Abstract
The paper deals with the problem of extrapolating data derived from sampling a \(C^m\) function at scattered sites on a Lipschitz region \(\varOmega \) in \(\mathbb R^d\) to points outside of \(\varOmega \) in a computationally efficient way. While extrapolation problems go back to Whitney and many such problems have had successful theoretical resolutions, practical, computationally efficient implementations seem to be lacking. The goal here is to provide one way of obtaining such a method in a solid mathematical framework. The method utilized is a novel two-step moving least squares procedure (MLS) where the second step incorporates an error term obtained from the first MLS step. While the utility of the extrapolation degrades as a function of the distance to the boundary of \(\varOmega \), the method gives rise to improved meshfree approximation error estimates when using the local Lagrange kernels related to certain radial basis functions.
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Notes
September 2017, Bonn.
As is well known, this is not a metric because it fails to obey the triangle inequality.
We are choosing N to be outward drawn, instead of inward drawn, as in [2].
Bases other than \(\{x^\gamma \}_{|\gamma | \le m-1}\) may be used for \(\pi _{m-1}\).
One interesting condition in [16, Propositions 2] is that the \(d_m\times \#X\) matrix with entries \(p_k(\xi )\) has rank \(d_m=\dim \pi _{m-1}(\mathbb R^d)=\left( {\begin{array}{c}d+m-1\\ m-1\end{array}}\right) \). This implies that the smallest possible unisolvent set in X will have exactly \(d_m\) points. For \(d=2\) and \(m=5\) this is just 15.
These are just column vectors here. We use f, g to suggest what will come later.
We are defining \(d_m\) as the dimension of \(V_{u,m}=\pi _{m-1}\), not \(\pi _m\).
The presence of \(\overline{P}_{u,m}^\perp \) is no surprise: \(\overline{E}\) annihilates \(\pi _{m-1}(\mathbb R^d)\).
We could just as well have used any \(0<\alpha <1\) and worked with \(B_{\alpha \delta }(u)\) instead of \(B_{\delta /2}(u)\).
References
Adams, R.A.: Sobolev spaces. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975). Pure and Applied Mathematics, Vol. 65
Alvarado, R., Brigham, D., Maz’ya, V., Mitrea, M., Ziadé, E.: On the regularity of domains satisfying a uniform hour-glass condition and a sharp version of the Hopf-Oleinik boundary point principle. pp. 281–360 (2011). 10.1007/s10958-011-0398-3. Problems in mathematical analysis. No. 57
Amir, A.: High order approximation to non-smooth multivariate functions. Ph.D. thesis, Tel-Aviv University (2017)
Amir, A., Levin, D.: Quasi-interpolation for near-boundary approximations. Jaen J. Approx. 11(1-2), 67–89 (2019)
Babuška, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numer. 12, 1–125 (2003). 10.1017/S0962492902000090.
Demanet, L., Townsend, A.: Stable extrapolation of analytic functions. Found. Comput. Math. 19(2), 297–331 (2019). 10.1007/s10208-018-9384-1.
Fefferman, C.: Whitney’s extension problem for \(C^m\). Ann. of Math. (2) 164(1), 313–359 (2006). 10.4007/annals.2006.164.313.
Fefferman, C.: \(C^m\) extension by linear operators. Ann. of Math. (2) 166(3), 779–835 (2007). 10.4007/annals.2007.166.779.
Fefferman, C.: Whitney’s extension problems and interpolation of data. Bull. Amer. Math. Soc. (N.S.) 46(2), 207–220 (2009). 10.1090/S0273-0979-08-01240-8.
Fefferman, C., Israel, A., Luli, G.: Fitting a Sobolev function to data I. Rev. Mat. Iberoam. 32(1), 275–376 (2016). https://doi.org/10.4171/RMI/887
Fefferman, C., Israel, A., Luli, G.: Fitting a Sobolev function to data II. Rev. Mat. Iberoam. 32(2), 649–750 (2016). https://doi.org/10.4171/RMI/897
Hangelbroek, T., Narcowich, F.J., Rieger, C., Ward, J.D.: Direct and Inverse Results on Bounded Domains for Meshless Methods via Localized Bases on Manifolds. In: J. Dick, F. Kuo, H. Wozniakowski (eds.) Contemporary Computational Mathematics - a celebration of the 80th birthday of Ian Sloan. Springer-Verlag (2018)
Hangelbroek, T., Narcowich, F.J., Rieger, C., Ward, J.D.: An inverse theorem for compact Lipschitz regions in \(\mathbb{R}^d\) using localized kernel bases. Math. Comp. 87(312), 1949–1989 (2018). 10.1090/mcom/3256.
Lancaster, P., Salkauskas, K.: Surfaces generated by moving least squares methods. Math. Comp. 37(155), 141–158 (1981). 10.2307/2007507.
Lehoucq, R.B., Narcowich, F.J., Rowe, S.T., Ward, J.D.: A meshless Galerkin method for non-local diffusion using localized kernel bases. Math. Comp. 87(313), 2233–2258 (2018). 10.1090/mcom/3294.
Levin, D.: The approximation power of moving least-squares. Math. Comp. 67, 1517–1531 (1998)
Wendland, H.: Local polynomial reproduction and moving least squares approximation. IMA J. Numer. Anal. 21(1), 285–300 (2001). 10.1093/imanum/21.1.285.
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge, UK (2005)
Whitney, H.: Functions differentiable on the boundaries of regions. Ann. of Math. (2) 35(3), 482–485 (1934). 10.2307/1968745.
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The authors wish to thank the anonymous referee for a very careful reading of the manuscript and for suggestions that greatly improved the paper. We also wish to thank Professor Arie Israel helpful discussions.
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Communicated by Hans Munthe-Kaas.
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Research supported by Grants DMS-1514789 and DMS-1813091 from the National Science Foundation.
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Amir, A., Levin, D., Narcowich, F.J. et al. Meshfree Extrapolation with Application to Enhanced Near-Boundary Approximation with Local Lagrange Kernels. Found Comput Math 22, 1–34 (2022). https://doi.org/10.1007/s10208-021-09507-x
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DOI: https://doi.org/10.1007/s10208-021-09507-x