On Multivariate Chebyshev Polynomials - From Group Theory to by rma97348

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									     Motivation: PDEs with discrete domain symmetries
Multivar. Chebyshev pol. and approximation on triangles
               Computing with Multivar. Chebyshev pol.




           On Multivariate Chebyshev Polynomials
                    From Group Theory to Numerical Analysis


                                       Hans Munthe-Kaas1
                                       1 Department
                                                  of Mathematics
                                        University of Bergen
                                  http://hans.munthe-kaas.no
                                    mailto:hans@math.uib.no


                                       MAIA, Ålesund 2007



                          H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Outline



  1     Motivation: PDEs with discrete domain symmetries


  2     Multivar. Chebyshev pol. and approximation on triangles


  3     Computing with Multivar. Chebyshev pol.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Motivation: PDEs and domain symmetries
                                                              2
                              Heat equation ut =                  u on sphere.
             2
                 commutes with rotations and reflections.
          Discretize respecting icosahedral symmetry group.




          Matrix exp(A), where AP = PA for domain symmetries P, is fast
          (by non-commutative Fourier analysis).
          We would like: Spectral Elements in space, Lie group integrators
          in time.
          HOWTO: Spectral elements based on triangular subdivisions?

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


icosahedral symmetries...
                                               G = C2 × A5




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Fast matrix exponential on the sphere
  We seek exp(A) where AP = PA for all permutation matrices
  induced by icosahedral symmetry group G = C2 × A5 .

  Irreducible representations of G = C2 × A5                                   ⇒
  Matrix block diagonalization ⇒
  3500 times faster matrix exponential.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Approximation on triangles; some references
  Approximation theory on triangles:
  J. Hesthaven, ’From Electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex’, SIAM J.
  Num. Anal., 1998.
  M. A. Taylor, B. A. Wingate and R. E. Vincent, ’An algorithm for computing Fekete points in the Triangle’, SIAM J.
  Num. Anal., 2000.
  F. X. Giraldo and T. Warburton, ’A nodal triangle-based spectral element method for shallow water equations on
  the sphere’, J. Comp. Phys., 2005.
  T. Lyche , K. Scherer, ’On the p-norm condition number of the multivariate triangular Bernstein basis’, J. of Comp.
  and Appl. Math., 2000.
  R. T. Farouki, T. N. T. Goodman, and T. Sauer, ’Construction of an orthogonal bases for polynomials in Bernstein
  form on triangular and simplex domains’, Comp. Aided Geom. Design, 2003.

  Multivariate Chebyshev polynomials:
  T. H. Koornwinder (1974), ’Orthogonal polynomials in two variables which are eigenfunctions of two algebraically
  independent partial differential operators I–IV’, Indiag. Math. 36.
  R. Eier and R. Lidl (1982), ’A class of orthogonal polynomials in k variables, Math. Ann. 260.
  M. E. Hoffman and W. D. Withers (1988), ’Generalized Chebyshev Polynomials Associated with Affine Weyl
  Groups’, Transactions of the AMS 308.

  Fast transforms for Multivariate Chebyshev pol.:
  H. Munthe-Kaas, ’Symmetric FFTs, a general approach’, in PhD thesis NTNU, 1989.
  M. Püschel and M. Rötteler, ’Cooley-Tukey FFT like algorithm for the discrete triangle transform, tech. rep. (2005).

  This talk:
  H. Z. Munthe-Kaas, ‘On group Fourier analysis and symmetry preserving discretizations of PDEs’, J. Phys. A 39
  (2006), no. 19, 5563–5584.

                           H. Z. Munthe-Kaas MAIA ’07          Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Classical Chebyshev pol. obtained by folding ...
          Consider periodic domain G = R/2πZ with Fourier basis

                                 (k , θ) = exp(ik θ),             for k ∈ G = Z.

          We say that (·, ·) : G × G → C is a pairing between dual abelian
          groups, satisfying
                (k , θ + θ ) = (k , θ) · (k , θ ),             (k + k , θ) = (k , θ) · (k , θ).


          Let a ’mirror group’ W = {1, −1} act on G, and define
          symmetrized Fourier basis
                                          1                          1
                          (k , θ)s =                   (k , gθ) =                (g ∗ k , θ)
                                         |W |                       |W |
                                                g∈W                        g∈W
                                      1
                                     = (exp(ik θ) + exp(−ik θ)) = cos(k θ).
                                      2

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Classical Chebyshev pol. obtained by folding ...
          Consider periodic domain G = R/2πZ with Fourier basis

                                 (k , θ) = exp(ik θ),             for k ∈ G = Z.

