Fourier series on triangles and tetrahedra

Transcription

1 Fourier series on triangles and tetrahedra Daan Huybrechs K.U.Leuven Cambridge, Sep 13, 2010

2 Outline Introduction The unit interval Triangles, tetrahedra and beyond

3 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval Triangles, tetrahedra and beyond

4 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

5 The Fourier extension problem Fourier extension/continuation (old) idea: want to avoid Gibbs phenomenon for non-periodic functions want to use FFT on general domains embed domain A into (simple) larger domain B represent function f on A as Fourier series on B How to construct Fourier extension of f f known on B: periodize using cut-off functions (outside A) f not known on B: best approximation in L 2 (A) g n = arg min f g L g G 2 (A) n(b)

6 An example in one dimension Represent f (x) = x on [ 1, 1] as Fourier series on [ 2, 2]

7 An example in one dimension (2) Represent f (x) = x on [ 1, 1] as Fourier series on [ 2, 2] f g n n

8 An example in two dimensions Represent f on A as Fourier series on B B A Very simple, highly accurate, but extremely ill-conditioned.

9 Related topics Fourier extension/continuation (Boyd, 2002), (Bruno, 2003), (Bruno, Han & Pohlman, 2007) Modified Fourier series expansions in Laplace-Neumann eigenfunctions Laplace-Neumann + Laplace-Dirichlet = Fourier series Fourier series on [ 2, 2] restricted to [ 1, 1] is a tight frame Extrapolation of band-limited functions Band-limited f sampled in part of its domain reconstruction of f Fourier extension problem Fast algorithms!

10 Two early warnings g n = arg min f g L g G 2 (A) n(b) 1. Interpretation as Fourier extension leads to intuition that is often wrong. Fourier extension g n converges on A but does not (necessarily) converge on B. 2. The exact solution to the least squares problem and its numerical approximation can be very different. Due to ill-conditioning of least squares problem and (implicit) regularization of standard solvers.

11 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

12 Properties of Fourier extension Properties of Fourier extension Fourier series amenable to FFT-based algorithms no Gibbs phenomenon exponential convergence good resolution of oscillatory functions

13 Resolution power of an approximation scheme With how many degrees of freedom can you represent an oscillatory function? more important than asymptotic convergence rates typically: minimal number of degrees of freedom per wavelength Resolution power: ndof s per wavelength before asymptotic convergence is seen for the test function on the interval [ 1, 1]. f (x) = exp(iπωx)

14 Chebyshev expansion in one dimension f (x) = exp(iπ20x)

15 The resolution power of approximation schemes Resolution power p Fourier series: p = 2 (optimal) polynomial approximation: p = π Fourier extension from [ 1, 1] to [ T, T ], with T > 1: g n = arg min f g L g G n([ T,T ]) 2 ([ 1,1]) exact solution: p = T 2 2 cos π T numerical solution: p = 2T

16 The resolution power of approximation schemes (2) Polynomials, numerical solution, exact solution

17 Outline Introduction The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond

18 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

19 Definition The Fourier extension problem g n = arg min f g L g span D 2 ([ 1,1]) n where D n = { cos π } n { T kx sin π } n k=0 T kx k=1 Exact solution found via orthogonal projection: orthogonalize Fourier basis of [ T, T ] on [ 1, 1] cosines and sines orthogonal to each other: treat separately orthogonal basis described using orthogonal polynomials

20 Interludium: Chebyshev polynomials Chebyshev polynomials of the first and second kinds: cos kx = T k (cos x) sin(k + 1)x sin x = U k (cos x) They are orthogonal polynomials on [ 1, 1]: 2 1 π 2 π T k (y)t l (y) dy = δ k l 1 y 2 U k (y)u l (y) 1 y 2 dy = δ k l

21 Towards a polynomial approximation problem Consider the even functions first (for T = 2): if cos kx = T k (cos x) then cos π 2 kx = T k(cos π 2 x) if y = cos π 2 x then span{cos π 2 kx}n k=0 = span{y k } n k=0 Assume orthogonal polynomial basis Tk h (y), then δ k l = = 4 π Tk h (cos π 2 x)t l h (cos π x) dx 2 Tk h (y)t l h 1 (y) dy 1 y 2

