Polynomial Approximation of Anisotropic Analytic Functions of Several Variables
Polynomial Approximation of Anisotropic Analytic Functions of Several Variables
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DOI：
10.1007/s00365-020-09511-4
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发表时间：
2020-07-16
影响因子：
2.7
通讯作者：
Petrova, Guergana
中科院分区：
文献类型：
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作者：
Bonito, Andrea;DeVore, Ronald;Petrova, Guergana
受求解参数偏微分方程组的数值方法的启发，研究了多元解析函数的代数多项式逼近问题。我们引入了各种基于泰勒展开式的各向异性模型类，并研究了它们在由下集Lambda描述的有限维多项式空间P-Lambda上的逼近。给定P-Lambda的维度预算n，我们证明了具有基数n的某些下集合Lambda(N)提供了某种意义上最优的可证明的逼近误差，并且这些下集合具有简单的单纯形定义。我们的主要目标是在变量数d很大甚至是无穷大时得到近似结果，因此我们几乎只关注d=无穷大的情况。我们还强调得到对全范围n>；=1成立的结果，而不是只有当n足够大时才成立的渐近结果。在应用程序中，人们通常希望n个小的，以符合计算预算。
Motivated by numerical methods for solving parametric partial differential equations, this paper studies the approximation of multivariate analytic functions by algebraic polynomials. We introduce various anisotropic model classes based on Taylor expansions, and study their approximation by finite dimensional polynomial spaces P-Lambda described by lower sets Lambda. Given a budget n for the dimension of P-Lambda, we prove that certain lower sets Lambda(n), with cardinality n, provide a certifiable approximation error that is in a certain sense optimal, and that these lower sets have a simple definition in terms of simplices. Our main goal is to obtain approximation results when the number of variables d is large and even infinite, and so we concentrate almost exclusively on the case d = infinity. We also emphasize obtaining results which hold for the full range n >= 1, rather than asymptotic results that only hold for n sufficiently large. In applications, one typically wants n small to comply with computational budgets.