Kernels, Sequences and Approximations P . Zinterhof Department of - PowerPoint PPT Presentation

Kernels, Sequences and Approximations P . Zinterhof Department of Computer Sciences University of Salzburg Smolenice, 07.09 - 09.09.16

Overview Introduction 1 Hausdorff-Distances and Dispersion 2 Quality of Interpolation 3 Computational complexity 4 P. Zinterhof : Kernels, Sequences and Approximations 2/28

Outline Introduction 1 Hausdorff-Distances and Dispersion 2 Quality of Interpolation 3 Computational complexity 4 P. Zinterhof : Kernels, Sequences and Approximations 3/28

Standard Problem Given a domain E and a function f ( x ) , find x 1 , ..., x N such that N � f ( x ) d µ ( x ) − 1 � | f ( x n ) | is small! N E n = 1 The answer depends on the domain E and the class H of functions f ∈ H . The problem of approximation and interpolation of functions is similar. Because of the generality of the questions the literature is knowingly infinite. P. Zinterhof : Kernels, Sequences and Approximations 4/28

I consider the following problem: Given a domain E and a class of functions H and points x 1 , x 2 , ... x N ∈ E . What can one say about the quality of the ”mesh” x 1 , ..., x N in view of integration, interpolation and approximation? P. Zinterhof : Kernels, Sequences and Approximations 5/28

A possible starting point let E = [ 0 , 1 ) s , s = 1 , 2 , 3 ... Let H α s = { f ( x 1 , x 2 , ... x s ) = ∞ � ˆ f ( m 1 , ... m s ) exp ( 2 π i ( m 1 x 1 + ... + m s x s )) } , m 1 ,... m s = −∞ such that for ¯ m = max ( 1 , | m | ) , m ∈ Z , holds c 1 � � | ˆ f ( m 1 , ... m s ) | ≤ ( m 1 , ... m s ) α = O , α > 1 , ( m 1 , ... m s ) α The exponent α is connected with differentiability properties of the function f . ”Korobow-classes” P. Zinterhof : Kernels, Sequences and Approximations 6/28

Discrepancy A mesh � x 1 , ..., � x N ∈ E will be good, if it is ”uniform distributed” in E . A classical measure for the quality of uniform distribution is essentially due to H. Weyl, who introduced the so called discrepancy, which we will not consider in this frame. P. Zinterhof : Kernels, Sequences and Approximations 7/28

Diaphony Some years ago I introduced the so called Diaphony F N of the mesh x N ∈ E [ 0 , 1 ) s : � x 1 , ..., � � x n = ( x 1 n , ..., x sn ) , n = 1 , ..., N : N ∞ F N := 1 1 1 � � ( m 1 , m 2 ... m s ) 2 e ( 2 π i ( m 1 ( x 1 k − x 1 l )+ ... + m s ( x sk − x sl ))) 2 N 2 k , l = 1 m = −∞ � m � = � � 0 P. Zinterhof : Kernels, Sequences and Approximations 8/28

It could be made much simpler: Let B 2 ( x ) = 1 − π 2 6 + π 2 1 2 ( 1 − 2 { x } ) 2 = � m 2 exp ( 2 π ixm ) m We get now N s F N = ( 1 1 � � B 2 ( x kj − x lj ) − 1 ) 2 N 2 k , l = 1 j = 1 P. Zinterhof : Kernels, Sequences and Approximations 9/28

curse of dimension N − > ∞ x n ) ∞ → 0 ⇔ ( � Theorem: F N − − − − n = 1 is u.d. mod 1 N | 1 � � x | ≤ CF α f ( � f ( � x ) d � Theorem: x n ) − N N E n = 1 � ln s N 1 � � � F N = O , F N = O Typically: N 1 − ǫ N Very Bad: cassical cartesian product rules: � 1 � F N = O , it is sharp! 1 N s P. Zinterhof : Kernels, Sequences and Approximations 10/28

