Speaker: H. Z. Munthe-Kaas

Place: FoCM 2011 Plenary lecture

Abstract: Classical univariate Chebyshev polynomials are fundamental objects in computational mathematics. Their ubiquity in applications is summarized by the quote: "Chebyshev polynomials are everywhere dense in numerical analysis" (perhaps due to George Forsythe). Most of the useful properties of Chebyshev polynomials arise from their tight connections to group theory. From this perspective, multivariate Chebyshev polynomials appear as natural generalizations, constructed by a kaleidoscope of mirrors acting upon R^n (i.e. affine Weyl groups). Multivariate Chebyshev polynomials were first introduced by Koornwinder already in 1974, but they have only very recently been applied in computational mathematics. The fact that the multivariate polynomials are orthogonal on domains related to triangles, tetrahedra and higher dimensional simplexes, rises the important question of their applicability in triangle-based spectral element methods for PDEs. We have developed general software tools with such applications in mind. Finally, we remark upon connections to the representation theory of compact Lie groups, which may lead to applications in very different areas of computational mathematics.

References:

1. •H. Z. Munthe-Kaas: On group Fourier analysis and symmetry preserving discretizations of PDEs, J. Phys. A: Math. Gen. 39 (2006) 5563–5584.
   

2. •H. Munthe-Kaas: Symmetric FFTs; a general approach, Tech. rep. Department of Math. Sciences, NTNU Trondheim, May 1989.
   

3. •H. Z. Munthe-Kaas, M. Nome, B. N. Ryland: Through the Kaleidoscope; Symmetries, Groups and Chebyshev Approximations from a Computational Point of View, to appear in FoCM Budapest proceedings.
   

4. •S. Christiansen, H. Z. Munthe-Kaas, B. Owren: Topics in structure preservation. Acta Numerica 2011.
   

5. •B.N. Ryland, H.Z. Munthe-Kaas: On multivariate Chebyshev polynomials and spectral approximation on triangles, Spectral and High Order Methods for Partial Differential Equations, Lecture Notes in computational science and engineering, Springer 2011.