Chebfun and equispaced data

Transcription

1 Chebfun and equispaced data Georges Klein University of Oxford SIAM Annual Meeting, San Diego, July 12, 2013

2 Outline 1 1D Interpolation, Chebfun and equispaced data 2 3 (joint work with R. Platte) G. Klein Chebfun and equispaced data 2/28

3 1D interpolation Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Given: a x 0 < x 1 <... < x n b, f (x 0 ), f (x 1 ),..., f (x n ), n + 1 distinct nodes and corresponding values. We study functions g from a finite-dimensional linear subspace of (C[a, b], ) which interpolate f between the nodes, g(x i ) = f (x i ) = f i, i = 0,..., n. There exists a unique polynomial p n of degree n, which interpolates the data and can be written in barycentric form p n (x) = n / n λ i λ / i f i with λ i = 1 (x i x k ). x x i x x i k i G. Klein Chebfun and equispaced data 3/28

4 Chebyshev points Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Polynomial interpolation with Chebyshev points Chebyshev points (red) is a successful and near optimal approach in the vector space of polynomials of degree n. G. Klein Chebfun and equispaced data 4/28

5 Exponential convergence Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Theorem If f is analytic in an open Bernstein ellipse E ρ, and f (z) M in E ρ for some M, then f p n 4Mρ n ρ 1. Bernstein ellipses, ρ = 1.1, 1.5, 2 1/(1 + 25x 2 ) G. Klein Chebfun and equispaced data 5/28

6 Condition and stability Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Theorem (condition) The Lebesgue constant Λ n, i.e., the condition number, associated with degree n polynomial interpolation with Chebyshev points satisfies Λ n = max a x b n Theorem (stability) / λ i n x x i λ i x x i 2 log(n + 1) + 1. π The polynomial interpolant with Chebyshev points can be evaluated stably when written in barycentric form n ( 1)i x x i f i / n ( 1)i x x i. G. Klein Chebfun and equispaced data 6/28

7 Condition and stability Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Theorem (condition) The Lebesgue constant Λ n, i.e., the condition number, associated with degree n polynomial interpolation with Chebyshev points satisfies Λ n = max a x b n Theorem (stability) / λ i n x x i λ i x x i 2 log(n + 1) + 1. π The polynomial interpolant with Chebyshev points can be evaluated stably when written in barycentric form n ( 1)i x x i f i / n ( 1)i x x i. G. Klein Chebfun and equispaced data 6/28

8 Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data When the data is provided at equispaced nodes only Polynomial interpolation with equispaced nodes is not reliable: it is severely ill-conditioned (Λ n exponentially fast), it leads to Runge s phenomenon Runge regions 1/(1 + 25x 2 ) exact function polynomial interpolation nodes G. Klein Chebfun and equispaced data 7/28

9 Impossibility Interpolation problem Polynomial interpolation with Chebyshev points Equispaced data Theorem (Platte Trefethen Kuijlaars 11) Let a compact set E containing [ 1, 1] in its interior be fixed, and suppose {φ n } is an approximation procedure based on equispaced n-grids such that, for some M < and σ > 1, f φ n (f ) [ 1,1] Mσ n f E 1 n < for all f B(E). Then the condition numbers for φ n satisfy κ n C n for some C > 1 and all sufficiently large n, with κ n = sup f lim sup ε 0 0< δf n-grid ε φ n (f + δf ) φ n (f ) [ 1,1] δf n-grid. G. Klein Chebfun and equispaced data 8/28

10 Floater Hormann interpolation Properties Funqui p n (x) = n n λ i x x i f i λ i n n w i x x i f i w i = r n (x) x x i x x i G. Klein Chebfun and equispaced data 9/28

