Minimum Sobolev norm interpolation of scattered derivative data

JOURNAL OF COMPUTATIONAL PHYSICS

卷:365

Article

Chandrasekaran, S.¹  Gorman, C. H.²  Mhaskar, H. N.³ 

[1] Univ Calif Santa Barbara, Dept Elect & Comp Engn, Santa Barbara, CA 93106 USA

[2] Univ Calif Santa Barbara, Dept Math, Santa Barbara, CA 93106 USA

[3] Claremont Grad Univ, Inst Math Sci, Claremont, CA 91711 USA

关键词: Minimum Sobolev norm;    Birkhoff interpolation;    Data-defined manifolds;    Diffusion polynomials;   

DOI  :  10.1016/j.jcp.2018.03.014

来源: Elsevier

【 摘 要 】

We study the problem of reconstructing a function on a manifold satisfying some mild conditions, given data of the values and some derivatives of the function at arbitrary points on the manifold. While the problem of finding a polynomial of two variables with total degree <= n given the values of the polynomial and some of its derivatives at exactly the same number of points as the dimension of the polynomial space is sometimes impossible, we show that such a problem always has a solution in a very general situation if the degree of the polynomials is sufficiently large. We give estimates on how large the degree should be, and give explicit constructions for such a polynomial even in a far more general case. As the number of sampling points at which the data is available increases, our polynomials converge to the target function on the set where the sampling points are dense. Numerical examples in single and double precision show that this method is stable, efficient, and of high-order. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free