# Analysis of Discrete \(L^2\) Projection on Polynomial Spaces with Random Evaluations

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## Abstract

We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from *random, noise-free* observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted \(L^2\) norm of the error committed by the random discrete projection is bounded with high probability from above by the best \(L^\infty \) error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function.

## Keywords

Approximation theory Error analysis Multivariate polynomial approximation Nonparametric regression Noise-free data Generalized polynomial chaos Point collocation## Mathematical Subject Classification

41A10 41A25 65N12 65N15 65N35## Notes

### Acknowledgments

The authors would like to recognize the support of the PECOS Center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project “Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar” and King Abdullah University of Science and Technology (KAUST) through the AEA projects “Predictability and Uncertainty Quantification for Models of Porous Media” and “Tracking Uncertainties in Computational Modeling of Reactive Systems” is also acknowledged. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification. The first and second authors were supported by the Italian Grant FIRB-IDEAS (Project RBID08223Z) “Advanced numerical techniques for uncertainty quantification in engineering and life science problems.” We are indebted to A. Cohen and R. DeVore for giving us valuable feedback on the convergence proof. We would also like to thank the anonymous referees for their useful comments that helped us to improve considerably the manuscript, and in particular for the suggestion in the proof of Theorem 2.

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