Abstract
In this paper, we present a rational RBF interpolation method to approximate multivariate functions with poles or other singularities on or near the domain of approximation. The method is based on scattered point layouts and is flexible with respect to the geometry of the problem’s domain. Despite the existing rational RBF-based techniques, the new method allows the use of conditionally positive definite kernels as basis functions. In particular, we use polyharmonic kernels and prove that the rational polyharmonic interpolation is scalable. The scaling property results in a stable algorithm provided that the method be implemented in a localized form. To this aim, we combine the rational polyharmonic interpolation with the partition of unity method. Sufficient number of numerical examples in one, two and three dimensions are given to show the efficiency and the accuracy of the method.
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References
Bayona, V.: An insight into RBF-FD approximations augmented with polynomials. Comput. Math. Appl. 77, 2337–2353 (2019)
Benzi, M., Golub, G.H., Lieson, J.: Numerical solution of saddle point problems. Acta Numer., 1–137 (2005)
Bozzini, M., Lenarduzzi, L., Schaback R.M.: R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199–215 (2015)
Buhmann, M.D.: Radial Basis Functions: Theory and Implementations. Cambridge University Press, Cambridge (2004)
Buhmann, M.D., DeMarchi, S., Perracchione, E.: Analysis of a new class of rational RBF expansions. IMA J. Nume. Anal. 40(3), 1972–1993 (2020)
Darani, M.R.A.: The RBF partition of unity method for solving the Klein-Gordon equation. Engineering with Computers In press (2020)
Davydov, O., Schaback, R.: Optimal stencils in Sobolev spaces. IMA J. Numer. Anal. 39, 398–422 (2019)
De Marchi, S., Martínez, A., Perracchione, E.: Fast and stable rational RBF-based partition of unity interpolation. J. Comput. Appl. Math. 349, 331–343 (2019)
Fasshauer, G.E.: Meshfree Approximations Methods with Matlab. World Scientific, Singapore (2007)
Gonnet, P., Pachòn, R., Trefethen, L.N.: Robust rational interpolation and least-squares. Elect. Trans. Numer. Anal. 38, 146–167 (2011)
Iske, A.: On the approximation order and numerical stability of local Lagrange interpolation by polyharmonic splines. In: International Series of Numerical Mathematics, vol. 145, pp 153–165. Birkhäuser Verlag (2003)
Iske, A.: On the construction of kernel-based adaptive particle methods in numerical flow simulation. In: Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM), pp 197–221. Springer, Berlin (2013)
Jabalameli, M., Mirzaei, D.: A weak-form RBF-generated finite difference method. Comput. Math. Appl. 79, 2624–2643 (2020)
Jakobsson, S., Andersson, B., Edelvik, F.: Rational radial basis function interpolation with applications to antenna design. J. Comput. Appl. Math. 233, 889–904 (2009)
Larsson, E., Shcherbakov, V., Heryudono, A.: A least squares radial basis function partition of unity method for solving PDEs. SIAM J. Sci. Comput. 39, A2538–A2563 (2017)
Mirzaei, D.: The direct radial basis function partition of unity (D-RBF-PU) method for solving PDEs. SIAM J. Sci. Comput. 43, A54–A83 (2021)
Nakatsukasa, Y., Sète, O., Trefethen, L.N.: The AAA algorithm for ratinal approximation. SIAM J. Sci. Comput. 40(3), A1494–A1522 (2018)
Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection—diffusion equations arising in financial applications. J. Sci. Comput. 64(2), 341–367 (2015)
Scott, A.S., Bai, Y.: A rational radial basis function method for accurately resolving discontinuities and steep gradients. Appl. Numer. Math. 130, 131–142 (2018)
Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23th National Conference, pp 517–523. ACM (1968)
Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis, 3rd edn. Springer, New York (2002)
Wendland, H.: Fast evaluation of radial basis functions: methods based on partition of unity. In: Approximation Theory, X: Wavelets, Splines, and Applications, pp 473–483. Vanderbilt University Press, Nashville (2002)
Wendland, H.: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
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We would like to thank the anonymous reviewers for their insightful comments which improved the quality of paper.
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The second author was in part supported by a Grant from IPM, No. 99650421.
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Communicated by: Robert Schaback
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Farazandeh, E., Mirzaei, D. A rational RBF interpolation with conditionally positive definite kernels. Adv Comput Math 47, 74 (2021). https://doi.org/10.1007/s10444-021-09900-8
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DOI: https://doi.org/10.1007/s10444-021-09900-8