New Multilevel Algorithms Based on Quasi-Monte Carlo Point Sets

Transcription

1 New Multilevel Based on Quasi-Monte Carlo Point Sets Michael Gnewuch Institut für Informatik Christian-Albrechts-Universität Kiel 1 February 13, 2012 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 GN 91/4-1. 1/14

2 Function Spaces of Integrands Analysis of multilevel algorithms on spaces of integrands depending on infinitely many variables. Spaces of integrands are constructed in the following way: Reproducing kernel Hilbert space H of univariate functions on [0, 1] Hilbert spaces H u of multivariate functions on [0, 1] u : 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 u f N : f u H u, u f N u f N } γu 1 f u 2 H u < 2/14

3 Integration, & Cost Model Integration functional I on H γ : I(f) := f(x)dx [0,1] N Admissable Quadratures: Q n (f) = n i=1 a i f(t (i) v i ; 0), where t (i) v i [0, 1] vi, v i f N Variable Subspace Sampling: [Creutzig, Dereich, Müller-Gronbach & Ritter 09] cost(q n ) := n maxv i i=1 3/14

4 Multilevel For levels k = 1,...,m: v k := {1,...,2 k }. n 1 n 2 n 3. Quasi-Monte Carlo : Q 1 (f) := 1 n 1 f(t v (j,1) n 1 ; 0) 1 and, for k 2, Q k (f) := 1 n k n k j=1 j=1 ( f(t (j,k) v k ; 0) f(t v (j,k) k 1 ; 0) ). QMC-Multilevel Algorithm: Q ML m := m Q k (f). k=1 Cost: cost(q ML m ) 2 m n k 2 k. k=1 4/14

5 Multilevel Multilevel algorithms have been used for finite-dimensional and infinite-dimensional integration, e.g., by Heinrich 98 & 01; Giles 08; Avikainen 09; Creutzig, Dereich, Müller-Gronbach & Ritter 09; Giles, Higham & Mao 09; Giles & Waterhouse 09; Müller-Gronbach & Ritter 09; Hickernell, Müller-Gronbach, Niu & Ritter 10; Kuo, Sloan, Wasilkowski & Woźniakowski 10; G. 10; Baldeaux 11; Dereich 11; Dereich & Heidenreich 11; Dereich & Neuenkirch 11; Niu, Hickernell, Müller-Gronbach & Ritter 11; Plaskota & Wasilkowski 11;... [List is from July last year and not objective, but mostly from the IBC point of view und thus completely incomplete.] 5/14

6 (Deterministic) Higher Order Convergence Reproducing kernel K α of H of smoothness α 2: For x, y [0, 1]: K α (x, y) = α 1 r=1 x r y r min{x,y} r! r! + (x t) α 1 (y t) α 1 dt 0 (α 1)! (α 1)! H consists of functions f L 2 ([0, 1]) with f(0) = 0 and f (1),..., f (α) L 2 ([0, 1]). K α induces an anchored decomposition on H γ. 6/14

7 (Deterministic) Higher Order Convergence We use QMC-multilevel algorithms Q ML m based on (completely!) deterministic polynomial lattice rules. Error criterion: worst case error e wor (Q ML m ; H γ) 2 = sup I(f) Q ML m (f) 2 f Hγ 1 Nth minimal worst case error: e wor (N) = inf{e wor (Q; H γ ) Q adm. alg., cost(q) N}. Convergence rate of e wor (N): λ wor = sup{t > 0 sup e wor (N)N t < }. N N 7/14

8 Finite-Intersection Weights Finite-order weights γ of order ω: 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. { decay := sup p R u f N γ 1/p u < }. Theorem [Dick, G. 11]. γ finite-intersection weights, decay > 1. Then decay 2α + 1: λ wor = α 2α + 1 > decay > 1: λ wor = decay 1 2 8/14

