Optimal Randomized Algorithms for Integration on Function Spaces with underlying ANOVA decomposition

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1 Optimal Randomized on Function Spaces with underlying ANOVA decomposition Michael Gnewuch 1 University of Kaiserslautern, Germany October 16, 2013 Based on Joint Work with Jan Baldeaux (UTS Sydney) & Josef Dick (UNSW Sydney) 1 Supported by the German Science Foundation DFG under Grant GN 91/3-1 and by the Australian Research Council ARC 1/18

2 ANOVA Decomposition ( -Variate Functions) Sequence space [0, 1] N, endowed with probability measure dx = j N dx j. For f L 2 ([0, 1] N ), u f N: f (x) := f(y)dy, [0,1] N f u (x) := f(x u,y N\u )dy N\u f v (x), [0,1] [N]\u v u where x u = (x j ) j u, y N\u = (y j ) j N\u. Then 1 0 f u (x)dx j = 0 for j u. This implies f = f u in L 2 ([0, 1] N ), and Var(f) = Var(f u ). u f N u f N 2/18

3 Function Spaces of Integrands Construction of spaces of integrands f : [0, 1] N R: Reproducing kernel Hilbert space H = H(k) of univariate functions f : [0, 1] R with 1 f(x)dx = 0. 0 Hilbert spaces H u of multivariate functions f u : [0, 1] u R: H u := j u H for u f N, where H = span{1}. Hilbert space H γ of functions of infinitely many variables: Weights γ = (γ u ) u f N with u f N γ u < { H γ := f u f u H u, f 2 H γ := u f N u f N } γu 1 f u 2 H u <, where H γ L 2 ([0, 1] N ) and f = u f N f u ANOVA decomposition. 3/18

4 Weights Product weights γ [Sloan & Woźniakowski 98]: Let γ 1 γ 2 γ 3 0. Then γ u := j u γ j. Finite-order weights γ of order ω [Dick, Sloan, Wang & Woźniakowski 06]: γ u = 0 for all u > ω. Finite-intersection weights γ of degree ρ: Finite-order weights with {u f N γ u > 0, u v } 1 + ρ for all v f N, γ v > 0. (Subclass of finite-intersection weights are the finite-diameter weights proposed by Creutzig.) { decay := sup p R u f N γ 1/p u < }. 4/18

5 , Algorithms & Cost Model functional I on H γ : I(f) := f(x)dx [0,1] N Admissable randomized algorithms: n Q n (f) = α i f(t (i) v i ;a), where t (i) v i [0, 1] vi, v i f N, a = 1/2 i=1 Nested Subspace Sampling [Creutzig, Dereich, Müller-Gronbach, Ritter 09]: Fix s 1. cost nest (Q n ) := n ( ) s maxvi i=1 Unrestricted Subspace Sampling [Kuo, Sloan, Wasilkowski, Woźniakowski 10]: Fix s 1. cost unr (Q n ) := n v i s i=1 5/18

6 Randomized Setting Error criterion: (worst case) randomized error e ran (Q; H γ ) 2 := sup E ( (I(f) Q(f)) 2) f Hγ 1 N th minimal randomized error: mod {nest, unr}, e ran mod(n) := inf{e ran (Q; H γ ) Q adm. rand. alg., cost mod (Q) N}. Convergence order of e ran mod (N): } λ ran mod {t := sup > 0 sup e ran mod (N) Nt <. N N 6/18

7 Nested Subspace Sampling: Multilevel Algorithms For levels k = 1,...,m: v k := {1,...,2 k }. n 1 n 2 n 3. (Unbiased) RQMC Algorithms: Q vk (g) := 1 n k n k j=1 g(t (j,k) v k ), t (j,k) v k [0, 1] v k. Projections: Ψ vk f(x) := f(x vk ;a) for k 1 and Ψ v0 f(x) := 0. RQMC-Multilevel Algorithm: Q ML m m (f) := Q vk (Ψ vk f Ψ vk 1 f). k=1 Cost: m cost nest (Q ML m ) = cost unr(q ML m ) 2 n k 2 ks. k=1 7/18

8 Nested Subspace Sampling: Multilevel Algorithms Projections: Ψ vk f(x) = f(x vk ;a) for k 1 and Ψ v0 f(x) = 0. RQMC-ML Algo.: Q ML m (f) = m Q vk (Ψ vk f Ψ vk 1 f). k=1 Then and E ( Q ML m (f) ) = m I(Ψ vk f Ψ vk 1 f) = I(Ψ vm f), k=1 ( (I(f) E Q ML m (f) ) ) 2 m = I(f) I(Ψ vm f) 2 + Var ( Q vk (Ψ vk f Ψ vk 1 f) ). k=1 8/18

9 Multilevel Algorithms Multilevel Monte Carlo algorithms were introduced in the context of integral equations and parametric integration by Heinrich (1998) and Heinrich and Sindambiwe (1999) and in the context of stochastic differential equations by Giles (2008). Multilevel quasi-monte Carlo algorithms were tested by Giles and Waterhouse (2009). Multilevel Monte Carlo and quasi-monte Carlo algorithms have been studied in a number of papers, see, e.g., the web page of Mike Giles community.html for more recent information. 9/18

