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Randomized Algorithms for Infinite- dimensional Optimal Randomized Algorithms for Integration Integration on Function Spaces with underlying ANOVA decomposition Michael Gnewuch 1 University of Kaiserslautern, Germany October 16, 2013 Based
Randomized Algorithms for Infinite- dimensional Integration
Michael Gnewuch1 University of Kaiserslautern, Germany October 16, 2013 Based on Joint Work with Jan Baldeaux (UTS Sydney) & Josef Dick (UNSW Sydney)
1Supported by the German Science Foundation DFG under Grant GN 91/3-1
and by the Australian Research Council ARC
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Randomized Algorithms for Infinite- dimensional Integration
Sequence space [0, 1]N, endowed with probability measure dx = ⊗j∈Ndxj. For f ∈ L2([0, 1]N), u ⊂f N: f∅(x) :=
fu(x) :=
fv(x) , where xu = (xj)j∈u, yN\u = (yj)j∈N\u. Then 1 fu(x) dxj = 0 for j ∈ u. This implies f =
fu in L2([0, 1]N) , and Var(f) =
Var(fu).
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Randomized Algorithms for Infinite- dimensional Integration
Construction of spaces of integrands f : [0, 1]N → R:
functions f : [0, 1] → R with 1
0 f(x) dx = 0.
Hu := ⊗j∈uH for u ⊂f N, where H∅ = span{1}.
Weights γ = (γu)u⊂f N with
u⊂fN γu < ∞
Hγ :=
u⊂f N
fu
Hγ :=
γ−1
u fu2 Hu < ∞
where Hγ ⊂ L2([0, 1]N) and f =
u⊂f N fu ANOVA decomposition.
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Randomized Algorithms for Infinite- dimensional Integration
Product weights γ [Sloan & Wo´ zniakowski’98]: Let γ1 ≥ γ2 ≥ γ3 ≥ · · · ≥ 0. Then γu :=
j∈u γj.
Finite-order weights γ of order ω [Dick, Sloan, Wang & Wo´ zniakowski’06]: γu = 0 for all |u| > ω. Finite-intersection weights γ of degree ρ: Finite-order weights with |{u ⊂f N | γu > 0, u ∩ v = ∅}| ≤ 1 + ρ for all v ⊂f N, γv > 0. (Subclass of finite-intersection weights are the “finite-diameter weights” proposed by Creutzig.) decay := sup
γ1/p
u
< ∞
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Randomized Algorithms for Infinite- dimensional Integration
Integration functional I on Hγ : I(f) :=
Admissable randomized algorithms: Qn(f) =
n
αif(t(i)
vi ; a),
where t(i)
vi ∈ [0, 1]vi, vi ⊂f N, a = 1/2
Nested Subspace Sampling [Creutzig, Dereich, M¨ uller-Gronbach, Ritter‘09]: Fix s ≥ 1. costnest(Qn) :=
n
s Unrestricted Subspace Sampling [Kuo, Sloan, Wasilkowski, Wo´ zniakowski‘10]: Fix s ≥ 1. costunr(Qn) :=
n
|vi|s
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Randomized Algorithms for Infinite- dimensional Integration
Error criterion: (worst case) randomized error eran(Q; Hγ)2 := sup
fHγ ≤1
E
Nth minimal randomized error: mod ∈ {nest, unr}, eran
mod(N) := inf{eran(Q; Hγ) | Q adm. rand. alg., costmod(Q) ≤ N}.
“Convergence order” of eran
mod(N):
λran
mod := sup
N∈N
eran
mod(N) · N t < ∞
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Randomized Algorithms for Infinite- dimensional Integration
For levels k = 1, . . . , m: vk := {1, . . . , 2k}. n1 ≥ n2 ≥ n3 ≥ · · · . (Unbiased) RQMC Algorithms: Qvk(g) := 1 nk
nk
g(t(j,k)
vk
), t(j,k)
vk
∈ [0, 1]vk. Projections: Ψvkf(x) := f(xvk; a) for k ≥ 1 and Ψv0f(x) := 0. RQMC-Multilevel Algorithm: QML
m (f) := m
Qvk(Ψvkf − Ψvk−1f). Cost: costnest(QML
m ) = costunr(QML m ) ≤ m
2 nk 2ks.
