Integration of permutation-invariant. functions

Transcription

1 permutation-invariant Markus Weimar Philipps-University Marburg Joint work with Dirk Nuyens and Gowri Suryanarayana (KU Leuven, Belgium) MCQMC2014, Leuven April 06-11, 2014 un

2 Outline Permutation-invariant subspaces Algorithms and worst case error Tractability and known results un un

3 Integration problem for We study multivariate integration Int d (f ) := f (x) dx, d N, [0,1] d for certain subspaces of some RKHS of (1-periodic) where we set F d (r α,β ) := { f L 2 ([0, 1] d ) f 2 d := h Z d f (h) 2 r α,β (h) } f d <, with α > 1/2 (smoothness), β > 0 (weight parameter), and r α,β (h) := d l=1 [ δ 0,hl + (1 δ 0,hl ) β 1 (2π h l ) 2α], h Z d. un

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

5 Symmetrizing the standard Fourier basis {e h := exp(2πi h ) h Z d} leads to an orthonormal basis {φ k k d }, φ k (x) := r α,β 1 (k) #S d M d (k)! e σ(k) (x), σ S d d := {k Z d k l1... k l#id }, of the subspace S Id (F d (r α,β )) of I d -permutation-invariant in F d (r α,β ). Here M d (k)! counts the repetitions in k (under S d ): M d (k)! := #{σ S d k = σ(k)}. un

6 We obtain a permutation-invariant reproducing kernel K perm (x, y) = k d 1 rα,β (k) M d (k)! σ S d e k (σ(x) y) for the subspace out of the classical reproducing kernel K d (x, y) = of F d (r α,β ) = H(K d ). This new kernel K perm is NOT of product form, NOT shift-invariant! h Z d r 1 α,β (h) e h(x y) un

7 Algorithms and worst case error Consider a general n-point cubature rule with nodes t (j) and weights w j : Q n,d (f ) := 1 n n 1 j=0 ( w j f t (j)). Its squared worst case error (see e.g. Novak, Woźniakowski) in any RKHS H d = H(K) is given by err(q n,d ; H d ) 2 := sup f H d f H d 1 Int d (f ) Q n,d (f ) 2 = IIK 2QIK + QQK. We are interested in the behavior of e wor (n, d) := inf Q n,d err(q n,d ; H(K perm )), n N, n(ε, d) := min{n N 0 e wor (n, d) ε}, ε (0, 1). un

8 Notions of tractability and known results Weak tractability: Polynomial tractability: n(ε, d) C d q ε p log n(ε, d) lim ε 1 +d ε 1 + d = 0 C, p > 0 and q 0 such that ε (0, 1), d N Strong polynomial tractability: polyn. tract. with q = 0 un

9 Notions of tractability and known results Weak tractability: Polynomial tractability: n(ε, d) C d q ε p log n(ε, d) lim ε 1 +d ε 1 + d = 0 C, p > 0 and q 0 such that ε (0, 1), d N Strong polynomial tractability: polyn. tract. with q = 0 Known results for I d = (i.e. no permutation-invariance) and dimension-dependent weights β = β(d): WT lim d β(d) = 0 PT β(d) log(d + 1) d un SPT β(d) 1 d

10 By averaging err(q n,d ; H(K perm )) 2 over all integration nodes t (j) [0, 1] d we conclude: Theorem (Nuyens, Suryanarayana, W ) For all n, d N, I d {1,..., d}, α > 1/2, and β > 0 there exists an equal weight cubature rule Qn,d such that e wor (n, d) 2 err(q n,d; H(K perm )) 2 C d,1 (r α,β ) n 1. un

11 By averaging err(q n,d ; H(K perm )) 2 over all integration nodes t (j) [0, 1] d we conclude: Theorem (Nuyens, Suryanarayana, W ) For all n, d N, I d {1,..., d}, α > 1/2, and β > 0 there exists an equal weight cubature rule Qn,d such that e wor (n, d) 2 err(q n,d; H(K perm )) 2 C d,1 (r α,β ) n 1. Therein the constant C d,1 (r α,β ) = k d \{0} r 1 α,β (k) = h Z d \{0} M d (h)! #S d r 1 α,β (h) grows exponentially with β/(2π) 2α and in (d #I d ), is polyn. bounded if β < c α and (d #I d ) O(ln d). = (strong) polynomial tractability, if the space allows un

