The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments How well do the Hermite-Pad´ e approximants reduce the Gibbs phenomenon? Ana C. Matos joint work with B. Beckermann, F. Wielonsky Laboratoire Paul Painlev´ e Universit´ e de Lille I Valery Kaliaguine University of Nishninovgorod Luminy, octobre 2009 Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Particular sequences of HP approximants Numerical experiments OUTLINE OF THE TALK the problem: accelerating partial sums of Fourier series of functions with jumps (spectral methods in PDE) definition of the Gibbs phenomenon and some classical approaches new approach: use of Hermite-Pad´ e forms definition of Hermite-Pad´ e approximants and motivation rate of convergence for a model problem comparison with Pad´ e approximants numerical experiments Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments The Gibbs phenomenon Problem Given a small number of coefficients of a real–valued Fourier series construct point values of f ( t ) = Re( � ∞ j =0 c j e ijt ) If we consider the partial sums n n � � c j e ijt ) = a 0 S n ( f )( t ) = Re( 2 + [ a j cos( jt ) + b j sin( jt )] , j =0 j =1 how good is the approximation? Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Convergence results f smooth and periodic: exponential accuracy − π ≤ x ≤ π | f ( x ) − f N ( x ) | ≤ e − α N , max α > 0 f ∈ C m − 1 ([ − π, π ]) periodic (ˆ f k = O ( | k | − ( m +1) ) , k → ±∞ − π ≤ x ≤ π | f ( x ) − f N ( x ) | = O ( N − m ) , N → ∞ max f discontinuous or non periodic: nonuniform convergence of the Fourier series � 1 � | f ( x 0 ) − f N ( x 0 ) | ∼ O away from discontinuity N [ − π,π ] | f ( x ) − f N ( x ) | doesn’t tend to 0 max ⇒ GIBBS PHENOMENON: oscillations and bad convergence Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Example:the saw-tooth function Partial Fourier sums 3 2 1 0 –3 –2 –1 1 2 3 t –1 –2 –3 La fonction 10 20 30 How to overcome this phenomenon? Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Some different approaches 1 Linear summation methods: Ces` aro means, De la Vall´ ee-Poussin means; 2 Gottlieb approach: to obtain exponential accuracy in the maximum norm in any interval of analiticity of a discontinuous piecewise analytic function - uses Gegenbauer polynomials; 3 Eckhoff approach: split a singular function into two parts, one presenting some regularity and the other corresponding to the singularities, modellized by some prototype functions (more details later) 4 Fourier-Pad´ e approach Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Definition of the Fourier-Pad´ e approximants Consider the following procedure (C. Brezinski, P. Wynn): construct S n ( f )( t ) + i � S n ( f )( t ) = G n ( f )( e it ) , with S n ( f )( t ) = � n � j =1 [ a j sin( jt ) − b j cos( jt )] , G n ( f ) the n th Taylor sum of the (formal) series � ∞ c 0 ( f ) = a 0 c j ( f ) z j , c j ( f ) = a j − ib j . G ( f )( z ) = 2 , j =0 compute Pad´ e approximants of this power series [ n + k / k ] f ( z ) Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Definition of the Fourier-Pad´ e approximants use the real part for approaching f ( t )= Re ( G ( f )( e it )). � � ǫ ( n ) 2 k ( t ) = p [ n + k / k ] G ( f ) ( e it ) q ( t ) = Re where p and q are trigonometric polynomials of degrees n + k and k respectively. we showed that for f ∈ L 2 and Q ( z ) � = 0 (denominator polynomial) for | z | ≤ 1 then f − p q is orthogonal to sin( jt ), cos( jt ) for j = 0 , 1 , ...., m + k . ⇔ non