An inverse theorem for compact Lipschitz regions in ℝ^{𝕕} using localized kernel bases

Hangelbroek, T.; Narcowich, F.; Rieger, C.; Ward, J.

doi:10.1090/mcom/3256

An inverse theorem for compact Lipschitz regions in $\mathbb {R}^d$ using localized kernel bases
HTML articles powered by AMS MathViewer

by T. Hangelbroek, F. J. Narcowich, C. Rieger and J. D. Ward PDF

Math. Comp. 87 (2018), 1949-1989 Request permission

Abstract:

While inverse estimates in the context of radial basis function approximation on boundary-free domains have been known for at least ten years, such theorems for the more important and difficult setting of bounded domains have been notably absent. This article develops inverse estimates for finite dimensional spaces arising in radial basis function approximation and meshless methods. The inverse estimates we consider control Sobolev norms of linear combinations of a localized basis by the $L_p$ norm over a bounded domain. The localized basis is generated by forming local Lagrange functions for certain types of RBFs (namely Matérn and surface spline RBFs). In this way it extends the boundary-free construction recently presented by Fuselier, Hangelbroek and Narcowich [Localized bases for kernel spaces on the unit sphere, SIAM J. Numer. Anal. 51 (2013), no. 5, 2358-2562].

References

Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, 3rd ed., Texts in Applied Mathematics, vol. 15, Springer, New York, 2008. MR 2373954, DOI 10.1007/978-0-387-75934-0
Carsten Carstensen and Birgit Faermann, Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind, Engineering analysis with boundary elements 25 (2001), no. 7, 497–509.
W. Dahmen, B. Faermann, I. G. Graham, W. Hackbusch, and S. A. Sauter, Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method, Math. Comp. 73 (2004), no. 247, 1107–1138. MR 2047080, DOI 10.1090/S0025-5718-03-01583-7
Ronald A. DeVore, Nonlinear approximation, Acta numerica, 1998, Acta Numer., vol. 7, Cambridge Univ. Press, Cambridge, 1998, pp. 51–150. MR 1689432, DOI 10.1017/S0962492900002816
Ronald A. DeVore and Robert C. Sharpley, Besov spaces on domains in $\textbf {R}^d$, Trans. Amer. Math. Soc. 335 (1993), no. 2, 843–864. MR 1152321, DOI 10.1090/S0002-9947-1993-1152321-6
Lawrence C. Evans, Partial differential equations, 2nd ed., Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2010. MR 2597943, DOI 10.1090/gsm/019
Birgit Faermann, Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case, Numer. Math. 92 (2002), no. 3, 467–499. MR 1930387, DOI 10.1007/s002110100319
E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, and G. B. Wright, Localized bases for kernel spaces on the unit sphere, SIAM J. Numer. Anal. 51 (2013), no. 5, 2538–2562. MR 3097032, DOI 10.1137/120876940
E. Fuselier, T. Hangelbroek, F. J. Narcowich, J. D. Ward, and G. B. Wright, Kernel based quadrature on spheres and other homogeneous spaces, Numer. Math. 127 (2014), no. 1, 57–92. MR 3191217, DOI 10.1007/s00211-013-0581-1
Emmanuil H. Georgoulis, Inverse-type estimates on $hp$-finite element spaces and applications, Math. Comp. 77 (2008), no. 261, 201–219. MR 2353949, DOI 10.1090/S0025-5718-07-02068-6
I. G. Graham, W. Hackbusch, and S. A. Sauter, Finite elements on degenerate meshes: inverse-type inequalities and applications, IMA J. Numer. Anal. 25 (2005), no. 2, 379–407. MR 2126208, DOI 10.1093/imanum/drh017
Michael Griebel, Christian Rieger, and Barbara Zwicknagl, Multiscale approximation and reproducing kernel Hilbert space methods, SIAM J. Numer. Anal. 53 (2015), no. 2, 852–873. MR 3327356, DOI 10.1137/130932144
J.-L. Guermond, The LBB condition in fractional Sobolev spaces and applications, IMA J. Numer. Anal. 29 (2009), no. 3, 790–805. MR 2520170, DOI 10.1093/imanum/drn028
