Approximation numbers of Sobolev embeddings - Sharp constants and tractability

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1 Approximation numbers of Sobolev embeddings - Sharp constants and tractability Thomas Kühn Universität Leipzig, Germany Workshop Uniform Distribution Theory and Applications Oberwolfach, 29 September - 5 October 2013 joint work with Winfried Sickel (Jena) and Tino Ullrich (Bonn) Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

2 Approximation numbers Approximation numbers of bounded linear operators T : X Y between two Banach spaces a n (T : X Y ) := inf{ T A : rank A < n} X, Y Hilbert spaces, T : X Y compact = a n (T ) = s n (T ) = n-th singular number of T Related to n-widths, s-numbers, entropy numbers,... Main interest - Asymptotic behaviour of a n (T ) as n - Two-sided estimates of a n (T ) for small n Typical examples For compact embeddings X Y of spaces of d-variate functions, information on a n (I : X Y ) is important in Numerical Analysis. Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

3 Sobolev spaces Sobolev spaces on the d-dimensional torus T d, of integer smoothness m N H m (T d ) consists of all f L 2 (T d ) such that D α f L 2 (T d ) for all multi-indices α N d 0 with α m. Natural norm ( f H m (T d ) := D α f L 2 (T d ) 2) 1/2 Modified natural norm f H m (T d ) := α m ( f L 2 (T d ) 2 + d m f j=1 x m j L 2 (T d ) 2 ) 1/2 Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

4 Fourier coefficients equivalent norms Fourier coefficients of f L 2 (T d ) c k (f ) := (2π) d/2 f (x)e ikx dx T d, k Z d Parseval s identity and c k (D α f ) = (ik) α c k (f ) = norms in H m (T d ) can be expressed in terms of c k (f ) For the natural norm one has equivalence f H m (T d ) ( d 1 + k j 2) m ck (f ) 2 k Z d j=1 with equivalence constants independent on d. For the modified natural norm one has even equality f H m (T d ) = ( d 1 + k j 2m) c k (f ) 2 k Z d Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18 j=1 1/2 1/2

5 Sobolev spaces of fractional smoothness s > 0 The norms are weighted l 2 -sums of Fourier coefficients. ( ) 1/2 f H s (T d ) := k Z d w(k) 2 c k (f ) 2 The weights for the (different equivalent) norms are w s (k) := w s (k) := w # s (k) := ( 1 + ( 1 + ( 1 + d k j 2) s/2 j=1 d k j 2 s) 1/2 j=1 natural norm modified natural norm d s k j ) auxiliary norm j=1 Note: w s (k) = w 1 (k) s and w # s (k) = w # 1 (k)s Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

6 Classical results for Sobolev spaces A.N. Kolmogorov, Über die beste Annäherung von Funktionen einer Funktionenklasse, Ann. Math. 37 (1936), a n (I : Ḣ m (T) L 2 (T)) = n m. J.W. Jerome, J. Math. Anal. App. 20 (1967), (non-periodic),..., books by V.N. Temlyakov 1993, by Edmunds and Triebel 1996 c s (d)n s/d a n (I d : H s (T d ) L 2 (T d )) C s (d)n s/d Problems considered in this talk Asymptotic behaviour of the constants c s (d), C s (d)? Does the limit lim n ns/d a n (I d : H s (T d ) L 2 (T d )) exist? Explicit two-sided estimates for a n (large n / small n) Consequences for tractability? Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

7 Tools diagonal operators Diagonal operators D : l 2 l 2, D(ξ n ) = (σ n ξ n ), where σ 1 σ 2... σ n...0. Then a n (D : l 2 l 2 ) = σ n. Now: more general index set Z d D(ξ k ) = (ω k ξ k ) with arbitrary ω k 0, k Z d. (σ n ) n N non-increasing rearrangement of ω = (ω k ) k Z d, then a n (D : l 2 (Z d ) l 2 (Z d )) = σ n ( ) We apply this for ω = 1/w s # (k) and its rearrangement (σ n) n N k Z ( d s Recall that w s # (k) = 1 + d k j ) are the weights used in the auxiliary norm on H s (T d ). j=1 Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

8 Tools discretization Commutative diagram I d H s,# (T d ) L 2 (T d ) A B l 2 (Z d D ) l 2 (Z d ) Af := (w s # (k) c k (f )) k Z d and Bξ = (2π) d/2 ξ k e ikx, k Z d D(ξ k ) = (ξ k /w s # (k)) = a n (I d ) = a n (D) = σ n Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

9 Tools combinatorics Lemma Let C(m, d) := {k Z d : d k j m}. Let s > 0 and d N. Then, for all m N and C(m 1, d) < n C(m, d) j=1 a n (I d : H s,# (T d ) L 2 (T d )) = (m + 1) s. Not very useful in this form. Recall: w s (k) = w 1 (k) s and w # s (k) = w # 1 (k)s = a n (I d : H s (T d ) L 2 (T d )) = a n (I d : H 1 (T d ) L 2 (T d )) s = enough to treat the case s = 1. This reduction is also possible for the #-norm, but not for the -norm! Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

