Measures of Error, Discrepancy, and Complexity

Measures of Error, Discrepancy, and Complexity
Fred J. Hickernell
Department of Applied Mathematics, Illinois Institute of Technology
[email protected] mypages.iit.edu/~hickernell
Thanks to the Guaranteed Automatic Integration Library (GAIL) team and friends
Supported by NSF-DMS-1522687
Working Group V, Ocotober 18, 2017

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Problem One-Dimensional Problems Multiple Dimensions References
Problem
Given an error tolerance, ε, and
a black-box, f, that accepts inputs, x ∈ Ω and produces outputs, f(x) ∈ R,
Carefully choose the number and positions of inputs, {xi}n
i=1
⊂ Ω, and then
use the output data, {f(xi)}n
i=1
, to construct an approximate solution, app {(xi, f(xi))}n
i=1
, ε
sol(f) =
Ω
f(x) dx or f(x)
app {(xi, f(xi))}n
i=1
, ε =
n
i=1
wif(xi) or
n
i=1
wi(x)f(xi)
satisfying sol(f) − app {(xi, f(xi))}n
i=1
, ε errn({xi}n
i=1
, f) ε
We focus on linear problems. For ﬁxed {xi}n
i=1
, app {(xi, f(xi))}n
i=1
, ε is linear, but the choices
of n, and the xi, may depend nonlinearly on f.
2/17

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(One-Dimensional) Quadrature by Simple Average
Suppose the functions belong to a Hilbert space H = f : Ω = [−1, 1] → R f
2
< ∞ with
f, g H
= f(0)g(0) +
1
2
1
−1
f (x)g (x) dx
3/17

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2
< ∞ with
f, g H
= f(0)g(0) +
1
2
1
−1
f (x)g (x) dx
The reproducing kernel K : [−1, 1] × [−1, 1] → R
is deﬁned by K(x, t) = 1 + |x| + |t| − |x − t|
N. Aronszajn. Theory of Reproducing Kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950), H., F. J. A
Generalized Discrepancy and Quadrature Error Bound. Math. Comp. 67, 299–322 (1998). 3/17

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2
< ∞ with
f, g H
= f(0)g(0) +
1
2
1
−1
f (x)g (x) dx
The reproducing kernel K : [−1, 1] × [−1, 1] → R is deﬁned by K(x, t) = 1 + |x| + |t| − |x − t|
To verify, note that
K(x, t) = K(t, x) ∀x, t ∈ [−1, 1]
K(·, t) ∈ H ∀t since K(·, t) 2
H
= |K(0, t)|2 +
1
2
1
−1
∂K
∂x
(x, t)
2
dx = 1 − 2 |t| < ∞
K(·, t), f H
= 1 × f(0) +
1
2
1
−1
[sign(x) − sign(x − t)] f (x) dx = f(t) ∀t ∈ [−1, 1]
N. Aronszajn. Theory of Reproducing Kernels. Trans. Amer. Math. Soc. 68, 337–404 (1950), H., F. J. A
Generalized Discrepancy and Quadrature Error Bound. Math. Comp. 67, 299–322 (1998). 3/17

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(One-Dimensional) Quadrature by Simple Average cont’d
f, g H
= f(0)g(0)+
1
2
1
−1
f (x)g (x) dx, K(x, t) = 1+|x|+|t|−|x − t| , K(·, t), f H
= f(t)
Let sol(f) =
1
−1
f(x) dx and quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
. Then by the Riesz
representation theorem and the Cauchy-Schwartz inequality
sol(f) − quadn
(f) = ηn, f H
, |sol(f) − quadn
(f)| ηn H
f H
,
4/17

