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Fourier Extension and Prolate Spheroidal R. Matthysen Wave Theory: - PowerPoint PPT Presentation

Fourier Extension and Prolate Spheroidal Wave Theory: Fast algorithms Fourier Extension and Prolate Spheroidal R. Matthysen Wave Theory: Fast algorithms Introduction PSWFS & FEs Cont. - Cont. Cont. - Discr. Discr. - Discr. Roel


  1. Fourier Extension and Prolate Spheroidal Wave Theory: Fast algorithms Fourier Extension and Prolate Spheroidal R. Matthysen Wave Theory: Fast algorithms Introduction PSWFS & FEs Cont. - Cont. Cont. - Discr. Discr. - Discr. Roel Matthysen Fast J. work with Daan Huybrechs Algorithms Frequency Commutation results University of Leuven Extensions & Open Problems ICERM Research Cluster on Sparse and Redundant Representations

  2. Fourier Extension and Introduction Prolate Spheroidal Wave Theory: Fourier Extension Fast algorithms Function f given on [ − 1 , 1], construct Fourier series on larger R. Matthysen domain [ − T , T ]. Introduction N PSWFS & FEs a n e i π n � T x || L 2 a : = arg a ∈ R 2 N +1 || f − min Cont. - Cont. [ − 1 , 1] Cont. - Discr. n = − N Discr. - Discr. Fast Algorithms Frequency Commutation results Extensions & Open Problems 0 -T -1 1 T

  3. Fourier Extension and Fourier Extension Prolate Spheroidal Wave Theory: • Formulation as a Least Squares problem: Fast algorithms Aa = b R. Matthysen . . ...     . . Introduction . . � 1 � 1 PSWFS & FEs     − 1 φ k ( x ) φ l ( x ) dx  a = − 1 f ( x ) φ k ( x ) dx Cont. - Cont.     Cont. - Discr.  .  .  ... . . Discr. - Discr. . . Fast Algorithms φ k ( x ) = e i π k T x , k = − N , . . . , N Frequency Commutation results • A is a subblock of the prolate matrix Extensions & Open 10 0 Problems λ i 10 − 14 • The exact solution of the LS problem is unbounded with N , but small norm solutions (TSVD) exist.

  4. Fourier Extension and Introduction Prolate Spheroidal Wave Theory: Fast Setting : Equispaced grid algorithms Samples f ( x l ) given, where x l = l / M , l = − M , . . . , M . R. Matthysen � 2 M � N Introduction a n e i π n � � T x l a : = arg min f ( x l ) − . PSWFS & FEs a ∈ R 2 N +1 Cont. - Cont. l = − M n = − N Cont. - Discr. Discr. - Discr. Fast Algorithms Linear Algebra problem Frequency Commutation results • Solve, in a least squares sense, Extensions & Open Problems A kl = e i π k T x l , Aa ≈ b , b l = f ( x l ) • Normal equations A ′ Aa = A ′ b worsen ill-conditioning • Convergence to machine precision ǫ proven for TSVD • Fast algorithms needed.

  5. Fourier Extension and Table of Contents Prolate Spheroidal Wave Theory: Fast algorithms 1 Introduction R. Matthysen Introduction 2 Prolate Spheroidal Wave Theory and FEs PSWFS & FEs Continuous - Continuous Cont. - Cont. Cont. - Discr. Continuous - Discrete Discr. - Discr. Fast Discrete - Discrete Algorithms Frequency Commutation 3 Fast Algorithms results Extensions & Exploiting frequency properties Open Problems Exploiting commutation with differential operator Numerical Results 4 Extensions & Open Problems

  6. Fourier Extension and PSWFs (Slepian, Landau, Pollak) Prolate Spheroidal Wave Theory: Given Fourier transform of f ( x ) in L 2 [ −∞ , ∞ ] , Fast algorithms R. Matthysen � ∞ f ( x ) e − 2 π ix ξ ds , F ( ξ ) = Introduction −∞ PSWFS & FEs Cont. - Cont. Define the time- and bandlimiting operators as Cont. - Discr. Discr. - Discr. � Ω � Fast f ( x ) | x | ≤ T Algorithms F ( ξ ) e i 2 πξ x d ξ, D f ( x ) = B f ( x ) = Frequency 0 | x | > T Commutation − Ω results Extensions & Then the PSWFs are the eigenfunctions of the operator BD . Open Problems λ i ψ i ( x ) = BD ψ i ( x ) � T sin(2 π Ω( x − s )) λ i ψ i ( x ) = ψ i ( s ) ds , π ( x − s ) − T 1 > λ 0 > λ 1 > · · · > 0.

