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Weakly Admissible Meshes and Discrete Extremal Sets

Published online by Cambridge University Press:  28 May 2015

Len Bos ,
Stefano De Marchi ,
Alvise Sommariva  and
Marco Vianello
Len Bos
Affiliation:
Department of Computer Science, University of Verona, Strada Le Grazie 15 - 37134 Verona, Italy
Stefano De Marchi
Affiliation:
Department of Pure and Applied Mathematics, University of Padua, Via Trieste 63 - 35121 Padua, Italy
Alvise Sommariva
Affiliation:
Department of Pure and Applied Mathematics, University of Padua, Via Trieste 63 - 35121 Padua, Italy
Marco Vianello
Affiliation:
Department of Pure and Applied Mathematics, University of Padua, Via Trieste 63 - 35121 Padua, Italy

Abstract

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We present a brief survey on (Weakly) Admissible Meshes and corresponding Discrete Extremal Sets, namely Approximate Fekete Points and Discrete Leja Points. These provide new computational tools for polynomial least squares and interpolation on multidimensional compact sets, with different applications such as numerical cubature, digital filtering, spectral and high-order methods for PDEs.

Keywords

65D0565D32Weakly admissible meshesApproximate Fekete pointsDiscrete Leja points
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 4 , Issue 1 , February 2011 , pp. 1 - 12
Copyright
Copyright © Global Science Press Limited 2011

References

[1] Berman, R., Boucksom, S. and Nyström, D. Witt, Fekete points and convergence towards equilibrium measures on complex manifolds, http://arxiv.org/abs/0907.2820, preprint, 2009.Google Scholar
[2] Białas-Ciez, L. and Calvi, J.-P., Pseudo Leja sequences, preprint, 2009.Google Scholar
[3] Bos, L., Caliari, M., De Marchi, S., Vianello, M. and Xu, Y., Bivariate Lagrange interpolation at the Padua points: the generating curve approach, J. Approx. Theory 143 (2006), 15–25.Google Scholar
[4] Bos, L., Calvi, J.-P., Levenberg, N., Sommariva, A. and Vianello, M., Geometric weakly admissible meshes, discrete least squares approximation and approximate Fekete points, Math. Comp., to appear (preprint online at: http://www.math.unipd.it/~marcov/CAApubl.html).Google Scholar
[5] Bos, L., De Marchi, S., Sommariva, A. and Vianello, M., Computing multivariate Fekete and Leja points by numerical linear algebra, SIAM J. Numer. Anal. (2010), to appear.CrossRefGoogle Scholar
[6] Bos, L. and Levenberg, N., On the approximate calculation of Fekete points: the univariate case, Electron. Trans. Numer. Anal. 30 (2008), 377–397.Google Scholar
[7] Bos, L., Levenberg, N. and Waldron, S., On the spacing of Fekete points for a sphere, ball or simplex, Indag. Math. 19 (2008), 163–176.CrossRefGoogle Scholar
[8] Bos, L., Sommariva, A. and Vianello, M., Least-squares polynomial approximation on weakly admissible meshes: disk and triangle, J. Comput. Appl. Math. 235 (2010), 660–668.Google Scholar
[9] Calvi, J.P. and Levenberg, N., Uniform approximation by discrete least squares polynomials, J. Approx. Theory 152 (2008), 82–100.Google Scholar
[10] Civril, A. and Magdon-Ismail, M., On selecting a maximum volume sub-matrix of a matrix and related problems, Theoretical Computer Science 410 (2009), 4801–4811.CrossRefGoogle Scholar
[11] Gassner, G.J., Lörcher, F., Munz, C.-D. and Hesthaven, J.S., Polymorphic nodal elements and their application in discontinuous Galerkin methods, J. Comput. Phys. 228 (2009), 1573–1590.Google Scholar
[12] Klimek, M., Pluripotentials theory, Oxford Univ. Press, 1992.Google Scholar
[13] Saff, E.B. and Totik, V, Logarithmic potentials with external fields, Springer, 1997.Google Scholar
[14] Schaback, R. and De Marchi, S., Nonstandard kernels and their applications, Dolomites Research Notes on Approximation 2 (2009) (http://drna.di.univr.it).Google Scholar
[15] Sommariva, A. and Vianello, M., Computing approximate Fekete points by QR factorizations of Vandermonde matrices, Comput. Math. Appl. 57 (2009), 1324–1336.Google Scholar
[16] Sommariva, A. and Vianello, M., Gauss-Green cubature and moment computation over arbitrary geometries, J. Comput. Appl. Math. 231 (2009), 886–896.Google Scholar
[17] Sommariva, A. and Vianello, M., Approximate Fekete points for weighted polynomial interpolation, Electron. Trans. Numer. Anal. 37 (2010), 1–22.Google Scholar
[18] Warburton, T., An explicit construction of interpolation nodes on the simplex, J. Engrg. Math. 56 (2006), 247–262.Google Scholar
[19] Wilhelmsen, D. R., A Markov inequality in several dimensions, J. Approx. Theory 11 (1974), 216–220.CrossRefGoogle Scholar