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f, g H
= f(0)g(0)+
1
2
1
−1
= f(t)
Let sol(f) =
1
−1
f(x) dx and quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
. Then by the Riesz
sol(f) − quadn
(f) = ηn, f H
, |sol(f) − quadn
(f)| ηn H
f H
,
and by the reproducing property, ηn(t) = K(·, t), ηn H
= sol(K(·, t)) − quadn
(K(·, t)), and
ηn
2
H
= ηn, ηn H
= sol(ηn) − quadn
(ηn)
= sol·· sol·(K(·, ··) − 2 sol·· quad·
n
(K(·, ··) + quad··
n
quad·
n
(K(·, ··)
=
16
3
−
4
n
n
i=1
2 + 2
2i − n − 1
n
−
2i − 1
2n
2
+
4
n2
n
i,k=1
1 +
2i − n − 1
n
+
2k − n − 1
n
−
2i − 2k
n
4/17

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f, g H
= f(0)g(0)+
1
2
1
−1
= f(t)
Let sol(f) =
1
−1
f(x) dx and quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
. Then by the Riesz
sol(f) − quadn
(f) = ηn, f H
, |sol(f) − quadn
(f)| ηn H
f H
,
(K(·, t)), and
ηn
2
H
= ηn, ηn H
= sol(ηn) − quadn
(ηn)
= sol·· sol·(K(·, ··) − 2 sol·· quad·
n
(K(·, ··) + quad··
n
quad·
n
(K(·, ··)
=
16
3
−
4
n
n
i=1
2 + 2
2i − n − 1
n
−
2i − 1
2n
2
+
4
n2
n
i,k=1
1 +
2i − n − 1
n
+
2k − n − 1
n
−
2i − 2k
n
=
4
3n2
4/17

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f, g H
= f(0)g(0)+
1
2
1
−1
= f(t)
Let sol(f) =
1
−1
f(x) dx and quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
. Then by the Riesz
sol(f) − quadn
(f) = ηn, f H
, |sol(f) − quadn
(f)| ηn H
f H
,
(K(·, t))
4/17

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f, g H
= f(0)g(0) +
1
2
1
−1
f (x)g (x) dx, K(x, t) = 1 + |x| + |t| − |x − t| , K(·, t), f H
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
is called the discrepancy, it also has a geometric interpretation
H., F. J. What Aﬀects the Accuracy of Quasi-Monte Carlo Quadrature?. in Monte Carlo and Quasi-Monte Carlo
Methods 1998 (eds Niederreiter, H. & Spanier, J.) (Springer-Verlag, Berlin, 2000), 16–55. 5/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
H., F. J. What Aﬀects the Accuracy of Quasi-Monte Carlo Quadrature?. in Monte Carlo and Quasi-Monte Carlo
Methods 1998 (eds Niederreiter, H. & Spanier, J.) (Springer-Verlag, Berlin, 2000), 16–55. 6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
Why doesn’t the midpoint rule have faster decay?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
Why doesn’t the midpoint rule have faster decay? H is not smooth enough
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
Can we extend this to arbitrary data sites, quadn
(f) =
2
n
n
i=1
f(xi)?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
Can we extend this to arbitrary data sites, quadn
(f) =
2
n
n
i=1
f(xi)? Yes
ηn
2
H
=
16
3
−
4
n
n
i=1
2 + 2 |xi| − |xi|2 +
4
n2
n
i,k=1
[1 + |xi| + |xk
| − |xi − xk
|]
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
Can we extend this to arbitrary weights, quadn
(f) =
n
i=1
wif(xi)?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
ηn
2
H
=
16
3
− 2
n
i=1
wi 2 + 2 |xi| − |xi|2 +
n
i,k=1
wiwk
[1 + |xi| + |xk
| − |xi − xk
|]
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
How do we choose n to make |sol(f) − quadn
(f)| ε?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
(f)| ε?
If f ∈ BR = {f : f H
R}, then choose n such that ηn H
ε/R. For the midpoint rule
n
√
3R/(2ε). Not easy, since f H
is unknown.
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
(f)| ε? Not easy, since f H
is unknown.
Is this the best order of error possible for this H?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
is unknown.
Is this the best order of error possible for this H? Coming up
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
is unknown.
What can we do for function approximation?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
is unknown.
What can we do for function approximation? Coming up
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
is unknown.
How do we extend to d dimensions?
6/17

