In this talk, we give an overview of results on numerical integration in Hermite spaces. These spaces contain functions defined on $\mathbb{R}^d$, and can be characterized by the decay of their Hermite coefficients. We consider the case of exponentially as well as polynomially decaying Hermite coefficients. For numerical integration, we either use Gauss-Hermite quadrature rules or algorithms based on quasi-Monte Carlo rules. We present upper and lower error bounds for these algorithms, and discuss their dependence on the dimension $d$. Furthermore, we comment on open problems for future research.

1. Numerical Integration in Hermite Spaces
Peter Kritzer
Johann Radon Institute for Computational and Applied Mathematics (RICAM)
Austrian Academy of Sciences
Linz, Austria
Joint work with C. Irrgeher (RICAM, Linz),
G. Leobacher (KFU Graz)
and F. Pillichshammer (JKU Linz)
SAMSI QMC Workshop 2017, August 2017
Research supported by the Austrian Science Fund, Project F5506-N26
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2. Johann Radon Institute for Computational and Applied Mathematics
Introduction
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3. Johann Radon Institute for Computational and Applied Mathematics
Approximate the d-dimensional integral
Id (f) :=
Rd
f(x)ϕd (x) dx
for f in a normed function space (Hd , · Hd
) by a linear algorithm
AN,d =
N
k=1
wk f(xk ),
where
ϕd (x) =
1
(2π)d/2
exp
−x · x
2
.
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4. Johann Radon Institute for Computational and Applied Mathematics
Integration error of AN,d for f ∈ Hd ,
err(AN,d , f) = Id (f) − AN,d (f).
Worst case integration error of AN,d ,
ewor
(AN,d , Hd ) := sup
f∈Hd
f Hd
≤1
|err(AN,d , f)| .
N-th minimal worst case error,
ewor
(N, Hd ) := inf
AN,d
ewor
(AN,d , Hd ).
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5. Johann Radon Institute for Computational and Applied Mathematics
Hermite spaces
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6. Johann Radon Institute for Computational and Applied Mathematics
Hermite polynomials Hk , k ∈ Nd
0 form an ONB of L2(Rd
, ϕd ).
The k-th (k ∈ N0) normalized univariate (probabilists’) Hermite
polynomial is
Hk (x) =
(−1)k
√
k!
exp(x2
/2)
dk
dxk
exp(−x2
/2).
Multivariate version:
For k = (k1, . . . , kd ) ∈ Nd
0 and x = (x1, . . . , xd ) ∈ Rd
,
Hk (x) =
d
j=1
Hkj
(xj ).
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7. Johann Radon Institute for Computational and Applied Mathematics
k-th Hermite coefficient of f ∈ L2(Rd
, ϕd )
ˆf(k) :=
Rd
f(x)Hk (x)ϕd (x) dx.
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8. Johann Radon Institute for Computational and Applied Mathematics
k-th Hermite coefficient of f ∈ L2(Rd
, ϕd )
ˆf(k) :=
Rd
f(x)Hk (x)ϕd (x) dx.
Define a positive function rd : Nd
0 → R,
rd (k) =
d
j=1
r(kj ),
where r : N0 → R is positive, r(0) = 1.
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9. Johann Radon Institute for Computational and Applied Mathematics
k-th Hermite coefficient of f ∈ L2(Rd
, ϕd )
ˆf(k) :=
Rd
f(x)Hk (x)ϕd (x) dx.
Define a positive function rd : Nd
0 → R,
rd (k) =
d
j=1
r(kj ),
where r : N0 → R is positive, r(0) = 1.
If r depends on j ∈ {1, . . . , d}: indicated by writing rj (kj ).
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10. Johann Radon Institute for Computational and Applied Mathematics
k-th Hermite coefficient of f ∈ L2(Rd
, ϕd )
ˆf(k) :=
Rd
f(x)Hk (x)ϕd (x) dx.
Define a positive function rd : Nd
0 → R,
rd (k) =
d
j=1
r(kj ),
where r : N0 → R is positive, r(0) = 1.
If r depends on j ∈ {1, . . . , d}: indicated by writing rj (kj ).
Two choices for rd in this talk:
rpol
d : polynomial decay of r, e.g., r(k) k−α
for α ∈ N.
rexp
d : exponential decay of r, e.g., r(k) ωk
for ω ∈ (0, 1).
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11. Johann Radon Institute for Computational and Applied Mathematics
Hermite space (depending on rd ):
Hd,rd
:= f : Rd
→ R : f continuous,
Rd
(f(x))2
ϕd (x) dx < ∞, f d,rd
< ∞ .
Norm:
f d,rd
:=


