### A Christoffel function weighted least squares algorithm for collocation approximations

Here, we propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.

- Publication Date:

- Report Number(s):
- SAND-2015-20768J

Journal ID: ISSN 0025-5718; PII: S002557182016031920

- Grant/Contract Number:
- AC04-94AL85000

- Type:
- Accepted Manuscript

- Journal Name:
- Mathematics of Computation

- Additional Journal Information:
- Journal Volume: 86; Journal Issue: 306; Journal ID: ISSN 0025-5718

- Research Org:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org:
- USDOE National Nuclear Security Administration (NNSA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

- OSTI Identifier:
- 1347352

```
Narayan, Akil, Jakeman, John D., and Zhou, Tao.
```*A Christoffel function weighted least squares algorithm for collocation approximations*. United States: N. p.,
Web. doi:10.1090/mcom/3192.

```
Narayan, Akil, Jakeman, John D., & Zhou, Tao.
```*A Christoffel function weighted least squares algorithm for collocation approximations*. United States. doi:10.1090/mcom/3192.

```
Narayan, Akil, Jakeman, John D., and Zhou, Tao. 2016.
"A Christoffel function weighted least squares algorithm for collocation approximations". United States.
doi:10.1090/mcom/3192. https://www.osti.gov/servlets/purl/1347352.
```

```
@article{osti_1347352,
```

title = {A Christoffel function weighted least squares algorithm for collocation approximations},

author = {Narayan, Akil and Jakeman, John D. and Zhou, Tao},

abstractNote = {Here, we propose, theoretically investigate, and numerically validate an algorithm for the Monte Carlo solution of least-squares polynomial approximation problems in a collocation framework. Our investigation is motivated by applications in the collocation approximation of parametric functions, which frequently entails construction of surrogates via orthogonal polynomials. A standard Monte Carlo approach would draw samples according to the density defining the orthogonal polynomial family. Our proposed algorithm instead samples with respect to the (weighted) pluripotential equilibrium measure of the domain, and subsequently solves a weighted least-squares problem, with weights given by evaluations of the Christoffel function. We present theoretical analysis to motivate the algorithm, and numerical results that show our method is superior to standard Monte Carlo methods in many situations of interest.},

doi = {10.1090/mcom/3192},

journal = {Mathematics of Computation},

number = 306,

volume = 86,

place = {United States},

year = {2016},

month = {11}

}