KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
King Abdullah University of Science and Technology (KAUST), CEMSE, Thuwal, Saudi Arabia.
Permanent link to this recordhttp://hdl.handle.net/10754/626523
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AbstractWeighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
CitationHaji-Ali, A.-L., Nobile, F., Tempone, R., & Wolfers, S. (2020). Multilevel weighted least squares polynomial approximation. ESAIM: Mathematical Modelling and Numerical Analysis, 54(2), 649–677. doi:10.1051/m2an/2019045
SponsorsF. Nobile received support from the Center for ADvanced MOdeling Science (CADMOS). R. Tempone and S. Wolfers are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. R. Tempone received support from the KAUST CRG3 Award Ref:2281, the KAUST CRG4 Award Ref:2584, and the Alexander von Humboldt foundation. We thank an anonymous referee for their help in improving Proposition 3.3.