9 Product Weights & Finite Product Weights γ 1 γ 2 γ 3. Product weights γ: γ u = j u γ j. Finite-product weights γ of order ω: As product weights, but γ u = 0 if u > ω. Theorem [Dick, G. 11]. γ be product or finite-product weights, and decay > 1. Then decay 4α: λ wor = α 4α > decay > 2: decay 4 2 decay > 1: λ wor = λ wor min decay 1 2 { α, } decay 1 2 [Anchored Decomposition & Product weights: Earlier related results in Kuo, Sloan, Wasilkowski & Woźniakowski 10 ; Niu, Hickernell, Müller-Gronbach & Ritter 11; G. 10; Plaskota & Wasilkowski 11] 9/14

10 Randomized Setting & ANOVA Decomposition Unanchored reproducing kernel K of H: For x, y [0, 1]: K(x, y) = x2 + y 2 max{x, y} 2 H consists of functions f L 2 ([0, 1]) with 1 0 f(x)dx = 0 and f (1) L 2 ([0, 1]). K induces ANOVA-decomposition on H γ. 10/14

11 Randomized Setting & ANOVA Decomposition We use RQMC-multilevel algorithms Q ML m based on fully scrambled digital (t, m, s)-nets and fully scrambled polynomial lattice rules. Error criterion: error in randomized setting e ran (Q ML m ; H γ) 2 = sup E ( (I(f) Q ML m (f))2) f Hγ 1 Nth minimal error in randomized setting: e ran (N) = inf{e ran (Q; H γ ) Q rand. adm. alg., cost(q) N}. Convergence rate of e ran (N): λ ran = sup{t > 0 sup e ran (N)N t < }. N N 11/14

12 Product Weights Theorem [Baldeaux, G. 11]. γ be product weights, and decay > 1. Then decay 6 : λ ran = 3/2 6 > decay > 2: λ ran decay 4 2 decay > 1: λ ran decay 1 2 Comparison with previously known results for product weights: [Hickernell, Niu, Müller-Gronbach, Ritter 10]: Multilevel algorithms Q ML m based on scrambled Niederreiter (t, m, s)-nets: decay 11: λ ran = 3/2 [Baldeaux 11]: Multilevel algorithms ˆQ ML m polynomial lattice rules: decay 10: λ ran = 3/2 based on scrambled 12/14

13 Finite-Intersection Weights Theorem [Baldeaux, G. 11]. γ be finite-intersection weights, decay > 1, Q ML m based on scrambled digital (t, m, s)-nets. Then decay 4: λ ran = 3/2 4 decay > 1: λ ran decay /14

14 (Some) References for the Settings Discussed Here 1 J. Baldeaux, Scrambled polynomial lattice rules for infinite-dimensional integration, To appear Proceedings of MCQMC J. Baldeaux, J. Dick, J. Greslehner, F. Pillichshammer, Construction algorithms for higher order polynomial lattice rules, J. Complexity 27 (2011), J. Baldeaux, J. Dick, G. Leobacher, D. Nuyens, F. Pillichshammer, Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules, To appear in Numer. Alg.. 4 J. Creutzig, S. Dereich, T. Müller-Gronbach, K. Ritter, Infinite-dimensional quadrature and approximation of distributions, Found. Comput. Math. 9 (2009), M. Gnewuch, Infinite-dimensional Integration on Weighted Hilbert Spaces, To appear in Math. Comp.. 6 F. J. Hickernell, T. Müller-Gronbach, B. Niu, K. Ritter, Multi-level Monte Carlo algorithms for infinite-dimensional integration on R N, J. Complexity 26 (2010), F. Y. Kuo, I. H. Sloan, G. W. Wasilkowski, H. Woźniakowski, Liberating the dimension, J. Complexity 26 (2010), B. Niu, F. J. Hickernell, T. Müller-Gronbach, K. Ritter, Deterministic multi-level algorithms for infinite-dimensional integration on R N, J. Complexity 27 (2011), L. Plaskota, G. W. Wasilkowski, Tractability of infinite-dimensional integration in the worst case and randomized settings, J. Complexity 27 (2011), /14