k∈Nd
0
1
rd (k)
(ˆf(k))2


1/2
.
Inner product:
f, g d,rd
:=
k∈Nd
0
1
rd (k)
ˆf(k)ˆg(k)
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12. Johann Radon Institute for Computational and Applied Mathematics
Hermite space Hd,rd
is reproducing kernel Hilbert space with kernel
Kd,rd
(x, y) :=
k∈Nd
0
rd (k)Hk (x)Hk (y).
Introduced in
C. Irrgeher, G. Leobacher. High-dimensional integration on the
Rd
, weighted Hermite spaces, and orthogonal transforms. J.
Complexity 31, 174–205, 2015.
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13. Johann Radon Institute for Computational and Applied Mathematics
The case rexp
d
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14. Johann Radon Institute for Computational and Applied Mathematics
Studied in the paper
C. Irrgeher, P. Kritzer, G. Leobacher, F. Pillichshammer.
Integration in Hermite spaces of analytic functions. J. Complexity
31, 380–404, 2015.
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15. Johann Radon Institute for Computational and Applied Mathematics
Studied in the paper
C. Irrgeher, P. Kritzer, G. Leobacher, F. Pillichshammer.
Integration in Hermite spaces of analytic functions. J. Complexity
31, 380–404, 2015.
Choose ω ∈ (0, 1) and two real sequences a = {aj }, b = {bj },
1 ≤ a1 ≤ a2 ≤ · · · and inf
j≥1
bj ≥ 1
and set
rexp
d (k) =
d
j=1
rj (kj ), with rj (k) = ωaj k
bj
.
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16. Johann Radon Institute for Computational and Applied Mathematics
Study integration in Hd,rexp
d
.
Hermite coefficients of functions in Hd,rexp
d
decrease exponentially
fast.
Hd,rexp
d
contains analytic functions.
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17. Johann Radon Institute for Computational and Applied Mathematics
First goal: show exponential error convergence.
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18. Johann Radon Institute for Computational and Applied Mathematics
First goal: show exponential error convergence.
Exponential convergence (EXP) if there exist q ∈ (0, 1) and
Cd , Md , pd > 0 for all d such that
ewor
(N, Hd,rexp
d
) ≤ Cd q (N/Md ) pd
for all N ∈ N. (1)
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19. Johann Radon Institute for Computational and Applied Mathematics
First goal: show exponential error convergence.
Exponential convergence (EXP) if there exist q ∈ (0, 1) and
Cd , Md , pd > 0 for all d such that
ewor
(N, Hd,rexp
d
) ≤ Cd q (N/Md ) pd
for all N ∈ N. (1)
Uniform exponential convergence (UEXP) if pd = p > 0 for all d ∈ N
in (1).
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20. Johann Radon Institute for Computational and Applied Mathematics
First goal: show exponential error convergence.
Exponential convergence (EXP) if there exist q ∈ (0, 1) and