  7. Fourier Extension and Concentration problem Prolate Spheroidal Wave Theory: Fast algorithms λ i ψ i ( x ) = BD ψ i ( x ) R. Matthysen PSWFs answer the question: “What is the maximum concentration of a bandlimited function inside a given Introduction PSWFS & FEs interval?” � T Cont. - Cont. − T ψ i ( x ) ψ i ( x ) dx Cont. - Discr. −∞ ψ i ( x ) ψ i ( x ) dx = λ i Discr. - Discr. � ∞ Fast Algorithms Exponential decay sets in after ∼ 2Ω T eigenvalues. Frequency Commutation results 10 0 Extensions & Open Problems 10 − 8 λ i 10 − 16 2Ω T i

  8. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 0 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen 0 Introduction PSWFS & FEs Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  9. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 1 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs 0 Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  10. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 2 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs 0 Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  11. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 3 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs 0 Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  12. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 4 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs Cont. - Cont. Cont. - Discr. Discr. - Discr. 0 Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  13. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 5 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs 0 Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  14. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 6 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs Cont. - Cont. Cont. - Discr. 0 Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  15. Fourier Extension and Properties Prolate Spheroidal Wave Theory: PSWF ψ 7 ( x ), 2Ω T ≈ 4 Fast algorithms R. Matthysen Introduction PSWFS & FEs 0 Cont. - Cont. Cont. - Discr. Discr. - Discr. Fast Algorithms Frequency Commutation -T T results Extensions & Open Problems Properties • The ψ i are orthogonal on both [ − T , T ] and [ −∞ , ∞ ] • ψ i has i zeros inside [ − T , T ] • ψ i is even and odd with i

  16. Fourier Extension and Other interesting properties Prolate Spheroidal Wave Theory: Fast Spectrum localisation algorithms The ψ i are eigenfunctions of the finite Fourier transform, R. Matthysen � Ω Introduction e i 2 π s ξ ψ i ( s ) ds = α i ψ i ( ξ ) . PSWFS & FEs Cont. - Cont. − Ω Cont. - Discr. Discr. - Discr. Fast Algorithms Commutation with 2nd order differential operator Frequency Commutation the differential operator results Extensions & � d 2 Open 1 − x 2 � dx 2 − 2 x d dx − (2 π Ω T ) 2 x 2 Problems P x = T 2 commutes with DB , i. e. for any bandlimited f P DB f = DB Pf , and P x ψ i ( x ) = χ i ψ i ( x ) .

  17. Fourier Extension and Bandlimited approximation Prolate Spheroidal Wave Theory: Fast • Problem : “find bandlimited ˜ f so that ˜ f agrees with f on algorithms the interval [ − T , T ] ” R. Matthysen • Solution : expand in PSWFs Introduction PSWFS & FEs ˜ � f = � f , ψ i � ψ i Cont. - Cont. Cont. - Discr. Discr. - Discr. k Fast Algorithms Frequency Commutation results Extensions & Open Problems 0 -T T

  18. Fourier Extension and Bandlimited approximation Prolate Spheroidal Wave Theory: Fast • Problem : “find bandlimited ˜ f so that ˜ f agrees with f on algorithms the interval [ − T , T ] ” R. Matthysen • Solution : expand in PSWFs Introduction PSWFS & FEs ˜ � f = � f , ψ i � ψ i Cont. - Cont. Cont. - Discr. Discr. - Discr. k Fast Algorithms Frequency Commutation results Extensions & Open Problems 0 -T T

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