We present results on efficient approximation of integrals with infinitely many variables. We provide new concepts of worst case truncation and superposition dimensions and show that, under modest error demands, these dimensions are very small for functions from weighted tensor product spaces. We also present Multivariate Decomposition Method that is almost as efficient as quadratures for univariate problems. The presentation is based on papers co-authored with A. Gilbert, M. Gnewuch, M. Hefter, P. Kritzer, F. Y. Kuo, F. Pillichshammer, L. Plaskota, K. Ritter, I. H. Sloan, and H. Wozniakowski.

1. G. W. Wasilkowski ∞-Variate Integration
∞-Variate Integration
G. W. Wasilkowski
Department of Computer Science
University of Kentucky
Presentation based on papers co-authored with
A. Gilbert, M. Gnewuch, M. Hefter,
A. Hinrichs, P. Kritzer, F. Y. Kuo,
D. Nuyens, F. Pillichshammer, L. Plaskota,
K. Ritter, I. H. Sloan, H. Wo´zniakowski
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2. G. W. Wasilkowski ∞-Variate Integration
∞-Variate Integration
Can Be EASY
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3. G. W. Wasilkowski ∞-Variate Integration
Consider a Banach space Fγ of functions
f : DN
→ R
with anchored decomposition
f(x) =
u
fu(xu) fu ∈ Fu,
where
u are finite subsets of N+
xu = (xj : j ∈ u)
Fu are Banach spaces of functions fu
fu(xu) = 0 if xj = 0 for some j ∈ u
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4. G. W. Wasilkowski ∞-Variate Integration
The weighted norm in Fγ is
f Fγ =
u
γ−p
u fu
p
Fu
1/p
(p ∈ [1, ∞])
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5. G. W. Wasilkowski ∞-Variate Integration
The weighted norm in Fγ is
f Fγ =
u
γ−p
u fu
p
Fu
1/p
(p ∈ [1, ∞])
For simplicity of presentation
D = [0, 1] and fu Fu = f(u)
Lp(D|u|)
Then
f Fγ =
u
γ−p
u f(u)
(·u; 0) p
Lp([0,1]|u|)
1/p
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6. G. W. Wasilkowski ∞-Variate Integration
Although there are results for general weights γu,
we concentrate on
Product Weights introduced by
[Sloan and Wo´zniakowski 1998]
γu = c
j∈u
j−β
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7. G. W. Wasilkowski ∞-Variate Integration
Integration Problem
APPROXIMATE:
I(f) =
DN
f(x) dN
x
= lim
d→∞ Dd
f(x1, . . . , xd, 0, 0, . . . ) d[x1, . . . , xd]
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8. G. W. Wasilkowski ∞-Variate Integration
Integration Problem
APPROXIMATE:
I(f) =
DN
f(x) dN
x
= lim
d→∞ Dd
f(x1, . . . , xd, 0, 0, . . . ) d[x1, . . . , xd]
We have
I =
u
γp∗
u /(1 + p∗
)|u|
1/p∗
ASSUMPTION: I < ∞
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9. G. W. Wasilkowski ∞-Variate Integration
How to Cope with So Many Variables?
(i) Truncate the dimension,
i.e., approximate the integral of
f(x1, . . . , xk, 0, 0, . . . )
or
(ii) use
Multivariate Decomposition Method
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10. G. W. Wasilkowski ∞-Variate Integration
Low Truncation Dimension
From [Kritzer, Pillichshammer, and W. 2016]
Let
fk(x1, . . . , xk) = f(x1, . . . , xk, 0, 0, . . . )
Given the error demand ε > 0,
ε-truncation dimension dimtrnc
(ε),
is the smallest k such that
|I(f) − I(fk)| ≤ ε f F for all f ∈ F
Our concept of Truncation Dimension
is different than the one in Statistics
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11. G. W. Wasilkowski ∞-Variate Integration
If
|I(f) − I(fk)| ≤ ε f F and |I(fk) − Qk(fk)| ≤ ε f F
then
|I(f) − Qk(fk)| ≤ 2 ε f F
Hence
the smaller dimtrnc
(ε) = k the better
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12. G. W. Wasilkowski ∞-Variate Integration
Recall that: γu = c j∈u j−β
dimtrnc
(ε) ≤ min ℓ :
j=ℓ+1
j−β p∗
≤
p∗
+ 1
cp∗ ln(1/(1 − εp∗
))
= O ε−1/(β−1+1/p)
for p > 1,
and
dimtrnc
(ε) =
c
ε
1/β
− 1 for p = 1.
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13. G. W. Wasilkowski ∞-Variate Integration
Specific Values of dimtrnc
(ε): for p = 1 and c = 1
ε 10−1
10−2
10−3
10−4
10−5
dimtrnc
(ε) 2 9 31 99 316 β = 2
dimtrnc
(ε) 2 4 9 21 46 β = 3
dimtrnc
(ε) 1 3 5 9 17 β = 4
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14. G. W. Wasilkowski ∞-Variate Integration
Specific Values of dimtrnc
(ε): for p = 1 and c = 1
ε 10−1
10−2
10−3
10−4
10−5
dimtrnc
(ε) 2 9 31 99 316 β = 2
dimtrnc
(ε) 2 4 9 21 46 β = 3
dimtrnc
(ε) 1 3 5 9 17 β = 4
For instance, for the error demand ε = 10−3
with β = 4,
it is sufficient to work with integrands that have
five variables instead of ∞-many!
The Worst Case Error of Quasi-Monte Carlo (QMC)
or Smolyak’s Methods is:
≤ O
ln4
n
n
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15. G. W. Wasilkowski ∞-Variate Integration
Multivariate Decomposition Method
Introduced in [Kuo, Sloan, W., and Wo´zniakowski 2010a]
MDM depends heavily on anchored decomposition.
Recall that
f(x) =
u
fu(xu), with fu ∈ Fu
where Fu is the Banach space of functions depending
only on xu and vanishing if xj = 0.
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16. G. W. Wasilkowski ∞-Variate Integration
All fu are equally important
but some fu are much more important
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17. G. W. Wasilkowski ∞-Variate Integration
Given the error demand ε > 0,
construct an “active set” Act(ε) of subsets u such that
I