22 Towards a polynomial approximation problem (2)... and the odd functions: if sin(k + 1)x = U k (cos x) sin x then sin π 2 kx = U k 1(cos π 2 x) sin π 2 x if y = cos π 2 x, then x = 2 π cos 1 y and sin π 2 x = 1 y 2 and span{sin π 2 kx}n k=1 = span{y k 1 y 2 } n 1 k=0 Assume orthogonal polynomial basis U h k (y) 1 y 2, then δ k l = = 4 π U h k (cos π 2 x) sin π 2 x Uh l (cos π 2 x) sin π 2 x dx U h k (y)uh l (y) 1 y 2 dy

23 ... and the exact solution to FEP is with n g n (x) = a k Tk h (cos π n 1 2 x) + b k Uk h (cos π 2 x) sin π 2 x k=0 k=0 a k = b k = f (x)tk h (cos π x) dx 2 f (x)u h k (cos π 2 x) sin π 2 x dx

24 ... and the exact solution to FEP is (2) n g n (x) = a k Tk h (cos π n 1 2 x) + b k Uk h (cos π 2 x) sin π 2 x k=0 k=0 or, with f (x) = f e (x) + f o (x) and y = cos( π 2 x): a k = 4 π b k = 4 π ( ) 2 f e π cos 1 y Tk h (y) 1 dy 1 y 2 ( f 2 o π cos 1 y ) U h 1 y 2 k (y) 1 y 2 dy f restricted to even and odd parts on [0, 1], otherwise inverse map x = 2 π cos 1 y not well defined

25 Comparison to Chebyshev expansions Fourier extensions are dual to Chebyshev expansions: Chebyshev expansions Fourier extension f polynomial basis Fourier basis f Fourier basis polynomial basis f (θ) = f (cos θ) ( f (y) = 2 fe π cos 1 y ) f (y) = fo( 2 π cos 1 y) 1 y 2

26 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

27 Convergence analysis Convergence of polynomial expansions on [ 1, 1] is known if f is analytic in ellipse with foci ±1 and sum of semiaxis lengths is ρ = ρ 1 + ρ 2 then polynomial expansion converges as ρ n ρ ρ 1

28 Convergence rate Approach: determine analyticity of f e ( T π cos 1 y) For the case T = 2: ρ = For general T : 1 ρ(t ) < 16 T 2 π 2 ρ(t ) = 3 + cos π T 1 cos π T + (3 + cos π ) 2 T 1 cos π 1 T Note: convergence rate (almost) independent of f can be slower if f has nearby singularities can be faster if f is periodic on [ T, T ]

29 An example in one dimension Convergence for f (x) = x with T = f g n n

30 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

31 Fourier extension of exp(iπωx) Resolution power What is resolution power of polynomial approximations of ( ) T f (y) = f e π cos 1 y = cos ωt cos 1 y Approach study Chebyshev expansions of f (y) use known bounds for n-th coefficient a n show exponential decay for increasing n if n pω

32 Fourier extension of exp(iπωx) (2) Bound for Chebyshev coefficients: a n 2M(r) r n, 1 r < M(r) is maximum of f along Bernstein ellipse E(r) E(r) := { 1 2 (r 1 e iθ + re iθ ) : θ [ π, π]}. mimimize M(r)/r n for given ω and n Result: p = T 2 2 cos π T

33 The resolution power of Fourier extensions Polynomials, numerical solution, exact solution

34 Exact representation of exp(iπωx) Recall the set D n = { cos π } n { T kx sin π } n k=0 T kx k=1 say ω = k/t is a multiple of 1/T then exp(iπωx) can be represented exactly if n k exp(iπωx) = exp(i π T kx) span D k resolution power is 2k/ω = 2kT /k = 2T?

35 Outline Introduction The unit interval Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

36 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

37 Multivariate generalizations Generalization of method and algorithms straightforward embed domain A in tensor-product domain B solve least squares problem short and simple implementation Analysis? generalization through eigenfunctions of Laplacian generalization through Chebyshev polynomials Both paths lead to the same result!

38 The central result 1. Let A be invariant under the action of a symmetry group W, 2. such that W is the Weyl group of a root system R 3. and the fundamental region B of R contains A in its interior. 4. Let M = {µ i } L i=1 be a complete set of irreducible representations for W with dimensions d i. 5. Let G n be a finite-dimensional space of (truncated) Fourier series, periodic on the lattice generated by the coroots of R. 6. Formulate the Fourier extension problem (FEP) as g n = arg min g G n f g L 2 (A)