The frequent occurrences of 2 and 1 2 call for Hilbert space approach!! Leitmotiv: Put ϕ n ( x ) = 1 nexp ( 2 π inx ) , n ∈ Z , f n | 2 < ∞ , � ˜ � | ˜ and put < ϕ n , ϕ m > = δ nm , H := { f ( x ) = f n ϕ n ( x ) } , n n ∈ R and you get a Hilbert space of continuous functions. P. Zinterhof : Kernels, Sequences and Approximations 11/28

More general Let E � = ∅ , K ( x , y ) : ExE → C strictly pos. def. matrix. Let < K ( x , y 1 ) , K ( x , y 2 ) > := K ( y 2 , y 1 ) , then span { a 1 K ( x , y 1 ) + a 2 K ( x , y 2 ) + ... + a n K ( x , y n ) } = H defines a Hilbertspace with rep. Kernel K(x,y): < f ( x ) , K ( x , y ) > = f ( y ) , f ∈ H f ⇒ f ( x ) being a cont. functional on H Bergman, Aronszajn, Moore, Zaremba, ... P. Zinterhof : Kernels, Sequences and Approximations 12/28

We define a metric on E := d ( x , y ) : || K ( t , x ) − K ( t , y ) || and assume compactness of the metric space (E, d). Let now ( x n ) ∞ n = 1 be a sequence of points in E , x n � = x m , n � = m . We apply Gram-Schmidt to K ( x , x 1 ) , K ( x , x 2 ) , ..., K ( x , x n ) , ... and get an orthonormal sequence τ n ( x ) , n ∈ N . ( τ n ( x )) ∞ n = 1 has the important property τ m ( x n ) = 0 for m > n . n = 1 ˜ f n τ n ( x ) ∈ H . We compute ˜ Let f ( x ) = � ∞ f n , n = 1 , 2 , ... recursively: ˜ f 1 = f ( x 1 ) /τ 1 ( x 1 ) , ˜ f 2 = ( f ( x 2 ) − ˜ f 1 τ 1 ( x 2 )) /τ 2 ( x 2 ) , ... ˜ f n = ( f ( x n ) − ˜ ˜ f 1 τ 1 ( x n ) − ... − f n − 1 τ n − 1 ( x n )) /τ n ( x n ) The τ n ( x n ) � = 0 , n ∈ N , by Cholesky’s theorem. e.g. P. Zinterhof : Kernels, Sequences and Approximations 13/28

Main question In which cases is the ONS ( τ n ( x )) ∞ n = 1 an ONB: ∞ � ˜ f n τ n ( x ) for all f ( x ) ∈ H ?? f ( x ) = n = 1 Theorem: ( τ n ( x )) n is an ONB ⇔ ( K ( t , x n )) n is total in H ⇔ ( f ( x n ) = 0 = > f = 0 ∀ f ∈ H ) Let now K N ( x , y ) := � N n = 1 τ n ( x ) τ n ( y ) , K ⊥ N ( x , y ) = K ( x , y ) − K N ( x , y ) . K N ( x , y ) reproduces H N = span { K ( x , x 1 ) , ..., K ( x , x N ) } , K ⊥ N ( x , y ) � H N = H ⊥ reproduces H - N . P. Zinterhof : Kernels, Sequences and Approximations 14/28

Def: The N-th totality of x 1 , ... x N , ... is y ∈ E || K ⊥ T N = max N ( x , y ) || = max || K N ( x , y ) || − || K N ( x , y ) || = y 1 1 2 − K N ( y , y ) 2 ) = max max y ( K ( y , y ) T N ( y ) y Theorem: ( x n ) ∞ N →∞ T N = 0. n = 1 is total in E iff lim The proof uses Arzela-Ascoli’s theorem. P. Zinterhof : Kernels, Sequences and Approximations 15/28

Outline Introduction 1 Hausdorff-Distances and Dispersion 2 Quality of Interpolation 3 Computational complexity 4 P. Zinterhof : Kernels, Sequences and Approximations 16/28