u/∈Act(ε)
fu

 ≤
ε
2
f F for all f ∈ F.
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18. G. W. Wasilkowski ∞-Variate Integration
Given the error demand ε > 0,
construct an “active set” Act(ε) of subsets u such that
I


u/∈Act(ε)
fu

 ≤
ε
2
f F for all f ∈ F.
For u∈Act(ε), choose nu and cubatures Qu,nu
to approximate integral of fu such that
u∈Act(ε)
|I(fu) − Qu,nu (fu)| ≤
ε
2
f F for all f ∈ F.
The cubatures Qu,nu could be QMC or Smolyak’s methods
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19. G. W. Wasilkowski ∞-Variate Integration
Then the MDM given by
Qε(f) :=
u∈Act(ε)
Qu,nu (fu)
has the ”worst case error” bounded by ε, i.e.,
|I(f) − Qε(f)| ≤ ε f F for all f ∈ F.
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20. G. W. Wasilkowski ∞-Variate Integration
Then the MDM given by
Qε(f) :=
u∈Act(ε)
Qu,nu (fu)
has the ”worst case error” bounded by ε, i.e.,
|I(f) − Qε(f)| ≤ ε f F for all f ∈ F.
Its cost is also small due to
Number of Integrals = |Act(ε)| − 1 = O(ε−1
)
and ([Plaskota and W. 2011])
Number of Variables ≤ d(ε) := max{|u| : u ∈ Act(ε)} = O (???)
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21. G. W. Wasilkowski ∞-Variate Integration
Then the MDM given by
Qε(f) :=
u∈Act(ε)
Qu,nu (fu)
has the ”worst case error” bounded by ε, i.e.,
|I(f) − Qε(f)| ≤ ε f F for all f ∈ F.
Its cost is also small due to
Number of Integrals = |Act(ε)| − 1 = O(ε−1
)
and ([Plaskota and W. 2011])
Number of Variables ≤ d(ε) = O
ln(1/ε)
ln(ln(1/ε))
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22. G. W. Wasilkowski ∞-Variate Integration
Very efficient algorithm to construct Act(ε) in [Gilbert and W. 2016]
Specific Values: for p = 1 and γu = j∈u j−β
ε 10−1
10−2
10−3
10−4
10−5
d(ε) and |Act(ε)| 2 6 3 22 3 112 4 534 5 2424 β = 2
d(ε) and |Act(ε)| 2 6 2 8 3 22 3 68 4 192 β = 3
d(ε) and |Act(ε)| 1 2 2 6 2 10 3 26 3 50 β = 4
For instance, for ε = 10−3
with β = 4
it is sufficient to approximate
10 integrals with at most 2 variables!
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23. G. W. Wasilkowski ∞-Variate Integration
Active Set Act(10−3
)
For β = 4
∅,
{1}, {2}, {3}, {4}, {5},
{1, 2}, {1, 3}, {1, 4}, {1, 5}
For β = 3
∅,
{1}, {2}, {3}, {4}, {5}, {6}, {7}, {8}, {9},
{1, 2}, {1, 3}, {1, 4}, {1, 5}, {1, 6}, {1, 7}, {1, 8}, {1, 9}, {2, 3}, {2, 4},
{1, 2, 3}, {1, 2, 4}
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24. G. W. Wasilkowski ∞-Variate Integration
REMARK:
We do NOT know fu terms. However, we can sample
them. Indeed due to [Kuo, Sloan, W., and Wo´zniakowski 2010b]
fu(xu) =
v⊆u
(−1)|u|−|v|
f([xv; 0])