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Randomized Algorithms for Infinite- dimensional Integration
Projections: Ψvkf(x) = f(xvk; a) for k ≥ 1 and Ψv0f(x) = 0. RQMC-ML Algo.: QML
m (f) = m
Qvk(Ψvkf−Ψvk−1f). Then E
m (f)
m
I(Ψvkf − Ψvk−1f) = I(Ψvmf), and E
m (f)
2 =|I(f) − I(Ψvmf)|2 +
m
Var
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Randomized Algorithms for Infinite- dimensional Integration
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 http://people.maths.ox.ac.uk/gilesm/mlmc community.html for more recent information.
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Randomized Algorithms for Infinite- dimensional Integration
Unanchored reproducing kernel k of H: For x, y ∈ [0, 1]: k(x, y) = 1 3 + x2 + y2 2 − max{x, y} H = H(k) consists of functions f ∈ L2([0, 1]) with f absolutely continuous, f (1) ∈ L2([0, 1]), and 1
0 f(x) dx = 0.
k induces ANOVA decomposition on Hγ: f =
fu, fu ∈ Hu, and 1 fu(x) dxj = 0 if j ∈ u.
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Randomized Algorithms for Infinite- dimensional Integration
Theorem [Baldeaux, G.‘12]. γ product weights, decay > 1. Then decay ≥ 1 + 3s : λran
nest = 3/2
1 + 3s ≥ decay > 1: λran
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, M¨ uller-Gronbach, Ritter‘10]: Multilevel algorithms ˜ QML
m
based on scrambled Niederreiter (t, m, s)-nets: decay ≥ 11: λran
nest = 3/2
[Baldeaux‘11]: Multilevel algorithms ˆ QML
m
based on scrambled polynomial lattice rules: decay ≥ 10: λran
nest = 3/2
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Randomized Algorithms for Infinite- dimensional Integration
Theorem [Baldeaux, G.‘12]. γ be finite-intersection weights, decay > 1. Then decay ≥ 1 + 3s : λran
nest = 3/2
1 + 3s ≥ decay > 1: λran
nest = decay −1
2s (Upper error bound achieved by multilevel algorithms based on scrambled polynomial lattice rules; lower error bound holds for general randomized algorithms.)
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Randomized Algorithms for Infinite- dimensional Integration
Anchored decomposition: f∅,a := f(a) and fu,a(x) := f(xu; a) −
fv,a(x). A changing dimension algorithm (or multivariate decomposition method) QCD is of the form QCD(f) =
Qu,nu(fu,a), Qu,nu using nu samples to approximate
QCD is linear if Qu,nus are linear: fu,a(x) =
v⊆u(−1)|u\v|f(xv; a)
[Kuo, Sloan, Wasilkowski, Wo´ zniakowski’10a] Cost for evaluating fu,a in unrestricted model: O(2|u||u|s).
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Randomized Algorithms for Infinite- dimensional Integration
Changing dimension algorithms (alias “multivariate decomposition methods”) for infinite-dimensional integration were introduced in [Kuo, Sloan, Wasilkowski, Wo´ zniakowski’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 & Wo´ zniakowski. A similar idea was used for multivariate integration in [Griebel & Holtz’10] (“dimension-wise quadrature methods”).
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Randomized Algorithms for Infinite- dimensional Integration
Unanchored reproducing kernel kχ of smoothness χ: x, y ∈ [0, 1] kχ(x, y) =
χ
Bτ(x) τ! Bτ(y) τ! + (−1)χ+1 B2χ(|x − y|) (2χ)! , where Bτ is Bernoulli polynomial of degree τ. H = H(kχ) consists of functions f ∈ L2([0, 1]) with f, f (1), . . . , f (χ−1) absolutely continuous, f (χ) ∈ L2([0, 1]), and 1
0 f(x) dx = 0.
kχ induces ANOVA decomposition on Hγ: f =
fu, fu ∈ Hu, and 1 fu(x) dxj = 0 if j ∈ u.
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Randomized Algorithms for Infinite- dimensional Integration
Theorem [Dick, G.‘13]. γ product weights or finite-intersection weights, decay > 1. Then decay ≥ 2(χ + 1) : λran
unr = χ + 1/2
2(χ + 1) ≥ decay > 1: λran
unr = decay −1
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(eσk) for some σ ∈ (0, ∞)!
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Randomized Algorithms for Infinite- dimensional Integration
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).
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).
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Randomized Algorithms for Infinite- dimensional Integration
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