Minimum Sobolev norm interpolation of scattered derivative data
| Autor: | Shivkumar Chandrasekaran, C. H. Gorman, Hrushikesh N. Mhaskar |
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| Rok vydání: | 2018 |
| Předmět: |
Numerical Analysis
Polynomial Physics and Astronomy (miscellaneous) Applied Mathematics Mathematical analysis Double-precision floating-point format 010103 numerical & computational mathematics Birkhoff interpolation 01 natural sciences Computer Science Applications Sobolev inequality 010101 applied mathematics Sobolev space Computational Mathematics Derivative (finance) Modeling and Simulation Norm (mathematics) Applied mathematics 0101 mathematics Mathematics PSPACE |
| Zdroj: | Journal of Computational Physics. 365:149-172 |
| ISSN: | 0021-9991 |
| DOI: | 10.1016/j.jcp.2018.03.014 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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