10 Nested Subspace Sampling Unanchored reproducing kernel k of H: For x, y [0, 1]: k(x, y) = x2 + y 2 max{x, y} 2 H = H(k) consists of functions f L 2 ([0, 1]) with f absolutely continuous, f (1) L 2 ([0, 1]), and 1 f(x)dx = 0. 0 k induces ANOVA decomposition on H γ : f = u f N f u, f u H u, and 1 0 f u (x)dx j = 0 if j u. 10/18

11 Nested Subspace Sampling: Product Weights Theorem [Baldeaux, G. 12]. γ product weights, decay > 1. Then decay 1 + 3s : λ ran nest = 3/ s decay > 1: λ ran nest = decay 1 2s (Upper error bound via multilevel algorithms based on scrambled polynomial lattice rules (scrambling: Owen 95; polynomial lattice rules: Niederreiter 92); lower error bound holds for general randomized algorithms.) Comparison with previously known results for s = 1: [Hickernell, Niu, Müller-Gronbach, Ritter 10]: Multilevel algorithms Q ML m based on scrambled Niederreiter (t, m, s)-nets: decay 11: λ ran nest = 3/2 ˆQ ML m [Baldeaux 11]: Multilevel algorithms based on scrambled polynomial lattice rules: decay 10: λ ran nest = 3/2 11/18

12 Nested Subspace Sampling: Finite-Intersection Weights Theorem [Baldeaux, G. 12]. γ be finite-intersection weights, decay > 1. Then decay 1 + 3s : λ ran nest = 3/ s decay > 1: λ ran decay 1 nest = 2s (Upper error bound achieved by multilevel algorithms based on scrambled polynomial lattice rules; lower error bound holds for general randomized algorithms.) 12/18

13 Unrestricted Subspace Sampling: CDAs (alias MDMs) Anchored decomposition: f,a := f(a) and f u,a (x) := f(x u ;a) v uf v,a (x). A changing dimension algorithm (or multivariate decomposition method) Q CD is of the form Q CD (f) = Q u,nu (f u,a ), u f N Q u,nu using n u samples to approximate [0,1] u f u,a (x u )dx u. Q CD is linear if Q u,nu s are linear: f u,a (x) = v u ( 1) u\v f(x v ;a) [Kuo, Sloan, Wasilkowski, Woźniakowski 10a] Cost for evaluating f u,a in unrestricted model: O(2 u u s ). 13/18

14 Changing Dimension Algorithms Changing dimension algorithms (alias multivariate decomposition methods ) for infinite-dimensional integration were introduced in [Kuo, Sloan, Wasilkowski, Woźniakowski 10] and refined in [Plaskota & Wasilkowski 11]. These algorithms have also been adapted to infinite-dimensional approximation problems, see the papers of Wasilkowski and of Wasilkowski & Woźniakowski. A similar idea was used for multivariate integration in [Griebel & Holtz 10] ( dimension-wise quadrature methods ). 14/18

15 Unrestricted Subspace Sampling Unanchored reproducing kernel k χ of smoothness χ: x, y [0, 1] k χ (x, y) = χ τ=1 B τ (x) B τ (y) τ! τ! where B τ is Bernoulli polynomial of degree τ. + ( 1) χ+1 B 2χ( x y ), (2χ)! H = H(k χ ) consists of functions f L 2 ([0, 1]) with f, f (1),..., f (χ 1) absolutely continuous, f (χ) L 2 ([0, 1]), and 1 0 f(x)dx = 0. k χ induces ANOVA decomposition on H γ : f = u f N f u, f u H u, and 1 0 f u (x)dx j = 0 if j u. 15/18

16 Unrestricted Subspace Sampling: Product Weights and Finite-Intersection Weights Theorem [Dick, G. 13]. γ product weights or finite-intersection weights, decay > 1. Then decay 2(χ + 1) : λ ran unr = χ + 1/2 2(χ + 1) decay > 1: λ ran decay 1 unr = 2 (Upper error bounds achieved by changing dimension algorithms based on interlaced scrambled polynomial lattice rules [Dick 11; Goda & Dick 13]; lower error bounds holds for (rather) general randomized algorithms.) Result of theorem still holds if cost of function evaluation in points with k active variables costs O(e σk ) for some σ (0, )! 16/18

17 Generalizations Similar results hold more generally for spaces H γ induced by more general kernels k : D D R, D R, ρ probability measure on D if k(x, y)ρ(dx) = 0 for all y D (ANOVA Case). D We have also results for 1 > s 0. The results can also be transferred to non-anova settings, as the anchored or alternative unanchored settings [Dick, G., Hefter, Hinrichs, Ritter] for product weights (relying on results from [Hefter, Ritter 13]) for more general weights (work in progress). 17/18

18 Thank you for your attention! 18/18