2014/01/05 by Zhou, Tao, Akil Narayan, Narayan, Akil +2
Decision Sciences · Engineering · Mathematics · #Algorithm #Applied mathematics #Chebyshev nodes #Chebyshev polynomials #Computer science #Control Systems and Identification #Discrete mathematics #Mathematical analysis #Mathematical optimization #Mathematics #Measure (data warehouse) #Polynomial #Probabilistic and Robust Engineering Design #Probability measure #Projection (relational algebra) #Structural Health Monitoring Techniques
paper · pdf · doi:10.48550/arxiv.1401.0894
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2014/01/05 · openalex created_date 2019/06/27 · openalex updated_date 2026/08/05
In this work, we discuss the problem of approximating a multivariate function by discrete least squares projection onto a polynomial space using a specially designed deterministic point set. The independent variables of the function are assumed to be random variables, stemming from the motivating application of Uncertainty Quantification (UQ). Our deterministic points are inspired by a theorem due to André Weil. We first work with the Chebyshev measure and consider the approximation in Chebyshev polynomial spaces. We prove the stability and an optimal convergence estimate, provided the number of points scales quadratically with the dimension of the polynomial space. A possible application for quantifying epistemic uncertainties is then discussed. We show that the point set asymptotically equidistributes to the product-Chebyshev measure, allowing us to propose a weighted least squares framework, and extending our method to more general polynomial approximations. Numerical examples are given to confirm the theoretical results. It is shown that the performance of our deterministic points is similar to that of randomly-generated points. However our construction, being deterministic, does not suffer from probabilistic qualifiers on convergence results. (E.g., convergence "with high probability".)