Interpolation and approximation of discontinuities via mapped - PowerPoint PPT Presentation

Interpolation and approximation of discontinuities via mapped polynomials and discontinous kernels 1 Stefano De Marchi Department of Mathematics “Tullio Levi-Civita” & Padua Neuroscience Center University of Padova (Italy) MACMAS19 - Granada, 11/09/2019 1 Joint work with W. Erb (Padova), F. Marchetti (Padova), E. Perracchione (Padova) and M. Rossini (Milano-Bicocca)

Data interpolation/approximation of discontinuous functions Approaches: optimal choice of interpolation points, rational approximation, sinc-approx, filtering, extrapolation,.... 2 of 54

Data interpolation in (medical) imaging CT, MRI, PET, SPECT, MPI, satellite images Approaches: geometric alignement, registration, reconstruction (CT, SPECT, MPI, satellite) , ... 3 of 54

Let’s start Interpolation by polynomials and rational functions of discontinuous functions is an historical approach and well-studied. Two well-known phenomena are the Runge and Gibbs effects [Runge 1901, Gibbs 1899]. 4 of 54

Let’s start Interpolation by polynomials and rational functions of discontinuous functions is an historical approach and well-studied. Two well-known phenomena are the Runge and Gibbs effects [Runge 1901, Gibbs 1899]. Interpolation by kernels, mainly Radial Basis Functions, are suitable for high-dimensional scattered data problems [Hardy 1971, MJD Powell 1977, Schaback 1993 and many more], solution of PDES [e.g. Kansa 1990], machine learning [Samuel 1950, Fasshauer&McCourt 1995], image registration [e.g. G´ omez-Garcia et al 2008], etc... 4 of 54

Let’s start Interpolation by polynomials and rational functions of discontinuous functions is an historical approach and well-studied. Two well-known phenomena are the Runge and Gibbs effects [Runge 1901, Gibbs 1899]. Interpolation by kernels, mainly Radial Basis Functions, are suitable for high-dimensional scattered data problems [Hardy 1971, MJD Powell 1977, Schaback 1993 and many more], solution of PDES [e.g. Kansa 1990], machine learning [Samuel 1950, Fasshauer&McCourt 1995], image registration [e.g. G´ omez-Garcia et al 2008], etc... Interpolation of discontinuos functions by mapped polynomials and discontinuous kernels is what we discuss today 1 S. De Marchi, F. Marchetti, E. Perracchione and D. Poggiali: Polynomial interpolation via mapped bases without resampling, J. Comp. Appl. Math. 2019 (online) 2 S. De Marchi, W. Erb, F. Marchetti, E. Perracchione and M. Rossini: Shape-driven interpolation with discontinuous kernels: error analysis, edge extraction and applications in MPI. Submitted/reviewed 4 of 54

PART I Polynomial interpolation with mapped bases: the ”fake nodes approach” 5 of 54

Outline PART I 1 Polynomial interpolation with mapped bases: the ”fake nodes approach” 2 Mapped bases The ”fake” nodes approach 3 Examples 4 Runge phenomenon Gibbs phenomenon Extension and higher dimensions 5 6 of 54

Inspiring ideas 1 In applications: samples are given. To resample (in 1d at Chebyshev points, or extract mock Chebyshev or other sets of good interpolation points that depend on applications ( such as Padua pts (2005), Approximate Fekete Pts, Discrete Leja Sequences (2010), Lissajous points (2015), (P or S)-greedy (2003/05), minimal energy (199?)...) 7 of 54

Inspiring ideas 1 In applications: samples are given. To resample (in 1d at Chebyshev points, or extract mock Chebyshev or other sets of good interpolation points that depend on applications ( such as Padua pts (2005), Approximate Fekete Pts, Discrete Leja Sequences (2010), Lissajous points (2015), (P or S)-greedy (2003/05), minimal energy (199?)...) 2 In [Adcock and Platte 2016] a similar idea was investigated for analytic functions on compact intervals by weighted least-squares of mapped polynomial basis via [Kosloff, Tal-Azer 1993] map m α ( x ) = sin( απ x / 2 ) sin( απ/ 2 ) , α ∈ ( 0 , 1 ] giving rise to the space P α n = { p ◦ m α , p ∈ P n } . 7 of 54

Some notations I = [ a , b ] ⊂ R , X n ⊂ I (interpolation points), f : I → R and F n := { f ( X n ) } ( f fnct to be reconstructed ) P n , f : interpolating polynomial, P n , f ∈ M n := span { 1 , x , . . . , x n } or using L n := span { l 0 , l 1 , . . . , l n } Lagrange basis n � P n , f ( x ) = l i ( x ) f ( x i ) , x ∈ I i = 0 with l i ( x ) = V i ( x 0 , ..., x i − 1 , x , x i , . . . , x n ) V ( x 0 , . . . , x n ) ratio of two Vandermonde determinants n � Lebesgue constant: Λ n := max | l i ( x ) | which is the stability x ∈ I i = 0 constant, norm of the projection on M n or conditioning of the interpolation problem. 8 of 54

Mapping the basis without resampling Let S : I → R be the map and P n , g : S ( I ) → R the interpolating polynomial at the mapped or “fake” nodes S ( X n ) , that is n � x i , ¯ P n , g (¯ c i ¯ x ) = x ∈ S ( I ) i = 0 for some g : S ( I ) → R ∈ C r ( I ) s.t. g | S ( X n ) = f | X n . ֒ → i.e. no resampling 9 of 54