39 The central result (2) 7. Then there exists a generalized cosine mapping y = cos x, 8. and L functions f i (y), i = 1,..., L, 9. and L sequences of multivariate, matrix-valued, orthogonal polynomials Π i = {p i,k }, 10. such that the exact solution of the FEP is determined by L expansions of f i in {p i,k }: P n f i (y) = f n i (y) = k a i,k p i,k (y). [ ] n g n (x) = a k Tk h (cos π n 1 2 x) + b k Uk h (cos π 2 x) sin π 2 x k=0 k=0

40 Conclusions Multivariate generalization for domains with symmetry connection to Lie groups concepts from representation theory new kinds of orthogonal polynomials

41 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

42 Generalized Chebyshev polynomials Overview Chebyshev polynomials: relate a periodic function on a lattice Q to a polynomial of a simpler periodic function i.e., we generalize the defining formulas cos kx = T k (cos x) and sin(k + 1)x sin x = U k (cos x) using a generalized cosine associated with the lattice Q y = cos x and using a generalization of odd and even functions via the generalized Fourier transform associated with finite groups

43 An example lattice Tiling of R 2 with equilateral triangle In this case W is the symmetry group of the triangle

44 The generalized cosine What is the generalized cosine: analogy to one-dimensional case 1 D n D periodicity e ikx e ikt x, k Q (dual lattice) symmetries [1, 1] W cosine cos kx = 1 ( 2 e ikx + e ikx) Cos k T x = 1 W w W eikt wx Cosine obtained by averaging over the group W.

45 The generalized Fourier transform Let M = {µ i } be a complete and orthogonal set of irreducible representations of W with dimension d i representation theory of finite groups they form an orthogonal basis for functions on the group representations may be matrix-valued! f µ (x) = 1 W f (wx) = 1 W w W d µ Tr µ M in 1D: f (x) = f e (x) + f o (x) µ(w)f (wx) [ ] µ(w) T f µ (wx) f e (x) = 1 2 [f (x) + f ( x)] and f o(x) = 1 [f (x) f ( x)] 2

46 Generalized Chebyshev polynomials, part I Generalization of Chebyshev polynomials of the first kind Cos k T x = 1 W w W e ikt wx = T k (Cos k T 1 x,..., Cos k T n x) = T k (cos x) k 1,..., k n : unit vectors of dual lattice Q 2 dimensions: (Koornwinder, 1974), (Lidl, 1975) n dimensions: (Eier and Lidl, 1982), (Hoffman and Withers, 1988a)

47 Generalized Chebyshev polynomials, part II Generalization of Chebyshev polynomials of the second kind Sin k T x = 1 W w W Sin k T x Sin k T 0 x = U k(cos x) det(w)e ikt wx det(w); determinant of w R n n as a matrix acting on R n k 0 : special element of Q (weight δ) n dimensions: (Hoffman and Withers, 1988b)

48 Generalized Chebyshev polynomials, part III Another generalization of Chebyshev polynomials Sin χ k T x = 1 W w W Sin χ k T x Sin χ k T χ,0 x = U χ,k(cos x) χ(w)e ikt wx χ is a linear character of the Weyl group W k χ,0 : special element of Q n dimensions: (Hoffman and Withers, 1988b)

49 Generalized Chebyshev polynomials, part IV A matrix-valued generalization of Chebyshev polynomials Sin µ k T x = 1 W i=1 w W µ(w)e ikt wx ( ) ( 1 d Sin µ k T x Sin µ kµ,ix) T = U µ,k (cos x) µ is a d-dimensional representation of the Weyl group W denominator is sum of rank-1 matrices 2, n dimensions: (H. and Munthe-Kaas, 2010?)

50 Example polynomial Root system A 2 fundamental region is equilateral triangle Weyl group corresponds to permutations of vertices fundamental region is itself invariant U W,[1,1] (x, y) = U W,[1,2] (x, y) = [ 1 3 x y 3 12 y 3 12 y [ xy x y y 3 24 xy 1 4 y ] 24 xy xy ]

51 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

52 Generalized Chebyshev versus Fourier extension Generalized cosine mapping does not preserve shapes Fourier space polynomial space

53 Outline Introduction The topic of this talk Highly oscillatory functions The unit interval The exact solution Asymptotic convergence rate Resolution power Triangles, tetrahedra and beyond What is the scope? Generalized Chebyshev polynomials An important point Numerical example

54 The equilateral triangle Fourier extension of f (x, y) = x + y on the equilateral triangle

55 The equilateral triangle Fourier extension of f (x, y) = x + y on the equilateral triangle

56 The equilateral triangle Absolute error

57 The end Thank you!