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f, g H
= f(0)g(0) +
1
2
1
−1
= f(t)
sol(f) =
1
−1
f(x) dx, quadn
(f) =
2
n
n
i=1
f
2i − n − 1
n
,
|sol(f) − quadn
(f)| ηn H
f H
=
2
√
3n
[f(0)]2 +
1
2
1
−1
[f (x)]2 dx
ηn H
(f) =
n
i=1
wif(xi)? Yes
is unknown.
How do we extend to d dimensions? Coming up
6/17

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Complexity of Integration
What is the best order of error possible for integrands in H?
Let app(·, ε) be any algorithm that guarantees 1
−1
f(x) dx − app({(xi, f(xi))}n
i=1
, ε) ε
for all f ∈ BR = {f : f H
R}.
Apply that algorithm to the zero function. Let {xi}n
i=1
be the particular data sites chosen.
Deﬁne the fooling function, f, that looks like the zero function at the data sites.
7/17

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Complexity of Integration
What is the best order of error possible for integrands in H?
Let app(·, ε) be any algorithm that guarantees 1
−1
f(x) dx − app({(xi, f(xi))}n
i=1
, ε) ε
for all f ∈ BR = {f : f H
R}.
Apply that algorithm to the zero function. Let {xi}n
i=1
be the particular data sites chosen.
Deﬁne the fooling function, f, that looks like the zero function at the data sites.
Then
ε
1
2
[|sol(f) − app({(xi, f(xi))}n
i=1
, ε)| + |sol(−f) − app({(xi, −f(xi))}n
i=1
, ε)|]
=
1
2
[|sol(f) − app({(xi, 0)}n
i=1
, ε)| + |− sol(f) − app({(xi, 0)}n
i=1
, ε)|]
|sol(f)|
f H
n
Thus, even the best algorithm must use at least
f H
ε
data sites. This is called the
complexity of the problem.
7/17

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Function Approximation Using Kernel Methods aka Kriging
The best approximation to f ∈ H is given by a weighted average of function values
f(x) = yTw(x), where w(x) = K−1k(x) (O(n3) operations)
y =



f(x1)
.
.
.
f(xn)



, k(x) =



K(x, x1)
.
.
.
K(x, xn)



, K =



K(x1, x1) · · · K(x1, xn)
.
.
.
...
.
.
.
K(xn, x1) · · · K(xn, xn)



8/17

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y =



f(x1)
.
.
.
f(xn)



, k(x) =



K(x, x1)
.
.
.
K(x, xn)



, K =



K(x1, x1) · · · K(x1, xn)
.
.
.
...
.
.
.
K(xn, x1) · · · K(xn, xn)



f(x) − f(x) K(x, x) − kT(x)K−1k(x) f H
=: Qn(x) f H
8/17

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y =



f(x1)
.
.
.
f(xn)



, k(x) =



K(x, x1)
.
.
.
K(x, xn)



, K =



K(x1, x1) · · · K(x1, xn)
.
.
.
...
.
.
.
K(xn, x1) · · · K(xn, xn)



f(x) − f(x) K(x, x) − kT(x)K−1k(x) f H
=: Qn(x) f H
Qn := Qn 2
=
1
−1
K(x, x) dx − trace(K−1A), A :=
1
−1
k(x)kT(x) dx
8/17

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Summary for One Dimension
Methods based on reproducing kernels can yield
sol(f) − app {(xi, f(xi))}n
i=1
, ε errn({xi}n
i=1
, f) ε,
with n f H
ε−r provided f H
can be bounded
where r depends on the smoothness of the kernel/H
Computing approximation and/or the quality measure of the data sites may require O(n2)
or O(n3) operations
Quality measures for the data sites only depend on the kernels and the data sites, not on
the function values at the data sites
9/17