Cd , Md , pd > 0 for all d such that
ewor
(N, Hd,rexp
d
) ≤ Cd q (N/Md ) pd
for all N ∈ N. (1)
Uniform exponential convergence (UEXP) if pd = p > 0 for all d ∈ N
in (1).
Interest in largest possible rate pd (or p).
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21. Johann Radon Institute for Computational and Applied Mathematics
First goal: show exponential error convergence.
Exponential convergence (EXP) if there exist q ∈ (0, 1) and
Cd , Md , pd > 0 for all d such that
ewor
(N, Hd,rexp
d
) ≤ Cd q (N/Md ) pd
for all N ∈ N. (1)
Uniform exponential convergence (UEXP) if pd = p > 0 for all d ∈ N
in (1).
Interest in largest possible rate pd (or p).
Second goal: Study dependence of ewor
(N, Hd,rexp
d
) on d.
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22. Johann Radon Institute for Computational and Applied Mathematics
Use Gauss-Hermite-rules.
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23. Johann Radon Institute for Computational and Applied Mathematics
Use Gauss-Hermite-rules.
Univariate case: Gauss-Hermite rule of order N:
GN(f) =
N
i=1
αi f(xi ).
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24. Johann Radon Institute for Computational and Applied Mathematics
Use Gauss-Hermite-rules.
Univariate case: Gauss-Hermite rule of order N:
GN(f) =
N
i=1
αi f(xi ).
Nodes x1, . . . , xN ∈ R zeros of HN.
Weights
αi =
1
NH2
N−1(xi )
.
Rule is exact for all polynomials of degree less than 2N.
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25. Johann Radon Institute for Computational and Applied Mathematics
d-variate case:
Use the product rule
GN = GN1
⊗ · · · ⊗ GNd
,
with N = N1 · · · Nd .
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26. Johann Radon Institute for Computational and Applied Mathematics
d-variate case:
Use the product rule
GN = GN1
⊗ · · · ⊗ GNd
,
with N = N1 · · · Nd .
Proposition 1
Let GN be a d-variate Gauss-Hermite product rule as above. Then
(ewor
(GN, Hd,rexp
d
))2
≤ −1 +
d
j=1
1 + ωaj (2Nj )
bj
√
8π
1 − ω2
.
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27. Johann Radon Institute for Computational and Applied Mathematics
Use the previous proposition and choose Nj = Nj (aj , bj ).
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28. Johann Radon Institute for Computational and Applied Mathematics
Use the previous proposition and choose Nj = Nj (aj , bj ).
Then GN can yield EXP, and UEXP if B :=
∞
j=1 b−1
j < ∞. E.g.,
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29. Johann Radon Institute for Computational and Applied Mathematics
Use the previous proposition and choose Nj = Nj (aj , bj ).
Then GN can yield EXP, and UEXP if B :=
∞
j=1 b−1
j < ∞. E.g.,
Theorem 1 (Irrgeher, K., Leobacher, Pillichshammer, 2015)
Let B < ∞. Let ε > 0 be given, and choose
Nj :=







