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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