          We say that (·, ·) : G × G → C is a pairing between dual abelian
          groups, satisfying
                (k , θ + θ ) = (k , θ) · (k , θ ),             (k + k , θ) = (k , θ) · (k , θ).


          Let a ’mirror group’ W = {1, −1} act on G, and define
          symmetrized Fourier basis
                                          1                          1
                          (k , θ)s =                   (k , gθ) =                (g ∗ k , θ)
                                         |W |                       |W |
                                                g∈W                        g∈W
                                      1
                                     = (exp(ik θ) + exp(−ik θ)) = cos(k θ).
                                      2

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


... Classical Chebyshev pol. and folding

          Define the change of variables

                                            x(θ) = (1, θ)s = cos(θ).

          This defines Chebyshev polynomials

                                Tk (x) = (k , θ)s            for k = {0, 1, . . .}.

  Note: Tk are polynomials because of recursion:

                                   T0 (x)           = 1
                                   T1 (x)           = x
                           2T1 (x)·Tk (x)           = Tk +1 (x) + Tk −1 (x),

  which follows from the properties of the dual pairing!

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


... Classical Chebyshev pol. and folding

          Define the change of variables

                                            x(θ) = (1, θ)s = cos(θ).

          This defines Chebyshev polynomials

                                Tk (x) = (k , θ)s            for k = {0, 1, . . .}.

  Note: Tk are polynomials because of recursion:

                                   T0 (x)           = 1
                                   T1 (x)           = x
                           2T1 (x)·Tk (x)           = Tk +1 (x) + Tk −1 (x),

  which follows from the properties of the dual pairing!

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Chebyshev recursion




                                                    1
                             x     = T1 =             (exp(iθ) + exp(−iθ))
                                                    2
                                           1
                           Tk      =         (exp(ik θ) + exp(−ik θ))
                                           2
                                           1
                         xTk       =         (Tk +1 + Tk −1 )
                                           2
                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Multivariate Chebyshev polynomials
                                Rootsystem A2


                                                                         General construction:
                                                                             1   Pick a reflection group W
                                                                                 acting on Rn , generating a
                                                                                 discrete translation group
                                                                                 Λ ⊂ Rn . Let G = Rn /Λ.
                                                                                 (G is parallelogram in
                                                                                 example).
                                                                             2   Exponentials on G fold to
                                                                                 ’cosine’ functions Tk (θ) on
                                                                                 fundamental domain (triangle
                                                                                 in example).
                                                                             3   Variable change xi = xi (θ),
                                                                                 where xi are the
                                                                                 W -symmetrized generators
                                                                                 of G = Zn , yields Multivar.
                                                                                    b
                                                                                 Chebyshev polynomials
                                                                                 Tk (x).
                                                                         Problem: Classify all possible such
                                                                         constructions!




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Root systems


  How to define ’mirror groups’ on general periodic domains?

  Definition
  A root system is a subset Φ of a euclidean space E such that
     1    Φ is finite, spans E and does not contain 0.
     2    If α ∈ Φ then the only multiples of α in Φ are ±α.
                                                                       T
                                                 αα
     3    If α ∈ Φ then the reflection σα = I − 2 αT α leaves Φ invariant.
                                        T
                             α β
     4    If α, β ∈ Φ then 2 αT α ∈ Z.

  (Theory originates from classification of semisimple Lie algebras).



                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Kaleidoscopes (the Affine Weyl group)



  Definition
  The group generated by the reflections W = σα |α ∈ Φ is called the
  Weyl group.

  The set Λ = θ → θ + α | α ∈ Φ is called the Root lattice.

  The affine Weyl group W = Λ W is the group generated by all
  these reflections and translations.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Classification of Root systems
  ( Cartan–Weyl–Coxeter–Dynkin)


                                             Dynkin diagram:
                                                     Nodes = generating mirrors.
                                                     Edges indicate mirror-angles
                                                             no edge : 90o
                                                           – one edge : 60o
                                                           = two edges : 45o
                                                           ≡ three edges : 30o
                                                     Arrow separates long and short
                                                     roots.