log
√
8π
1−ω2
π2
6
j2
log(1+ε2)
aj 2bj log ω−1


1/bj





.
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30. Johann Radon Institute for Computational and Applied Mathematics
Use the previous proposition and choose Nj = Nj (aj , bj ).
Then GN can yield EXP, and UEXP if B :=
∞
j=1 b−1
j < ∞. E.g.,
Theorem 1 (Irrgeher, K., Leobacher, Pillichshammer, 2015)
Let B < ∞. Let ε > 0 be given, and choose
Nj :=







log
√
8π
1−ω2
π2
6
j2
log(1+ε2)
aj 2bj log ω−1


1/bj





.
Then
ewor
(GN, Hd,rexp
d
) ≤ ε,
and for any δ > 0 there exists Cδ > 0 such that
N ≤ Cδ logB+δ
(1 + ε−1
).
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31. Johann Radon Institute for Computational and Applied Mathematics
Further results on EXP/UEXP and tractability in the paper.
Related results: L2-approximation in Hd,rexp
d
:
C. Irrgeher, P. Kritzer, F. Pillichshammer, H. Wo´zniakowski.
Approximation in Hermite spaces of smooth functions. J. Approx.
Th. 207, 98–126, 2016.
C. Irrgeher, P. Kritzer, F. Pillichshammer, H. Wo´zniakowski.
Tractability of multivariate approximation defined over Hilbert
spaces with exponential weights. J. Approx. Th. 207, 301–338,
2016.
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32. Johann Radon Institute for Computational and Applied Mathematics
The case rpol
d
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33. Johann Radon Institute for Computational and Applied Mathematics
J. Dick, C. Irrgeher, G. Leobacher, F. Pillichshammer. On the
optimal order of integration in Hermite spaces with finite
smoothness. Submitted, 2017. https://arxiv.org/abs/1608.06061
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34. Johann Radon Institute for Computational and Applied Mathematics
J. Dick, C. Irrgeher, G. Leobacher, F. Pillichshammer. On the
optimal order of integration in Hermite spaces with finite
smoothness. Submitted, 2017. https://arxiv.org/abs/1608.06061
Let α ∈ {1, 2, . . .} and set
rpol
d (k) =
d
j=1
rj (kj ), with rj (k)
1
kα
(precise definition of rj in the paper).
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35. Johann Radon Institute for Computational and Applied Mathematics
J. Dick, C. Irrgeher, G. Leobacher, F. Pillichshammer. On the
optimal order of integration in Hermite spaces with finite
smoothness. Submitted, 2017. https://arxiv.org/abs/1608.06061
Let α ∈ {1, 2, . . .} and set
rpol
d (k) =
d
j=1
rj (kj ), with rj (k)
1
kα
(precise definition of rj in the paper).
Functions in Hd,rpol
d
are α times (weakly) differentiable; norm can be
re-written as Sobolev-type norm using derivatives.
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36. Johann Radon Institute for Computational and Applied Mathematics
Theorem 2 (Dick, Irrgeher, Leobacher, Pillichshammer, 2017)
Let d, α ∈ N. Then for all N ∈ N it is true that
ewor
(N, Hd,rpol
d
) ≥ Cd,α
(log N)
d−1
2
Nα
,
where Cd,α depends on d and α.
Proof: Fooling function argument.
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37. Johann Radon Institute for Computational and Applied Mathematics
Upper bound:
Truncate the domain Rd
to [−b, b] with b = (b, b, . . . , b) and
b = 2 α log N.
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38. Johann Radon Institute for Computational and Applied Mathematics
Upper bound:
Truncate the domain Rd
to [−b, b] with b = (b, b, . . . , b) and
b = 2 α log N.
Linear transformation T : [0, 1]d
→ [−b, b].
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39. Johann Radon Institute for Computational and Applied Mathematics
Upper bound:
Truncate the domain Rd
to [−b, b] with b = (b, b, . . . , b) and
b = 2 α log N.
Linear transformation T : [0, 1]d
→ [−b, b].
Integration nodes: xk = T(zk ), k = 1, . . . , N, where zk stem from
a digital higher-order net in [0, 1]d
.
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40. Johann Radon Institute for Computational and Applied Mathematics
Upper bound:
Truncate the domain Rd
to [−b, b] with b = (b, b, . . . , b) and
b = 2 α log N.
Linear transformation T : [0, 1]d
→ [−b, b].
Integration nodes: xk = T(zk ), k = 1, . . . , N, where zk stem from
a digital higher-order net in [0, 1]d
.
Integration weights wk = (2b)d
ϕd (T(zk ))/N, k = 1, . . . , N in
AN,d =
N
k=1
wk f(xk ).
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41. Johann Radon Institute for Computational and Applied Mathematics
Theorem 3 (Dick, Irrgeher, Leobacher, Pillichshammer, 2017)
Let d, α ∈ N. Then
ewor
(AN,d , Hd,rpol
d
) ≤ Cd,α
(log N)d 2α+3
4 − 1
2
Nα
,
where Cd,α depends on d and α.
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42. Johann Radon Institute for Computational and Applied Mathematics
Theorem 3 (Dick, Irrgeher, Leobacher, Pillichshammer, 2017)
Let d, α ∈ N. Then
ewor
(AN,d , Hd,rpol
d
) ≤ Cd,α
(log N)d 2α+3
4 − 1
2
Nα
,
where Cd,α depends on d and α.
Optimal main convergence order N−α
, but presumably
non-optimal power of (log N)-term,
Cd,α depends on d.
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43. Johann Radon Institute for Computational and Applied Mathematics
Open problems
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44. Johann Radon Institute for Computational and Applied Mathematics