Cont’ without resampling We are then interested to the function n � R S n , f ( x ) := P n , g ( S ( x )) = c i S i ( x ) , (1) i = 0 R S n , f is the interpolant at the original nodes X n and function values F n spanned by mapped basis S n := � S i ( · ) , i = 0 , . . . , n � . 10 of 54

Cont’ without resampling We are then interested to the function n � R S n , f ( x ) := P n , g ( S ( x )) = c i S i ( x ) , (1) i = 0 R S n , f is the interpolant at the original nodes X n and function values F n spanned by mapped basis S n := � S i ( · ) , i = 0 , . . . , n � . Equivalence mapped bases approach on I : interpolate f on the set X n via R s n , f in the function space S n . The “fake” nodes approach on S ( I ) : interpolate g on the set S ( X n ) via P n , g in the polynomial space M n . 10 of 54

Admissible S maps Definition S is admissible if the resulting interpolation process has unique solution, that is the Vandermonde-like V S := V ( S ( x 0 ) , . . . , S ( x n )) � 0 11 of 54

Admissible S maps Definition S is admissible if the resulting interpolation process has unique solution, that is the Vandermonde-like V S := V ( S ( x 0 ) , . . . , S ( x n )) � 0 Necessity: S ( x i ) � S ( x j ) , ∀ x i � x j , i.e. S is injective on X n . V S = σ ( X n , S ) · V ( x 0 , . . . , x n ) with S j − S i � σ ( S , X n ) = . (2) x j − x i 0 ≤ i < j ≤ n here S r = S ( x r ) . 11 of 54

Admissible S maps: cont’ Proposition [DeM et al. JCAM 2019] Let l i be the classical i -th Lagrange polynomial and let l S i be the S-Lagrange. Then, l S i ( x ) = γ i ( x ) l i ( x ) , x ∈ I , (3) det ( V s i ( x )) where γ i ( x ) ≔ σ ( S , X n ) det ( V i ( x )) , with σ ( S , X n ) as before. Actually, γ i ( x ) = β i ( x ) , with α i S ( x ) − S j S i − S j � � β i ( x ) := , α i := . x − x j x i − x j 0 ≤ j ≤ n 0 ≤ j ≤ n j � i j � i 12 of 54

Mapped Interpolant n � R S f i l S n , f ( x ) = i ( x ) , x ∈ I i = 0 We can define the S-Lebesgue constant n � Λ S | l S n := max i ( x ) | . (4) x ∈ S ( I ) i = 0 so that � f − R S n , f � ∞ ≤ ( 1 + Λ s n ) E s ,⋆ n ( f ) , (5) 13 of 54

Observations Obs 1 [Gross and Richards 1986] gave a remarkable formula for points in the unit disk of C for analytic map S c ( z ) = ( 1 − z ) − c , c ≥ 1. Lettting s ∈ [ − a , a ] n the determinant of the coefficient matrix can be factored as � det ( I − susu − 1 ) − ( c + n − 1 ) du det M n ( s , s ) = c n V ( s ) V ( s ) U ( n ) where U ( n ) the group of unitary complex matrices (see also [Bos,DeM,Levenberg 2014] where we discussed Fekete points for ridge functions). Obs 2 Generalized Vandermonde determinants give rise to similar factorization [DeM 2001, 2002] 14 of 54

Lebesgue bound Theorem [DeM et al. JCAM 2019] We have that � L � n Λ S Λ n , n ≤ D where � � S ( x ) − S j � � L = max max � � , � � � x − x j � � j x ∈ I � � � S i − S j � � D = min min � � � , � � x i − x j � � i j � i � 15 of 54

Lebesgue bound Theorem [DeM et al. JCAM 2019] We have that � L � n Λ S Λ n , n ≤ D where � � S ( x ) − S j � � L = max max � � , � � � x − x j � � j x ∈ I � � � S i − S j � � D = min min � � , � � � x i − x j � � i j � i � � S ( x ) − S j � Sketch. We proceed by giving an upper bound for | β i | : | β i ( x ) | ≤ � � � L j i , where L j i ≔ max � � . Thus, � � � x − x j x ∈ I � � � 0 ≤ j ≤ n j � i � Si − Sj � | β i ( x ) | ≤ L n i , We then give a lower bound for | α i | obtaining | α i | ≥ D n � � i , where D i ≔ min j � i � . We have that � � xi − xj � � � � L i � n | ℓ s i ( x ) | ≤ | ℓ i ( x ) | . D i Therefore, defining L ≔ max i L i , D ≔ min i D i and considering the sum of the Lagrange polynomials, we obtain � L � n 15 of 54 Λ S n ≤ Λ n . � D

The ”fake” nodes approach: summary Constructing R S n , f is equivalent to build the polynomial interpolant at the ”fake” nodes. If l i is the Lagrange polynomial at S ( X n ) , then at ¯ x = S ( x ) ∈ S ( I ) S ( x ) − S ( x j ) � S ( x i ) − S ( x j ) = l S l i (¯ x ) = l i ( S ( x )) = i ( x ) , x ∈ I . 0 ≤ j ≤ n j � i As a consequence, we obtain Λ S n ( I ) = Λ n ( S ( I )) . 16 of 54