10 Asymptotic behaviour of the constants Theorem (K, Sickel, T. Ullrich 2013) Let s > 0 and d N. Then one has (i) lim n ns/d a n (I d : H s,# (T d ) L 2 (T d )) = ( 2 ) s ( 2e ) s d d! d (ii) (iii) ( 1 ) s lim n ns/d a n (I d : H s (T d ) L 2 (T d )) = vol(b2 d ) s/d d lim n ns/d a n (I d : H s, (T d ) L 2 (T d )) = vol(b2s) d s/d 1 d B d p = {x R d : d j=1 x j p 1} (i) auxiliary norm, (ii) natural norm, (iii) modified natural norm Proof: Combinatorics in (i), volume estimates in (ii) and (iii) Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

11 Estimates for large n Theorem (KSU 2013) (i) If n 6 d /3 then 1 d s n s/d a n (I d : H s,# (T d ) L 2 (T d )) (ii) If n 11 d e d/(2s) then ( 4e ) sn s/d d 1 2es n s/d a n (I d : H s, (T d ) L 2 (T d )) 4 s e(d + 2s) d n s/d (iii) If n 11 d e d/2 ( 1 ) s/2n s/d a n (I d : H s (T d ) L 2 (T d )) 4 s( 2e ) s/2n s/d e(d + 2) d Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

12 Estimates for small n Theorem (KSU 2013) Let s > 0, d N and 2 n 2 d. Then one has ( 1 ) s ( an (I d : H s,# (T d ) L 2 (T d log2 (2d + 1) ) s )). 2 + log 2 n log 2 n n s/d is replaced by (log 2 n) s, which is better in the range n 2 d. There remains a gap of order (log d) s. Upper bound is nontrivial for n 2d + 2. Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

13 The approximation problem in IBC Approximation problem Let s > 0. For all d N, consider the embeddings I d : H s (T d ) L 2 (T d ) where the Sobolev spaces are equipped with one of our three norms. A linear algorithm that uses n arbitrary informations is of the form A n f = n L i (f )g i i=1 where the L i are linear functionals on H s (T d ) and g i L 2 (T d ). These are just the operators A n : H s (T d ) L 2 (T d ) of rank n. Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

14 Information complexity The optimal worst case error of the approximation problem is then e(n, I d ) = a n+1 (I d : H s (T d ) L 2 (T d )). In our case, the normalized error criterion e(0, I d ) = I d = 1 holds for all s > 0 and d N, and all three equivalent norms. Information complexity n(ε, d) = min{n N : a n+1 (I d ) ε} Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

15 Tractability notions There are many different notions of tractability, see the monographs by Erich Novak and Henryk Woźniakowski ( ) For our results we need only a few of them, not the full spectrum. Quasi-polynomial tractability There is C > 0 such that Weak tractability Intractability log n(ε, d) C(1 + ln(ε 1 ))(1 + ln d) log n(ε, d) lim ε 1 +d ε 1 + d = 0 = no weak tractability Curse of dimension There are c, γ, ε 0 > 0 such that for all ε (0, ε 0 ] and infinitely many d N n(ε, d) c(1 + γ) d. Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

16 Tractability results Let s > 0. We consider the following approximation problems: I d : H s (T d ) L 2 (T d ) I d : H s, (T d ) L 2 (T d ) I d : H s,# (T d ) L 2 (T d ) (natural norm) (modified natural norm) (auxiliary norm) Theorem (KSU 2013 bad news) For every s > 0, none of the above approximation problems is quasi-polynomially tractable. Theorem (KSU 2013 good news) For every s > 0, none of the above approximation problems suffers from the curse of dimensionality. Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

17 Weak tractability vs intractability Theorem (KSU 2013) The approximation problem I d : H s,# (T d ) L 2 (T d ) is weakly tractable, if s > 1 intractable, if 0 < s 1. (auxiliary norm) Theorem (KSU 2013) The approximation problem I d : H s (T d ) L 2 (T d ) is weakly tractable, if s > 2 intractable, if 0 < s 1. (natural norm) The case 1 < s 2 is open. Conjecture: weakly tractable Theorem (KSU 2013) The approximation problem I d : H s, (T d ) L 2 (T d ) is intractable for all s > 0. (modified natural norm) Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18

18 Final comments Let H m (T d ) be the classical Sobolev spaces of integer smoothness m N with the natural norm (using all derivatives up to order m). For the approximation problem I d : H m (T d ) L 2 (T d ) one has no curse, no quasi-polynomial tractability intractability for m = 1 weak tractability for m 3 We have similar results on approximation numbers for embeddings of periodic Sobolev spaces with dominating mixed smoothness. Many open problems Change the source spaces, take e.g. C k or H s p with p 2. Change the target space, take e.g. L Study the non-periodic case Reference: T. Kühn, W. Sickel and T. Ullrich, Approximation numbers of Sobolev embeddings sharp constants and tractability. J. Complexity (2013, online) Thomas Kühn (Leipzig) Sobolev embeddings Oberwolfach / 18