Johnny Guzmán, Abner J. Salgado, and Francisco-Javier Sayas, A note on the Ladyženskaja-Babuška-Brezzi condition, J. Sci. Comput. 56 (2013), no. 2, 219–229. MR 3071173, DOI 10.1007/s10915-012-9670-z
T. Hangelbroek, F. Narcowich, C. Rieger, and J. Ward, Direct and inverse results on bounded domains for meshless methods via localized bases on manifolds, arXiv:1406.1435. (to appear in Festschrift for the 80th birthday of Ian Sloan)
T. Hangelbroek, F. J. Narcowich, X. Sun, and J. D. Ward, Kernel approximation on manifolds II: the $L_\infty$ norm of the $L_2$ projector, SIAM J. Math. Anal. 43 (2011), no. 2, 662–684. MR 2784871, DOI 10.1137/100795334
T. Hangelbroek, F. J. Narcowich, and J. D. Ward, Kernel approximation on manifolds I: bounding the Lebesgue constant, SIAM J. Math. Anal. 42 (2010), no. 4, 1732–1760. MR 2679593, DOI 10.1137/090769570
T. Hangelbroek, F. J. Narcowich, and J. D. Ward, Polyharmonic and related kernels on manifolds: interpolation and approximation, Found. Comput. Math. 12 (2012), no. 5, 625–670. MR 2970852, DOI 10.1007/s10208-011-9113-5
R. B. Lehoucq and S. T. Rowe, A radial basis function Galerkin method for inhomogeneous nonlocal diffusion, Comput. Methods Appl. Mech. Engrg. 299 (2016), 366–380. MR 3434919, DOI 10.1016/j.cma.2015.10.021
Svitlana Mayboroda and Marius Mitrea, Sharp estimates for Green potentials on non-smooth domains, Math. Res. Lett. 11 (2004), no. 4, 481–492. MR 2092902, DOI 10.4310/MRL.2004.v11.n4.a7
Jens Markus Melenk, On approximation in meshless methods, Frontiers of numerical analysis, Universitext, Springer, Berlin, 2005, pp. 65–141. MR 2178270, DOI 10.1007/3-540-28884-8_{2}
H. N. Mhaskar, F. J. Narcowich, J. Prestin, and J. D. Ward, $L^p$ Bernstein estimates and approximation by spherical basis functions, Math. Comp. 79 (2010), no. 271, 1647–1679. MR 2630006, DOI 10.1090/S0025-5718-09-02322-9
H. N. Mhaskar, F. J. Narcowich, and J. D. Ward, Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature, Math. Comp. 70 (2001), no. 235, 1113–1130. MR 1710640, DOI 10.1090/S0025-5718-00-01240-0
Francis J. Narcowich, Stephen T. Rowe, and Joseph D. Ward, A novel Galerkin method for solving PDEs on the sphere using highly localized kernel bases, Math. Comp. 86 (2017), no. 303, 197–231. MR 3557798, DOI 10.1090/mcom/3097
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting, Math. Comp. 74 (2005), no. 250, 743–763. MR 2114646, DOI 10.1090/S0025-5718-04-01708-9
Francis J. Narcowich, Joseph D. Ward, and Holger Wendland, Sobolev error estimates and a Bernstein inequality for scattered data interpolation via radial basis functions, Constr. Approx. 24 (2006), no. 2, 175–186. MR 2239119, DOI 10.1007/s00365-005-0624-7
Christian Rieger, Sampling inequalities and applications, Ph.D. thesis, Universität Göttingen, 2008. http://hdl.handle.net/11858/00-1735-0000-0006-B3B9-0
Robert Schaback and Holger Wendland, Inverse and saturation theorems for radial basis function interpolation, Math. Comp. 71 (2002), no. 238, 669–681. MR 1885620, DOI 10.1090/S0025-5718-01-01383-7
Hans Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkhäuser Verlag, Basel, 1983. MR 781540, DOI 10.1007/978-3-0346-0416-1
Hans Triebel, Theory of Function Spaces. II, Monographs in Mathematics, vol. 84, Birkhäuser Verlag, Basel, 1992.
John Paul Ward, $L^p$ Bernstein inequalities and inverse theorems for RBF approximation on $\Bbb R^d$, J. Approx. Theory 164 (2012), no. 12, 1577–1593. MR 3020939, DOI 10.1016/j.jat.2012.09.003
Holger Wendland, Scattered data approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17, Cambridge University Press, Cambridge, 2005. MR 2131724