Main open problem:
Can we vanquish the curse of dimensionality for the case rpol
d ?
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45. Johann Radon Institute for Computational and Applied Mathematics
Main open problem:
Can we vanquish the curse of dimensionality for the case rpol
d ?
The upper bound in Theorem 3 was obtained by analyzing
(1) the error of approximating the integral outside of [−b, b] by 0,
(2) the error of approximating the integral within [−b, b] by AN,d .
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46. Johann Radon Institute for Computational and Applied Mathematics
Main open problem:
Can we vanquish the curse of dimensionality for the case rpol
d ?
The upper bound in Theorem 3 was obtained by analyzing
(1) the error of approximating the integral outside of [−b, b] by 0,
(2) the error of approximating the integral within [−b, b] by AN,d .
Common way to reduce the curse of dimensionality in QMC: Use
weights in the sense of Sloan/Wo´zniakowski.
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47. Johann Radon Institute for Computational and Applied Mathematics
Main open problem:
Can we vanquish the curse of dimensionality for the case rpol
d ?
The upper bound in Theorem 3 was obtained by analyzing
(1) the error of approximating the integral outside of [−b, b] by 0,
(2) the error of approximating the integral within [−b, b] by AN,d .
Common way to reduce the curse of dimensionality in QMC: Use
weights in the sense of Sloan/Wo´zniakowski.
Natural approach: Use weights and adjust the bounds of the box
[−b, b] coordinate-wise.
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48. Johann Radon Institute for Computational and Applied Mathematics
Main open problem:
Can we vanquish the curse of dimensionality for the case rpol
d ?
The upper bound in Theorem 3 was obtained by analyzing
(1) the error of approximating the integral outside of [−b, b] by 0,
(2) the error of approximating the integral within [−b, b] by AN,d .
Common way to reduce the curse of dimensionality in QMC: Use
weights in the sense of Sloan/Wo´zniakowski.
Natural approach: Use weights and adjust the bounds of the box
[−b, b] coordinate-wise.
Problem: Choosing the weights to reduce (1) increases (2), and vice
versa.
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49. Johann Radon Institute for Computational and Applied Mathematics
Solutions to this problem?
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50. Johann Radon Institute for Computational and Applied Mathematics
Solutions to this problem?
Use a different way of estimating (1) and (2) to obtain better
bounds. Is this possible at all?
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51. Johann Radon Institute for Computational and Applied Mathematics
Solutions to this problem?
Use a different way of estimating (1) and (2) to obtain better
bounds. Is this possible at all?
Approximate the integral outside of [−b, b] by something more
sophisticated than 0. What is “something more sophisticated” ?
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52. Johann Radon Institute for Computational and Applied Mathematics
Solutions to this problem?
Use a different way of estimating (1) and (2) to obtain better
bounds. Is this possible at all?
Approximate the integral outside of [−b, b] by something more
sophisticated than 0. What is “something more sophisticated” ?
Use a different approach altogether. Which approach would
work?
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53. Johann Radon Institute for Computational and Applied Mathematics
Conclusion
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54. Johann Radon Institute for Computational and Applied Mathematics
We studied numerical integration in Hermite spaces.
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55. Johann Radon Institute for Computational and Applied Mathematics
We studied numerical integration in Hermite spaces.
Case of exponentially decaying Hermite coefficients: many
results (even for function approximation).
Peter Kritzer Numerical Integration in Hermite Spaces 27

56. Johann Radon Institute for Computational and Applied Mathematics
We studied numerical integration in Hermite spaces.
Case of exponentially decaying Hermite coefficients: many
results (even for function approximation).
Case of polynomially decaying Hermite coefficients: higher order
digital nets can be used.
Peter Kritzer Numerical Integration in Hermite Spaces 27

57. Johann Radon Institute for Computational and Applied Mathematics
We studied numerical integration in Hermite spaces.
Case of exponentially decaying Hermite coefficients: many
results (even for function approximation).
Case of polynomially decaying Hermite coefficients: higher order
digital nets can be used.
Curse of dimensionality remains unresolved problem, not clear
which strategy will work.
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58. Johann Radon Institute for Computational and Applied Mathematics
Thank you very much for your attention.
Peter Kritzer Numerical Integration in Hermite Spaces 28