Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-02-05T22:50:42.486Z Has data issue: false hasContentIssue false

9 - Through the Kaleidoscope: Symmetries, Groups and Chebyshev-Approximations from a Computational Point of View

Published online by Cambridge University Press:  05 December 2012

By
H. Munthe-Kaas ,
M. Nome  and
B. N. Ryland
Edited by
Felipe Cucker ,
Teresa Krick ,
Allan Pinkus  and
Agnes Szanto
H. Munthe-Kaas
Affiliation:
University of Bergen
M. Nome
Affiliation:
University of Bergen
B. N. Ryland
Affiliation:
University of Bergen
Felipe Cucker
Affiliation:
City University of Hong Kong
Teresa Krick
Affiliation:
Universidad de Buenos Aires, Argentina
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Agnes Szanto
Affiliation:
North Carolina State University
Get access

Summary

Abstract

In this paper we survey parts of group theory, with emphasis on structures that are important in design and analysis of numerical algorithms and in software design. In particular, we provide an extensive introduction to Fourier analysis on locally compact abelian groups, and point towards applications of this theory in computational mathematics. Fourier analysis on non-commutative groups, with applications, is discussed more briefly. In the final part of the paper we provide an introduction to multivariate Chebyshev polynomials. These are constructed by a kaleidoscope of mirrors acting upon an abelian group, and have recently been applied in numerical Clenshaw-Curtis type numerical quadrature and in spectral element solution of partial differential equations, based on triangular and simplicial subdivisions of the domain.

Introduction

Group theory is the mathematical language of symmetry. As a mature branch of mathematics, with roots going almost two centuries back, it has evolved into a highly technical discipline. Many texts on group theory and representation theory are not readily accessible to applied mathematicians and computational scientists, and the relevance of group theoretical techniques in computational mathematics is not widely recognized.

Nevertheless, it is our conviction that knowledge of central parts of group theory and harmonic analysis on groups is invaluable also for computational scientists, both as a language to unify, analyze and generalize computational algorithms and also as an organizing principle of mathematical software construction.