                           H. Z. Munthe-Kaas MAIA ’07         Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Non-separable 2D cases: A2 , B2 and G2
                                Rootsystem A2




                                                                                 Blue dots: Roots.
                                                                                 Blue arrows: Basis for root
                                                                                 system.
                                                                                 Red arrows: Fundamental
                                                                                 dominant weights.
                                                                                 Dotted lines: Mirrors in affine
                                                                                 Weyl group.
                                                                                 Yellow triangle: Fundamental
                                                                                 domain of affine Weyl group.
                                                                                 Red circles: Weights lattice.
                                                                                 Black dots: Downscaled root
                                                                                 lattice.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Root system B2 :
                               Rootsystem B2




                                                                                 Blue dots: Roots.
                                                                                 Blue arrows: Basis for root
                                                                                 system.
                                                                                 Red arrows: Fundamental
                                                                                 dominant weights.
                                                                                 Dotted lines: Mirrors in affine
                                                                                 Weyl group.
                                                                                 Yellow triangle: Fundamental
                                                                                 domain of affine Weyl group.
                                                                                 Red circles: Weights lattice.
                                                                                 Black dots: Downscaled root
                                                                                 lattice.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Root system G2 :
                              Rootsystem G2




                                                                                 Blue dots: Roots.
                                                                                 Blue arrows: Basis for root
                                                                                 system.
                                                                                 Red arrows: Fundamental
                                                                                 dominant weights.
                                                                                 Dotted lines: Mirrors in affine
                                                                                 Weyl group.
                                                                                 Yellow triangle: Fundamental
                                                                                 domain of affine Weyl group.
                                                                                 Red circles: Weights lattice.
                                                                                 Black dots: Downscaled root
                                                                                 lattice.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Multivariate Chebyshev polynomials

  Let Φ be a d-dimensional root system, W be the Weyl group and Λ
  the root lattice.
  Let G = Rd /Λ be the ’root-periodic’ domain, G = Zd the dual group
  and (k , θ) = exp(ik T θ) be the Fourier basis (dual pairing).

  Definition
  Multivariate Chebyshev polynomials Tk (x) are defined as follows:

                                          1                          1
                    (k , θ)s      =                   (k , gθ) =                 (g T k , θ)
                                         |W |                       |W |
                                                g∈W                        g∈W

                     xk (θ) = (ek , θ)s                    ek = (0, . . . , 1, . . . , 0)T
                     Tk (x) = (k , θ)s .




                           H. Z. Munthe-Kaas MAIA ’07       Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Recurrence relations

  The ’Mother of all’ recurrence relations between Tk (x) is the
  following:

                              T0 (x) = 1                                                        (1)
                             Tej (x) = xj                                                       (2)
                    Tk (x)T (x) =                          αk , (m)|Wm|Tm (x),                  (3)
                                                  m∈W \G
                                                       b


  where α is given by a convolution

                                                 1
                            αk , (m) =                                δgk +h    ,m .            (4)
                                               |W |2
                                                            g,h∈W
                                                                c




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Recurrence example:



  A2 recurrence:


                              z = x1 ,              z = x2
                       T−1,0 = z,                  T0,0 = 1,        T1,0 = z
                         Tn,0 = 3zTn−1,0 − 3zTn−2,0 + Tn−3,0
                        Tn,m = (3Tn,0 Tm,0 − Tn−m,0 )/2.




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


The example: A2




  Fundamental domain of affine Weyl group mapped to Deltoid by
  variable change θ → x.

                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


The Deltoid
  The Deltoid, or 3-cusp Steiner hypocycloide, was first introduced by
  L. Euler in 1745 in discussion of caustic patterns in optics like this
  reflection in a bathroom mirror:




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Deltoid drawn by Spirograph
  The deltoid can be constructed as a ’Spirograph’ curve obtained by
  drawing the perimeter of a circle of radius 2/3 rolling inside a circle of
  radius 1. If the rolling is stopped at regular angles 2π/k , we get lines
  that cross each other at 1-D Chebyshev–Gauss and
  Chebyshev–Gauss–Lobatto points.
                                             The deltoid circumscribes the diameter of a rolling circle



                                  2.5

                                    2

                                  1.5

                                    1

                                  0.5

                                    0

                                 -0.5

                                  -1

                                 -1.5

                                  -2

                                 -2.5


                                        -3   -2          -1              0            1             2     3
                           H. Z. Munthe-Kaas MAIA ’07                        Multivariate Chebyshev Polynomials
       Motivation: PDEs with discrete domain symmetries
  Multivar. Chebyshev pol. and approximation on triangles
                 Computing with Multivar. Chebyshev pol.


