2019/04/27 by Andrea Bonito, Ronald DeVore, Bonito, Andrea +7
Engineering · Mathematics · #41A10 #41A58 #41A63 #65N15 #Advanced Numerical Analysis Techniques #FOS: Mathematics #Mathematical Approximation and Integration #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1904.12105
openalex publication_date 2019/04/27 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
Motivated by numerical methods for solving parametric partial differential\nequations, this paper studies the approximation of multivariate analytic\nfunctions by algebraic polynomials. We introduce various anisotropic model\nclasses based on Taylor expansions, and study their approximation by finite\ndimensional polynomial spaces calP\Λ described by lower sets\n\Λ. Given a budget n for the dimension of calP\Λ, we\nprove that certain lower sets \Λn, with cardinality n, provide a\ncertifiable approximation error that is in a certain sense optimal, and that\nthese lower sets have a simple definition in terms of simplices. Our main goal\nis to obtain approximation results when the number of variables d is large\nand even infinite, and so we concentrate almost exclusively on the case\nd=\∞. We also emphasize obtaining results which hold for the full range\nn\≥ 1, rather than asymptotic results that only hold for n sufficiently\nlarge. In applications, one typically wants n small to comply with\ncomputational budgets.\n