Abstract
In many numerical algorithms, integrals or derivatives of functions have to be approximated by linear combinations of function values at nodes. This ranges from numerical integration to meshless methods for solving partial differential equations. The approximations should use as few nodal values as possible and at the same time have a smallest possible error. For each fixed set of nodes and each fixed Hilbert space of functions with continuous point evaluation, e.g. a fixed Sobolev space, there is an error–optimal method available using the reproducing kernel of the space. But the choice of the nodes is usually left open. This paper shows how to select good nodes adaptively by a computationally cheap greedy method, keeping the error optimal in the above sense for each incremental step of the node selection. This is applied to interpolation, numerical integration, and numerical differentiation. The latter case is particularly important for the design of meshless methods with sparse generalized stiffness matrices. The greedy algorithm is described in detail, and numerical examples are provided. In contrast to the usual practice, the greedy method does not always use nearest neighbors for local approximations of function values and derivatives. Furthermore, it avoids multiple points from clusters and it is better conditioned than choosing nearest neighbors.
Similar content being viewed by others
References
Beatson, R., Cherrie, J., Mouat, C.: Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration. Adv. Comput. Math. 11, 253–270 (1999)
Beatson, R., Goodsell, G., Powell, M.: On multigrid techniques for thin plate spline interpolation in two dimensions. In: The Mathematics of Numerical Analysis of Lectures in Applied Mathematics, vol. 32, pp. 77–97. American Mathematical Society, Providence (1996)
Beatson, R., Powell, M.: An iterative method for thin plate spline interpolation that employs approximations to Lagrange functions. In: Griffits, D., Watson, G.A. (eds.) Numerical Analysis 1993, Number 303 in Pitman Research Notes Mathematical Series, pp. 17–39. Longman Science Technology, Harlow (1994)
Bellman, R., Casti, J.: Differential quadrature and long-term integration. J. Math. Anal. Appl. 34, 235–238 (1971)
Bellman, R., Kashef, B.G., Casti, J.: Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations. J. Comput. Phys. 10, 40–52 (1972)
Brown, D., Ling, L., Kansa, E., Levesley, J.: On approximate cardinal preconditioning methods for solving PDEs with radial basis functions. Eng. Anal. Bound. Elem. 19, 343–353 (2005)
Davydov, O., Schaback, R.: Error bounds for kernel-based numerical differentiation. Draft (2013)
Driscoll, T., Fornberg, B.: Interpolation in the limit of increasingly flat radial basis functions. Comput. Math. Appl. 43, 413–422 (2002)
Fasshauer, G.: Solving partial differential equations by collocation with radial basis functions. In: LeMéhauté, A., Rabut, C., Schumaker, L. (eds.) Surface Fitting and Multiresolution Methods, pp. 131–138. Vanderbilt University Press, Nashville (1997)
Franke, C., Schaback, R.: Convergence order estimates of meshless collocation methods using radial basis functions. Adv. Comput. Math. 8, 381–399 (1998)
Franke, R., Salkauskas, K.: Localization of multivariate interpolation and smoothing methods. Paper presented at SIAM Geometric Modeling Conference. Nashville (1995)
Golomb, M., Weinberger, H.: Optimal approximation and error bounds. In: Langer, R. (ed.) On Numerical Approximation, pp. 117–190. The University of Wisconsin Press, Madison (1959)
Mirzaei, D., Schaback, R., Dehghan, M.: On generalized moving least squares and diffuse derivatives. IMA J. Numer. Anal. 32(3), 983–1000 (2012). doi:10.1093/imanum/drr030
Montès, P.: Local kriging interpolation: application to scattered data on the sphere. In: Laurent, P., LeMéhauté, A., Schumaker, L. (eds.) Curves and Surfaces, pp. 325–329. Academic, Boston (1991)
Mouattamid, M., Schaback, R.: Recursive kernels. Anal. Theory Appl. 25, 301–316 (2009)
Müller, S., Schaback, R.: A Newton basis for kernel spaces. J. Approx. Theory 161, 645–655 (2009). doi:10.1016/j.jat.2008.10.014
Pazouki, M., Schaback, R.: Bases for kernel-based spaces. Comput. Appl. Math. 236, 575–588 (2011)
Sard, A.: Linear Approximation, vol. 9. AMS, Providence (1963)
Šarler, B.: From global to local radial basis function collocation method for transport phenomena. In: Advances in Meshfree Techniques of Computer Methods of Applied Science, vol. 5, pp. 257–282. Springer, Dordrecht (2007)
Schaback, R.: Reconstruction of multivariate functions from scattered data. Manuscript, available via http://www.num.math.uni-goettingen.de/schaback/research/group.html (1997)
Schaback, R.: Limit problems for interpolation by analytic radial basis functions. J. Comp. Appl. Math. 212, 127–149 (2008)
Schaback, R.: A computational tool for comparing all linear PDE solvers. Submitted, http://www.num.math.uni-goettingen.de/schaback/research/group.html (2013)
Schaback, R.: Direct discretizations with applications to meshless methods for PDEs. Dolomites Research Notes on Approximation, Proceedings of DWCAA12 6, 37–51 (2013)
Shen, Q.: Local RBF-based differential quadrature collocation method for the boundary layer problems. Eng. Anal. Bound. Elem. 34(3), 213–228 (2010)
Shu, C., Ding, H., Yeo, K.S.: Computation of incompressible Navier-Stokes equations by local RBF-based differential quadrature method. CMES Comput. Model. Eng. Sci. 7(2), 195–205 (2005)
Stevens, D., Power, H., Lees, M., Morvan, H.: The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems. J. Comput. Phys. 228(12), 4606–4624 (2009)
Wirtz, D., Haasdonk, B.: A vectorial kernel orthogonal greedy algorithm. Preprint, Stuttgart Research Centre for Simulation Technology (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schaback, R. Greedy sparse linear approximations of functionals from nodal data. Numer Algor 67, 531–547 (2014). https://doi.org/10.1007/s11075-013-9806-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-013-9806-8