On lower bounds for integration of multivariate permutation-invariant functions

Transcription

1 On lower bounds for of multivariate permutation-invariant functions Markus Weimar Philipps-University Marburg Oberwolfach October 2013 Research supported by Deutsche Forschungsgemeinschaft DFG (DA 360/19-1)

2 Outline Permutation-invariant subspaces

3 Integration problem for We study multivariate Int d (f ) = f (x) dx [0,1] d for periodic, complex-valued functions in the Korobov class E d,α = { f L 1 ([0, 1] d ) f := f } E d,α < where d N, α > 1, and ( ) α f E d,α = sup f (k) k 1... k d. k Z d For k = (k 1,..., k d ) Z d we set k m := max {1, k m } and f (k) := f, e 2πik L 2.

4 Without loss of generality, we consider linear cubature rules A N,d (f ) := N n=1 ( w n f t (n)), N N 0, with nodes t (n) [0, 1] d and weights w n C, n = 1..., N. As usual the Nth minimal worst case error of Int = (Int d ) d N is defined by e(n, d; Int d, E d,α ) := inf A N,d f sup E d,α 1 Int d (f ) A N,d (f ).

5 The problem is well-scaled: e(0, d; Int d, E d,α ) = 1 for all d N. In 1997 Sloan and Woźniakowski [SW97] showed that for every d N e(n, d; Int d, E d,α ) = e(0, d; Int d, E d,α ) provided that N < 2 d. Hence, we have the curse of dimensionality since n(ε, d) := n(ε, d; Int d, E d,α ) := min {N N 0 e(n, d; Int d, E d,α ) ε} 2 d for all ε (0, 1), d N.

6 Notions of tractability Polynomial tractability: C, p > 0 and q 0 s.t. n(ε, d) C d q ε p, d N, ε (0, 1) Strong polynomial tractability: polynomial tractability with q = 0

7 Notions of tractability Polynomial tractability: C, p > 0 and q 0 s.t. n(ε, d) C d q ε p, d N, ε (0, 1) Strong polynomial tractability: polynomial tractability with q = 0 (s, t)-weak tractability: s, t (0, 1] s.t. log n(ε, d) lim ε 1 +d ε s + d t = 0 (Classical) weak tractability: (1, 1)-weak tractability Uniform weak tractability: (s, t)-weak tractability for all s, t (0, 1] (sufficient: s = t (0, 1])

8 Permutation-invariant subspaces For each d N take a subset of coordinates I d {1,..., d}. A function f E d,α is called I d -permutation-invariant, if for all x [0, 1] d and every permutation σ S d of I d 1 x 2 σ(x) f (x) = f (σ(x)). x x 1

9 Let S Id (E d,α ) denote the subspace of all I d -permuationinvariant functions in E d,α, d N, and consider Int = (Int d ) d N restricted to these sets. New kind of additional structure (in contrast to weights) Motivated by applications (wave functions and Pauli principle in quantum mechanics) Handsome: f (k) = f (σ(k)) for all f S Id (E d,α ), every k Z d, and each σ S d. Successfully used for (tensor product) approximation problems

10 Does (full) permuation-invariance make trivial? 1 x 2 σ(x) x x 1

11 Does (full) permuation-invariance make trivial? 1 x 2 σ(x) x x 1 NO!

12 Theorem (W., 2013) Let N := N (d, I d ) := (#I d + 1) 2 d #I d, d N. Then, for every N < N, and e(n, d; Int d, S Id (E d,α )) = 1 ( e(n, d; Int d, S Id (E d,α )) 1 + ζ(α) ) d 2 α 1 1 for all d N and α > 1. Consequently, for all d N. lim α e(n, d; Int d, S Id (E d,α )) = 0

13 N is sharp (at least for large smoothness α) Reformulation: n(ε, d; Int d, S Id (E d,α )) N = (#I d + 1) 2 d #I d for all d N and every ε (0, 1). If I d = then N = 2 d (generalization of [SW97]) If I d = {1,..., d} then still N d + 1

14 Corollary (W., 2013) Let α > 1 and set b d := d #I d for all d N. If Int is polynomially tractable with the constants C, p, q then q 1 and (b d ) d N O(ln d). = NO strong polynomial tractability! If Int is uniformly weakly tractable then (b d ) d N o(d t ) for all t (0, 1]. If Int is (s, t)-weakly tractable for some s, t (0, 1] then (b d ) d N o(d t ). In particular, weak tractability implies (b d ) d N o(d). If (b d ) d N / o(d) then we have the curse of dimensionality. In turn, already the absence of the curse implies (b d ) d N o(d).

15 of the proof In [SW97], i.e. without permutation-invariance (I d = ): For k Z d set e k = exp(2πi k ). Choose a bijection Ψ: {0,..., 2 d 1} {0, 1} d. For N < N = 2 d solve the system N a n e ψ(n) (t (j) ) = 0, j = 1,..., N, n=0 such that a n = 1 = max n=0,1,...,n a n for some n. Fooling function N f N (x) := e ψ(n )(x) a n e ψ(n) (x). n=0 satisfies f N 1, A N,d (f N ) = 0, and Int d (f N ) = 1. Upper bound via 2 d -point product-rectangle rule.

16 Proof in the permutation-invariant case Symmetrize the old proof using (S Id e k )(x) := 1 #S Id instead of e k and replace σ S Id e k (σ(x)), x [0, 1] d, {0, 1} d by {k {0, 1} d k 1... k #Id }. A lot of nasty calculations caused by the permutations.

17 Further results for more than only one symmetry set, weighted spaces (product weights). Work in progress: Upper bounds for related spaces endowed with l 2 -norm. Conclusion: Integration of permutation-invariant functions is not trivial (no strong polynomial tractability)!

18 Further results for more than only one symmetry set, weighted spaces (product weights). Work in progress: Upper bounds for related spaces endowed with l 2 -norm. Conclusion: Integration of permutation-invariant functions is not trivial (no strong polynomial tractability)! Open problem Do we have polynomial tractability for of (fully) permutation-invariant functions?

19 Novak, Woźniakowski: Tractability of Multivariate s. Vol. I III. European Mathematical Society (EMS), Zürich, Sloan, Woźniakowski: An intractability result for multiple. Math. Comp. 66, (1997). W.: On lower bounds for of multivariate permutation-invariant functions. To appear in J. Complexity (2013?!). W.: Several Approaches to Break the Curse of Dimensionality. Ph.D. thesis, FSU Jena, Thank you!