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Overall Picture
Methods based on reproducing kernels can be extended to multiple dimensions to yield
sol(f) − app {(xi, f(xi))}n
i=1
, ε errn({xi}n
i=1
, f) ε,
with n f H
C(d)ε−r(d) provided f H
can be bounded
Hilbert spaces of functions having derivatives of up to total order s typically have r(d) s/d
Hilbert spaces of functions having mixed derivatives of up to order s in each direction
typically have r(d) s
Even if r is independent of d, C(d) can be worse than polynomial
Computing approximation and/or the quality measure of the data sites may require
O(dn2) or O(dn3) operations
Quality measures for the data sites only depend on the kernels and the data sites, not on
the function values at the data sites
Novak, E. & Woźniakowski, H. Tractability of Multivariate Problems Volume I: Linear Information. EMS Tracts in
Mathematics 6 (European Mathematical Society, Zürich, 2008), Novak, E. & Woźniakowski, H. Tractability of
Multivariate Problems Volume II: Standard Information for Functionals. EMS Tracts in Mathematics 12 (European
Mathematical Society, Zürich, 2010), Novak, E. & Woźniakowski, H. Tractability of Multivariate Problems: Volume
III: Standard Information for Operators. EMS Tracts in Mathematics 18 (European Mathematical Society, Zürich,
2012). 10/17

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Product Kernels with Weights
To avoid the curse of dimensionality in the exponent of ε−1 or the coeﬃcient, we typically must
appeal to weighted spaces, i.e., spaces where not all coordinates are equally important, e.g,
Kd(x, t) =
u⊂1:d
γu
j∈u
xj + tj − xj − tj , γu → 0 as max j : j ∈ u or |u| → ∞
f, g H
=
f(0)g(0)
γ∅
+
1
2γ{1}
1
−1
∂f
∂x1
(x1, 0, · · · )
∂g
∂x1
(x1, 0, · · · ) dx1
+
1
2γ{2}
1
−1
∂f
∂x2
(0, x2, 0, · · · )
∂g
∂x2
(0, x1, 0, · · · ) dx2 + · · ·
+
1
4γ{1,2}
1
−1
1
−1
∂2f
∂x1∂x2
(x1, x2, 0, · · · )
∂2g
∂x1∂x2
(x1, x2, 0, · · · ) dx1dx2 + · · ·
The discrepancy, ηn H
, tends to zero as n → ∞ for your favorite data sites if γu → 0 fast
enough. For f H
to be of reasonable size as γu → 0, the importance of the coordinates in u
must vanish. Problem: How do you know which coordinates are most important a priori?
11/17

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These Are a Few of My Favorite Data Sites
Good for integration, not sure how good for function approximation
Niederreiter, H. Random Number Generation and Quasi-Monte Carlo Methods. (SIAM, Philadelphia, 1992),
Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press, Oxford, 1994), Dick, J. &
Pillichshammer, F. Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration.
(Cambridge University Press, Cambridge, 2010). 12/17

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Approximation when Linear Functionals Can Be Evaluated
To understand how diﬃcult things can be in higher dimensions, let us make the problem both
harder (function approximation) and easier (we can evaluate L(f) for any linear functional L we
like). If f(x) =
∞
m=1
fmφm(x) for L2(Ω) orthonormal functions φm, which are simultaneously
orthonormal in H with φm H
= λ−1
m
, and λ1 λ2 · · · , then
f H
=
fm
λm
∞
m=1 2
, Kd(x, t) =
∞
m=1
λ2
m
φm(x)φm(t),
and the optimal algorithm using n linear functionals is
f =
n
m=1
fφm, f − f
2
= fm
∞
m=n+1 2
=
fm
λm
· λm
∞
m=n+1 2
λn+1
f H
13/17

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f(x) =
∞
m=1
fmφm(x), f H
=
fm
λm
∞
m=1 2
, K(x, t) =
∞
m=1
λ2
m
φm(x)φm(t),
f =
n
m=1
fφm, f − f
2
= fm
∞
m=n+1 2
=
fm
λm
· λm
∞
m=n+1 2
λn+1
f H
If Kd =
u⊂1:d
γu
j∈u
˜
Kj(xj, tj), then the λm and φm are of product form, and bounds on the
decay rate of λn+1
in terms of the decay rates of the ˜
λj and the γu are possible.
Mathematics 6 (European Mathematical Society, Zürich, 2008). 14/17