11 Floater Hormann interpolation Properties Funqui Construction of Floater Hormann interpolation for arbitrary nodes - Given n + 1 nodes, a = x 0 < x 1 <... < x n = b, and corresponding function values, f 0,..., f n, choose an integer d {0, 1,..., n}, blending parameter, - for i = 0,..., n d, define p i (x), the polynomial of low degree d interpolating f i, f i+1,..., f i+d. The d-th interpolant of the family is a blend of the p i (x), r n (x) = n d λ i (x)p i (x) n d λ i (x), with λ i (x) = Notice that for d = n, r n simplifies to p n. ( 1) i (x x i )... (x x i+d ). G. Klein Chebfun and equispaced data 10/28

12 Linear barycentric rational form Floater Hormann interpolation Properties Funqui For its evaluation, we write r n in linear barycentric form n d n w i λ i (x)p i (x) f i x x i r n (x) = =. n d n w i λ i (x) x x i For equispaced nodes, the weights w i do not depend on f, and oscillate in sign with absolute values 1, 1,..., 1, 1, d = 0, 1 2, 1, 1,..., 1, 1, 1 2, d = 1, 1 4, 3 4, 1, 1,..., 1, 1, 3 4, 1 4, d = 2, 1 8, 4 8, 7 8, 1, 1,..., 1, 1, 7 8, 4 8, 1 8, d = 3, d 4. G. Klein Chebfun and equispaced data 11/28

13 Floater Hormann interpolation Properties Funqui Algebraic convergence and polynomial reproduction Theorem (Floater Hormann 07) Let 1 d n and f C d+2 [a, b], h = max (x i+1 x i ), then 0 i n 1 f r n Kh d+1, where K depends only d, b a and derivatives of f ; the analytic rational function r n has no real poles; r n reproduces polynomials of degree d if n d is even and of degree d + 1 otherwise. Remark: No Runge phenomenon. G. Klein Chebfun and equispaced data 12/28

14 Condition number Floater Hormann interpolation Properties Funqui Theorem (Bos De Marchi Hormann K. 12) Let 0 d n and the nodes x i, i = 0,..., n, be equispaced. Then 2 d 2 ( n ) d + 1 log d 1 Λ n,d 2 d 1 (2 + log n) Figure: Lebesgue function for Floater Hormann interpolation with equispaced nodes in [ 1, 1] with d = 2 and d = 5 and n = 40. G. Klein Chebfun and equispaced data 13/28

15 Floater Hormann interpolation Properties Funqui Approximation of derivatives and integrals Theorem (K. Berrut 12) Suppose n, d n, and k d are positive integers and f C d+1+k [a, b]. If the nodes are equispaced or almost equispaced, then f (k) (x i ) r (k) n (x i ) Kh d+1 k, 0 i n. Theorem (Güttel K. 13) Suppose n and d, d n/2 1, are positive integers, f C d+3 [a, b] and the nodes are equispaced. Then for any x [a, b], x x f (y) dy r n (y) dy Khd+2. a a G. Klein Chebfun and equispaced data 14/28

16 Adaptive choice of d Floater Hormann interpolation Properties Funqui Adaptive choice of d via cross validation remove a few data values, compute the interpolant for various values of d, compare the errors at the removed data points and keep the d that gives the smallest error, interpolate the entire data set with this d, represent that analytic interpolant in Chebfun ( Funqui). Remark: not limited to analytic functions nor equispaced data. G. Klein Chebfun and equispaced data 15/28

17 Examples on [ 1, 1] Floater Hormann interpolation Properties Funqui 10 0 sin(100x) funqui chebfun x funqui chebfun entire function singularity at x = 1.05 G. Klein Chebfun and equispaced data 16/28

18 Examples on [ 1, 1] Floater Hormann interpolation Properties Funqui 1/( x 2 ) 1/( sin(7x) 2 ) funqui chebfun 10 0 funqui chebfun singularities at x = ± 1 10 i x = ± arcsinh(1/4) 7 i ± kπ 7 G. Klein Chebfun and equispaced data 17/28