linear Fourier-Pad´ e approximants numerical examples for functions with jumps show very good acceleration properties and strong reduction of Gibbs oscillations Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Numerical example n=0,k=8 t .1e2 –3 –2 –1 1 2 3 1. .1 .1e–1 .1e–2 .1e–3 1e–05 1e–06 1e–07 1e–08 1e–09 1e–10 1e–11 1e–12 1e–13 1e–14 1e–15 1e–16 |function - approximant Re([n+k|k](exp(i*t)))| |function - partial sum of order n+2*k| |function - Cesaro/Fejer mean of partial sum| |function - de la Vallee Poussin mean of partial sum| s ( t ) = π + t for t ∈ ( − π, 0] , s ( t ) = − π + t for t ∈ (0 , π ] , the 2 π periodic saw tooth function, having one jump of absolute value 2 π at t = 0 in [ − π, π ), approximants computed with 18 Fourier coefficients Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Convergence results (B. Beckermann,AM, F.Wielonsky, 2008) we consider a class of test functions � � � � α + 1 , 1 � G ( α,β ) ( z ) = 2 F 1 � z , , α, β > − 1 α + β + 2 some examples are f ( t ) = sign(cos( t )) f ( t ) = | sin( t / 2) | ∈ C 0 \ C 1 f ( t ) = (1 − cos( s )) s ( t ) Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Convergence results Convergence for columns � � �� � � [ n | k ]( e it ) � = O ( n − 2 k ) n →∞ max � f ( t ) − Re for fixed k t ∈ I even after perturbation of f with C m function, m sufficiently large. Convergence of ray sequences : for some γ ≥ 1 � � �� 1 / k � � [ n | k ]( e it ) k →∞ , n = γ k max lim � f ( t ) − Re < 1 . � t ∈ I Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Hermite-Pad´ e approach Definition Particular sequences of HP approximants Some different approaches Numerical experiments Why does it work so well? � � ∞ � sin( jt ) G ( s )( e it ) s ( t ) = − 2 = Re , with j j =1 � 1 ∞ � z j dx G ( s )( z ) = 2 i j = − 2 i log(1 − z ) = 2 iz 1 − xz . 0 j =1 � 1 G ( s 1 )( z ) = 2 π − z 1 − x dx s 1 ( t ) = | sin( t 2 ) | , √ x 1 − xz , π 0 � 1 d σ ( x ) Stieltjes functions g σ ( z ) = 1 − zx are very well approximated by 0 Pad´ e approximants in compact subsets of C \ [1 , + ∞ ), in particular all poles are on (1 , + ∞ ) Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Definition of Hermite-Pad´ e approximation Hermite-Pad´ e approach Model problems: Nikishin systems Particular sequences of HP approximants Results on potential theory Numerical experiments Rate of convergence Why Hermite-Pad´ e forms ? Hypothesis: We know location of singularity but not amplitude of jumps! If f ∈ C n 1 +1 ([ − π, 0) ∪ (0 , π ]) periodic has left- and right-hand side derivatives of order 0 , 1 , ..., n 1 at t = 0: n 1 � d j sin j ( t ) s ( t ) ∈ C n 1 +1 ([ − π, π ]) ∃ d j ∈ R : e ( t ) = f ( t ) − j =0 In terms of z = e it : n 1 � d j sin j ( t ) = Re ( i f ( t ) = Re ( F ( z )) , 2 p 1 ( z )) , j =0 � 2 � s ( t ) = Re i log(1 − z ) Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
The Gibbs phenomenon Definition of Hermite-Pad´ e approximation Hermite-Pad´ e approach Model problems: Nikishin systems Particular sequences of HP approximants Results on potential theory Numerical experiments Rate of convergence Why Hermite-Pad´ e forms ? thus reasonable approximation: � � f ( t ) ≈ Re − p 0 ( z ) − p 1 ( z ) log(1 − z ) with p 0 ∈ P n 0 , p 1 ∈ P n 1 , p 0 ( z ) + p 1 ( z ) log(1 − z ) + F ( z ) = O ( z n 0 + n 1 +2 ) z → 0 . particular case of Hermite-Pad´ e approximants Ana C. Matos HP approximants and the reduction of the Gibbs phenomenon
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