Hausdorff-Distances and Dispersion The Hausdorff-Distance between { x 1 , ... x N } ⊆ E and E itself is defiend by δ ( x 1 , ... x N ) = max n = 1 ,... N d ( x , x n ) min x Mostly δ ( x 1 , ... x N ) is called the dispersion of x 1 , ..., x N in E . Well known: The sequence ( x n ) ∞ n = 1 is dense in E iff lim N →∞ δ N = 0 The following theorem holds: 0 ≤ T N ≤ δ N P. Zinterhof : Kernels, Sequences and Approximations 17/28

Corollary If ( x n ) ∞ n = 1 is dense in E, it is total as well. I have a Hilbert space and total sequences with arbitrary bad density properties. Furthermore I have a Hilbert space (E,K,H), where T N → 0 ⇔ δ N → 0 . It means, a sequence ( x n ) ∞ n = 1 is total iff it is dense in that special space E . P. Zinterhof : Kernels, Sequences and Approximations 18/28

Outline Introduction 1 Hausdorff-Distances and Dispersion 2 Quality of Interpolation 3 Computational complexity 4 P. Zinterhof : Kernels, Sequences and Approximations 19/28

Problem We want to find a function f N ( x ) , such that f N ( x n ) = f ( x n ) , n = 1 , ..., N , and f N ( x ) ∈ span { K ( x , x n ) , n = 1 , ..., N } = H N . Let l 1 ( x ) , l 2 ( x ) , ..., l H ( x ) ∈ H N , such that l m ( x n ) = δ nm . The l 1 ( x ) , ..., l H ( x ) are the dual base to K ( x , x 1 ) , ..., K ( x , x N ) . So, f N ( x ) = � N n = 1 f ( x n ) l n ( x ) ∈ H n fulfills the requirements. On the k = 1 ˜ other hand f ( x n ) = � N f k τ k ( x n ) , because τ k ( x n ) = 0 for k > N . So we get f N ( x ) = � N n = 1 ˜ f n τ n ( x ) , because f N ( x ) and f ( x ) coincide at x 1 , ..., x N . P. Zinterhof : Kernels, Sequences and Approximations 20/28

It follows: ∞ ∞ ∞ f n | 2 � 1 | τ n ( x ) | 2 � 1 � 2 � � ˜ � | ˜ � 2 | f ( x ) − f N ( x ) | = | f n τ n ( x ) | ≤ n = N + 1 n = N + 1 n = N + 1 ∞ 1 � | ˜ f n | 2 ) T N = || f ⊥ x ∈ E = | f ( x ) − f N ( x ) | ≤ ( N ( x ) || · T N ≤ || f || T N max 2 n = N + 1 ∞ 1 | ˜ � f n | 2 ) and || f − f N || = ( 2 n = N + 1 means that the ”Lagrange”-Interpolation function f N ( x ) is the best approximation in Hilbert space sense! P. Zinterhof : Kernels, Sequences and Approximations 21/28

Outline Introduction 1 Hausdorff-Distances and Dispersion 2 Quality of Interpolation 3 Computational complexity 4 P. Zinterhof : Kernels, Sequences and Approximations 22/28

Computational complexity The computation of ˜ f n , τ n ( x ) , n = 1 , ..., N , K N ( x , y ) needs O ( N 2 ) operations. Important special case: E = G, compact abelian group with Haar measure λ and K ( x , y ) = k ( x − y ) , k ( x ) = f ( x ) ⋆ f ( − x ) , f ( x ) ∈ L 2 ( G ) , and x n = x 1 · n , n = 0 , ..., N − 1 is a cyclic subgroup of G. Then one can apply FFT with N log(N) operations. P. Zinterhof : Kernels, Sequences and Approximations 23/28

I have similar results for numercical Integration of the type I g ( f ) = < f , g > using f ( x 1 ) , ..., f ( x H ) . ”But that’s another story” would Rudyard Kipling say.... P. Zinterhof : Kernels, Sequences and Approximations 24/28