19 Examples on [ 1, 1] Floater Hormann interpolation Properties Funqui exp( 1/x 2 ) x 3 funqui chebfun 10 2 funqui chebfun C function C 2 function G. Klein Chebfun and equispaced data 18/28

20 Schemes for equispaced interpolation Results with other schemes for equispaced interpolation joint work with Rodrigo Platte from ASU G. Klein Chebfun and equispaced data 19/28

21 Schemes for equispaced interpolation Results Fourier extension (Bruno et al., Boyd, Huybrechs, Adcock) Let G N be the space of 2T -periodic functions, T > 1, G N = span{e i kπ T x : k N} The solution of the discrete least squares problem g N = arg min g G N n f i g(x i ) 2, is the Fourier extension of f to the interval [ T, T ]. G. Klein Chebfun and equispaced data 20/28

22 Schemes for equispaced interpolation Results Hybrid windowed Fourier (Platte, Gelb) Multiply f by w(x) = exp( αx 2λ ) so that wf and its derivatives become periodic to machine precision on the interval; approximate wf by its truncated Fourier series F N [wf ]; divide out w, F N[wf ] w ; use polynomial least squares near the ends, where w is too small. G. Klein Chebfun and equispaced data 21/28

23 Map and least squares Kosloff and Tal-Ezer map Schemes for equispaced interpolation Results Equispaced Mapped Chebyshev x = g(y; α) = arcsin(αy), 0 α < 1 x, y [ 1, 1], arcsin(α) Chebyshev pts more evenly spaced pts, singularities at ±1/α. g(y; 1): Chebyshev pts equispaced pts, singular map. g 1 : equispaced nodes points that cluster toward the ends of the interval, and approximate by polynomial least squares. Several strategies for choosing α, e.g., for given n and target relative precision, 2 α = ε 1/n + ε 1/n. G. Klein Chebfun and equispaced data 22/28

24 Wang Moin Iaccarino 10 Schemes for equispaced interpolation Results Consider an approximation f (x) = n a i (x)f i, i=1 where a i (x) are some basis functions which sum to 1 for all x. If f C N+1, it can be represented at each node x i with its degree N Taylor expansion around x, so that the approximation error becomes f N ( n (x) f (x) = f (k) (x) a i (x) (x i x) k ) k! k=1 i=1 n + f (N+1) (ξ i ) (a i (x) (x i x) N+1 ). (N + 1)! i=1 G. Klein Chebfun and equispaced data 23/28

25 Examples on [ 1, 1] Schemes for equispaced interpolation Results sin(100x) LS 10 0 funqui FE hybrid mapping spline 10 5 Wang etal chebfun x LS funqui FE hybrid mapping spline Wang etal chebfun entire function singularity at x = 1.05 G. Klein Chebfun and equispaced data 24/28

26 Examples on [ 1, 1] Schemes for equispaced interpolation Results /( x 2 ) 1/( sin(7x) 2 ) 10 0 LS funqui FE hybrid mapping 10 5 spline Wang etal chebfun LS funqui FE hybrid mapping spline Wang etal chebfun singularities at x = ± 1 10 i x = ± arcsinh(1/4) 7 i ± kπ 7 G. Klein Chebfun and equispaced data 25/28

27 Examples on [ 1, 1] Schemes for equispaced interpolation Results exp( 1/x 2 ) x 3 LS funqui FE hybrid mapping spline Wang etal chebfun LS 10 2 funqui FE hybrid mapping 10 4 spline Wang etal chebfun C function C 2 function G. Klein Chebfun and equispaced data 26/28

28 Summary We have seen Chebfun is powerful, builds on near-best polynomial approximation this is impossible to achieve with equispaced data funqui comes close with certain examples comparison with other schemes for equispaced interpolation Future work theory for funqui ongoing comparison with other schemes for equispaced interpolation G. Klein Chebfun and equispaced data 27/28

29 Thank you for your attention! G. Klein Chebfun and equispaced data 28/28