Type
Chapter
Information
Foundations of Computational Mathematics, Budapest 2011 , pp. 188 - 229
Publisher: Cambridge University Press
Print publication year: 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] K., Åhlander, M., Haveraaen and H., Munthe-Kaas, On the role of mathematical abstractions for scientific computing. In The Architecture of Scientific Software, R. F., Boisvert and P. T. P., Tang, Editors, IFIP Advances in Information and Communication Technology, 60, 145–158, 2001.
[2] K., Åhlander and H., Munthe-Kaas, Applications of the generalized Fourier transform in numerical linear algebra. BIT Numerical Mathematics, 45, (2005), 819–850.Google Scholar
[3] E. L., Allgower, K., Georg, R., Miranda and J., Tausch, Numerical exploitation of equivariance. ZAMM, 78, (1998), 185–201.Google Scholar
[4] L., Auslander, Lecture Notes on Nil-Theta Functions. Regional Conference Series in Mathematics, 34, AMS, 1977.
[5] L., Auslander and R., Tolimieri, Is computing with the finite Fourier transform pure or applied mathematics. Notices AMS, 1, 1979.Google Scholar
[6] R. J., Beerends, Chebyshev polynomials in several variables and the radial part of the Laplace—Beltrami operator. Trans. AMS, 328, (1991), 779–814.Google Scholar
[7] A., Bossavit, Symmetry, groups, and boundary value problems. a progressive introduction to noncommutative harmonic analysis of partial differential equations in domains with geometrical symmetry. Comput. Methods Appl. Mech. Engrg., 56, (1986), 167–215.Google Scholar
[8] D., Bump, Lie Groups. Springer Verlag, 2004.
[9] S. H., Christiansen, H. Z., Munthe-Kaas and B., Owren, Topics in structure-preserving discretization. Acta Numerica, 20, (2011), 1–119.Google Scholar
[10] J. H., Conway, N. J. A., Sloane and E., Bannai, Sphere Packings, Lattices, and Groups. Springer Verlag, 1999.
[11] C. C., Douglas and J., Mandel, Abstract theory for the domain reduction method. Computing, 48, (1992), 73–96.Google Scholar
[12] M., Dubiner, Spectral methods on triangles and other domains. J. Scientific Computing, 6, (1991), 345–390.Google Scholar
[13] R., Eier and R., Lidl, A class of orthogonal polynomials in k variables. Math. Ann., 260, (1982), 93–99.Google Scholar
[14] K., Engo, A., Marthinsen and H. Z., Munthe-Kaas, Diffman: An object-oriented matlab toolbox for solving differential equations on manifolds. Applied Numer. Math., 39, (2001), 323–347.Google Scholar
[15] A. F., Fässler and E., Stiefel, Group Theoretical Methods and their Applications. Birkhäuser, Boston, 1992.
[16] K., Georg and R., Miranda, Exploiting symmetry in solving linear equations. In Bifurcation and Symmetry, E. L., Allgower, K., Böhmer and M., Golubisky, editors, 104 of ISNM, 157–168, Birkhäuser, Basel, 1992.
[17] F. X., Giraldo and T., Warburton, A nodal triangle-based spectral element method for the shallow water equations on the sphere. J. Comput. Phys., 207, (2005), 129–150.Google Scholar
[18] R., Hadani and A., Singer, Representation theoretic patterns in three dimensional cryo-electron microscopy I: the intrinsic reconstitution algorithm. Annals Math., 174, (2011), 1219–1241.Google Scholar
[19] R., Hadani and A., Singer, Representation theoretic patterns in three-dimensional cryo-electron microscopy II—the class averaging problem. Found. Comput. Math., 11, (2011), 589–616.Google Scholar
[20] J. S., Hesthaven and T., Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer-Verlag, New York, 2008.
[21] M. E., Hoffman and W. D., Withers, Generalized Chebyshev polynomials associated with affine Weyl groups. Trans. AMS, 308, (1988), 91–104.Google Scholar
[22] D., Huybrechs, On the Fourier extension of non-periodic functions. SIAM J. Numer. Anal., 47, (2010), 4326–4355.Google Scholar
[23] A., Iserles, H., Munthe-Kaas, S. P., Nørsett and A., Zanna, Lie-group methods. Acta Numerica, 9, (2000), 215–365.Google Scholar
[24] G., James and M., Liebeck, Representations and Characters of Groups. Cambridge University Press, 2nd edition, 2001.
[25] T., Koornwinder, Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators I–IV. Indiag. Math., 36, (1974), 48–66 and 357–381.Google Scholar
[26] R., Lidl, Tchebyscheffpolynome in mehreren Variabelen. J. Reine Angew. Math, 273, (1975), 178–198.Google Scholar
[27] J. S., Lomont, Applications of Finite Groups. Academic Press, New York, 1959.
[28] H., Munthe-Kaas, Symmetric FFTs; a general approach. In Topics in linear algebra for vector- and parallel computers, PhD thesis. NTNU, Trondheim, Norway, 1989. Available at: http://hans.munthe- kaas.no.
[29] H. Z., Munthe-Kaas, On group Fourier analysis and symmetry preserving discretizations of PDEs. J. Physics A: Mathematical and General, 39, (2006), 5563.Google Scholar
[30] H., Munthe-Kaas and T., Sørevik, Multidimensional pseudo-spectral methods on lattice grids. To appear in Applied Numerical Mathematics.
[31] W., Rudin, Fourier Analysis on Groups. Wiley-Interscience, 1990.
[32] B. N., Ryland and H. Z., Munthe-Kaas, On multivariate Chebyshev polynomials and spectral approximations on triangles. In Spectral and High Order Methods for Partial Differential Equations, J. S., Hesthaven and E. M., Rønquist, Editors, Lecture Notes Comp. Sci. and Eng., 76, 19–41, 2011.
[33] J. P., Serre, Linear Representations of Finite Groups. Springer, 1977.
[34] R. J., Stanton and P. A., Tomas, Polyhedral summability of Fourier series on compact Lie groups. Amer. J. Math., 100, (1978), 477–493.Google Scholar
[35] S., Thangavelu, Harmonic Analysis on the Heisenberg Group. Birkhauser, 1998.
[36] G., Travaglini, Polyhedral summability of multiple Fourier series. Colloq. Math, 65, (1993), 103–116.Google Scholar
[37] Y., Xu, Fourier series and approximation on hexagonal and triangular domains. Constr. Approx., 31, 2010, 115–138.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×