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f(x) =
∞
m=1
fmφm(x), f H
=
fm
λm
∞
m=1 2
, K(x, t) =
∞
m=1
λ2
m
φm(x)φm(t),
f =
n
m=1
fφm, f − f
2
= fm
∞
m=n+1 2
=
fm
λm
· λm
∞
m=n+1 2
λn+1
f H
If Kd =
u⊂1:d
γu
j∈u
˜
Kj(xj, tj), then the λm and φm are of product form, and bounds on the
decay rate of λn+1
in terms of the decay rates of the ˜
λj and the γu are possible.
Proposed research direction:
Assume f is in the union of unit balls of diﬀerent weighted Hilbert spaces with most γu
vanishing, e.g., γu = 0 for all but O(d) pairs u, so one can hope for n d
Use an initial linear functional values to screen for the important coordinate directions.
Assume that high order terms do not occur unless lower order terms occur, etc.
Use further linear functional values to construct an accurate approximation
Mathematics 6 (European Mathematical Society, Zürich, 2008). 14/17

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How Do We Find a Criteria for Sequential Choice of Data Sites?
The existing criteria that for the quality of data sites do not depend on function values. Here is
an untried possibility:
The error depends on the norm of the function, f H
.
Although we do not know f H
, we can compute the norm of the kriging approximation,
f
H
= yK−1y, which no greater than f H
, and perhaps close
If you can partition the domain into regions, you can compute f
H
= yK−1y based
on only a subset of the data from a particular region. Place more points where the norm is
greater. E.g.,
region [−1, 0) [0, 1] [−1, 1]
f
H
= yK−1y 4.76 1.87 5.08
15/17

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A Few More Thoughts
There is a catalog of Hilbert spaces and their kernels
There are connections of low discrepancy designs to traditional designs measures
Berlinet, A. & Thomas-Agnan, C. Reproducing Kernel Hilbert Spaces in Probability and Statistics. (Kluwer
Academic Publishers, Boston, 2004).
H., F. J. & Liu, M. Q. Uniform Designs Limit Aliasing. Biometrika 89, 893–904 (2002). 16/17

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Thank you

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N. Aronszajn. Theory of Reproducing Kernels. Trans. Amer. Math. Soc. 68, 337–404
(1950).
H., F. J. A Generalized Discrepancy and Quadrature Error Bound. Math. Comp. 67,
299–322 (1998).
H., F. J. What Aﬀects the Accuracy of Quasi-Monte Carlo Quadrature?. in Monte Carlo
and Quasi-Monte Carlo Methods 1998 (eds Niederreiter, H. & Spanier, J.)
(Springer-Verlag, Berlin, 2000), 16–55.
Novak, E. & Woźniakowski, H. Tractability of Multivariate Problems Volume I: Linear
Information. EMS Tracts in Mathematics 6 (European Mathematical Society, Zürich,
2008).
Novak, E. & Woźniakowski, H. Tractability of Multivariate Problems Volume II: Standard
Information for Functionals. EMS Tracts in Mathematics 12 (European Mathematical
Society, Zürich, 2010).
17/17

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Novak, E. & Woźniakowski, H. Tractability of Multivariate Problems: Volume III: Standard
Information for Operators. EMS Tracts in Mathematics 18 (European Mathematical
Society, Zürich, 2012).
Niederreiter, H. Random Number Generation and Quasi-Monte Carlo Methods. (SIAM,
Philadelphia, 1992).
Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press,
Oxford, 1994).
Dick, J. & Pillichshammer, F. Digital Nets and Sequences: Discrepancy Theory and
Quasi-Monte Carlo Integration. (Cambridge University Press, Cambridge, 2010).
Berlinet, A. & Thomas-Agnan, C. Reproducing Kernel Hilbert Spaces in Probability and
Statistics. (Kluwer Academic Publishers, Boston, 2004).
H., F. J. & Liu, M. Q. Uniform Designs Limit Aliasing. Biometrika 89, 893–904 (2002).
17/17

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Measures of Error, Discrepancy, and Complexity

Measures of Error, Discrepancy, and Complexity

Fred J. Hickernell

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