12 (Un) rank-1 lattice rules Since we deal with, (rank-1) lattice rules, completely determined by their generating vector z (from Z d n = {0, 1,..., n 1} d ), are the method of choice: Qn,d(f lat ) = 1 n 1 ({ f z j }). n n j=0 Using the character property we conclude err(q lat n,d; H(K perm )) 2 = h L \{0} r 1 α,β (h) #S d σ S d 1 σ(h) L, where L denotes the dual lattice induced by z, i.e. the set of all h Z d with h z 0 (mod n). un

13 A negative result Theorem (Nuyens, Suryanarayana, W ) Let α > 1/2 and β > 0. Then, independently of the amount of permutationinvariance, the nth minimal worst case error among all un lattice rules satisfies e wor lat (n, d) 2 = h Z d \{0} for all d N and n N prime. r 1 α,β (nh) ( ζ(2α) ) d β (2πn) 2α 1 c α,β d n 2α un = The class of un lattice rules is too small to obtain strong polynomial tractability!

14 (Randomly) Shifted lattice rules Let us enlarge the class of algorithms under consideration by adding random shifts [0, 1) d to given lattice rules Q lat Qn,d sh lat (f ) := (Qn,d lat + )(f ) = 1 n 1 f n Proposition For all Q lat n,d E(Q lat n,d) 2 := and j=0 the average over all shifts satisfies err(q lat [0,1] d n,d + ; H(K perm )) 2 d = err(q lat n,d; H(K sh,perm )) 2 = c α,β max{d #I d, 1} n 2α ({z j n + }). h L \{0} M d (h)! r 1 #S d n,d : α,β (h) E(Q lat n,d) 2 err(q lat n,d; H(K perm )) 2. un It grows exponentially with β/(2πn) 2α and in (d #I d ).

15 Theorem (Nuyens, Suryanarayana, W ) Let d N, I d {1,..., d}, as well as n N prime, and set C d,λ (r α,β ) = [ ] λ Md (h)! 1/λ rα,β 1 #S (h), λ 1. h Z d d \{0} Then there ex. Q lat n,d such that for all 1 λ < 2α E(Q lat n,d ) 2 2 λ C d,λ (r α,β ) n λ, lat there ex. Qn,d, constructed component-by-component, such that for all 1 λ < 2α E(Q lat n,d ) 2 2 λ d λ 1 C d,λ (r α,β ) n λ. un

16 Conclusions works = strong polynomial tractability, if the space allows (e.g. for I d = {1,..., d} and β < c α ) Upper bounds for Qn,d lat sh,perm with respect to H(K ) imply the existence of good shifts such that the same bounds hold for Qn,d sh lat = Qn,d lat + with respect to H(K perm ). The () CBC-construction realizes the random sampling error bound up to a factor of 2. Un lattice rules cannot achieve such a bound. un

17 Final remarks For I d = our bounds match well-known results. The CBC-algorithm depends on the target dimension, i.e. the obtained lattice rules are not extendable. For higher-order convergence n λ, λ > 1, we lose a factor of d λ 1 using the CBC-approach, but we (almost) obtain the optimal rate 2α, C d,λ (r α,β ) grows exponentially for all I d and constant β, C d,λ (r α,β ) is polynomially upper bounded for (d #I d ) O(ln d) and β = β(d) 1 d λ 1. Finally, everything is semi-constructive. un

18 Based on results for integration and approximation w.r.t. the average case setting (see Hickernell & Woźniakowski and Wasilkowski) we can show (semi-constructively): Theorem (Nuyens, Suryanarayana, W ) For all α > 1/2 and δ > 0 there exists Q n,d such that e wor (n, d) 2 err(q n,d; H(K perm )) 2 C α,β,δ (d) n 2α+δ. Therein the constant satisfies C α,β,δ (d) c α,β,δ d q for some q 0, provided that #I d and β 1 are large enough. un = (strong) polynomial tractability with (almost) opt. rate if the space allows

19 Nuyens, Suryanarayana, and W.: Rank-1 lattice rules for multivariate integration in spaces of permutation-invariant : error bounds and tractability. Manuscript (2014). Nuyens, Suryanarayana, and W.: Construction of quasi-monte Carlo rules for multivariate integration in spaces of. Manuscript (2014). Novak, Woźniakowski: Tractability of Multivariate s. Vol. I III. European Mathematical Society (EMS), Zürich, Hickernell, Woźniakowski: Integration and approximation in arbitrary dimensions. Advances in comp. math. 12, (2000). Wasilkowski: Integration and approximation of multivariate : Average case complexity with isotropic wiener measure. J. Approx. Theory 77, (1994). W.: On lower bounds for integration of multivariate. J. Complexity 30(1), (2014). un Thank you!