requires
2|u|
samples of f
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25. G. W. Wasilkowski ∞-Variate Integration
REMARK:
We do NOT know fu terms. However, we can sample
them. Indeed due to [Kuo, Sloan, W., and Wo´zniakowski 2010b]
fu(xu) =
v⊆u
(−1)|u|−|v|
f([xv; 0])
requires
2|u|
samples of f
but from [Plaskota and W. 2011]
2|u|
= O ε
−1
ln(ln(1/ε)) is small for modest ε.
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26. G. W. Wasilkowski ∞-Variate Integration
GENERALIZATIONS
More General Domain:
Any interval D including D = R
More General Distributions µ on D:
e.g., Exponential, Gaussian
More General Integrals: RN f(x) µN
(dx)
Other Linear Solution Operators: S(f) =???
e.g., Function Approximation, PDE’s
General Information about f:
L1(f), L2(f), . . . , Ln(f), Lj ∈ F∗
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27. G. W. Wasilkowski ∞-Variate Integration
Bayesian Approach:
Endowing F with
Gaussian probability measure PROB
and studying average case errors:
F
S(f) − Alg(L1(f), . . . , Ln(f)) p
S(F) PROB(df)
Similar results in [W. 2014]
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28. G. W. Wasilkowski ∞-Variate Integration
How About ANOVA Spaces?
f ∈ FAVOVA
iff
f(x) =
w
fu,A(xu)
with D fu,A(xu) dxj = 0 and
f FANOVA =
u
γ−p
u f
(u)
u,A
p
Lp
1/p
< ∞
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29. G. W. Wasilkowski ∞-Variate Integration
ANOVA decomposition terms fu,A
cannot be sampled, i.e.,
low truncation dimension and MDM might
not be applicable.
Even worse: the ‘easiest’ (constant) term
is NOT known;
it is the integral we want:
f∅,A = I(f)
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30. G. W. Wasilkowski ∞-Variate Integration
Equivalence of anchored and ANOVA Spaces
For product weights γu = j∈u j−β
F = FANOVA
as sets.
For the imbedding ı : F ֒→ FANOVA
we have
ı = ı−1
≤
∞
j=1
1 + j−β
EQUIVALENCE iff β > 1
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31. G. W. Wasilkowski ∞-Variate Integration
Research direction initiated in [Hefter and Ritter 2014],
Hilbert spaces setting p = 2 and product weights
[Hefter, Ritter and W. 2016]
p ∈ {1, ∞} and general weights,
[Hinrichs and Schneider 2016]
p ∈ (1, ∞),
[Gnewuch, Hefter, Hinrichs, Ritter, and W. 2016]
more general spaces,
[Kritzer, Pillichshammer, and W. 2017]
sharp lower bounds,
[Hinrichs, Kritzer, Pillichshammer, and W. 2017] most general
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32. G. W. Wasilkowski ∞-Variate Integration
THANK YOU FOR THE ATTENTION
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