Straightening the deltoid
   We have constructed a coordinate map which straightens the deltoid to a triangle. The map has analytically
   computable jacobian. It is well behaved away from the corners, but has corner singularities due to the cusps of the
   deltoid. Interpolation points are given analytically.




                            H. Z. Munthe-Kaas MAIA ’07          Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Lebesgue constant in triangular interpolation
  Lebesgue constant: L = ||I||∞ , where I is the (multivariate) interpolation operator in the given nodes. Slow growth
  of the Lebesgue constant is necessary for spectral convergence.
  Define Lebesgue function:                                X
                                                 λ(x) =        | i (x)|,
                                                                                  i∈I

  where i (x) is Lagrangian cardinal polynominal at node i, then L = ||λ(x)||∞ .
                                                                                    No. of nodal points
                           0            50        100          150     200           250               300
                      40



                      35           Uniform                 Hesthaven                                         Bottom curve: Cheby-Lobatto
                                                                                                             points on Deltoid.    All other
                      30                                                                                     curves: Interpolation points on
                                                                                                             triangle:
                                                                                           Chebyshev ∆ pts
                      25                                                                                            Fekete points.
  Lebesgue constant




                                                                                                                    Image of C–L points by
                      20                                                                                            straightening Deltoid to
                                                                                                                    triangle.
                      15                                                 Fekete                                     Hesthaven electrostatic
                                                                                                                    points.
                      10
                                                                                                                    Uniform meshpoints on
                                                                                  Chebyshev on Deltoid              triangle.
                       5



                       0
                               5             10           15                      20Polynomial degree

                                                  H. Z. Munthe-Kaas MAIA ’07            Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Lebesgue function on Deltoid
    0.8



    0.6



    0.4



    0.2



      0


                                                                                            5
   -0.2
                                                                                            4

   -0.4                                                                                     3

                                                                                            2
   -0.6
                                                                                            1

   -0.8

          -0.8   -0.6   -0.4    -0.2    0     0.2   0.4     0.6      0.8    1      1.2


  The Lebesgue function is nice and nearly uniform over the domain.
                               H. Z. Munthe-Kaas MAIA ’07         Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Lebesgue function on Deltoid straightened to Triangle




  The Lebesgue function is still nice and quite uniform.
                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Unified treatment of all Chebyshev families



                                        Dynkin diagram
                                                ⇓
                                          Cartan matrix
                                                ⇓
                                   Matrix representation of W
                                                ⇓
                                        Symmetric FFTs




                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Symmetric FFTs for multivar. Chebyshev
  Cooley–Tukey is based on duality theory of abelian groups:
                        If H < G then H = G/H ⊥ and G/H = H ⊥ ,
  thus Cooley-Tukey can be based on any subgroup decomposition of
  the lattice.
  Algorithm:
          Pick G by downscaling root lattice with factor Mk = c · 2k .
          Pick sequence of subgrids H1 < H2 < · · · < Hk −1 < G, where Hj
          is root lattice downscaled by factor Mj .
          Divide-and-conquer in C–T style. Take care of symmetries in
          primal space, and dual symmetries in dual space. Compute only
          in fundamental domains. Cosets of Hj + q contain either all the
          original symmetries, or are equivalent to other cosets Hj + r .

                     5
         Cost:         N log2 (N),           (N: gridpoints in fundamental domain).
                     2
                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials
      Motivation: PDEs with discrete domain symmetries
 Multivar. Chebyshev pol. and approximation on triangles
                Computing with Multivar. Chebyshev pol.


Beautiful properties of multivar. Chebyshev:
          Fast algorithms for interpolation, differentiation and integration,
          based on group theory (crystallographic groups and duality).
          Continuous and discrete orthogonalities, both for
          Chebyshev–Gauss and Chebyshev–Gauss–Lobatto points, thus
          high order integration rules.
          Lebesque constant for interpolation problem grows
          logartihmically, just like 1-D case. Thus the discrete nodes are
          good interpolation points.
          Optimal ∞-norm characterization of Tk generalize from 1D to 2D
          (and higher?)
          Recurrence relations.
          The root lattice of A1 , A2 , A3 , D4 , D5 , E6 , E7 , E8 are densest
          lattice packings for these dimensions. Thus optimal in terms of
          sampling theory. (Densest in primal space implies loosest in
          dual space.)
                           H. Z. Munthe-Kaas MAIA ’07      Multivariate Chebyshev Polynomials

								
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