Fred J. Hickernell
June 19, 2018
79

# Ian Sloan 80th Birthday

Presentation on lattice rules at a conference to honor Ian Sloan on the occasion of his 80th birthday

June 19, 2018

## Transcript

1. The Advantages of Sampling with Integration Lattices
Fred J. Hickernell
Department of Applied Mathematics
Center for Interdisciplinary Scientiﬁc Computation
Illinois Institute of Technology
[email protected] mypages.iit.edu/~hickernell
Happy Birthday and Congratulations to Ian!
Thanks to the organizers, the GAIL team, NSF-DMS-1522687 and NSF-DMS-1638521 (SAMSI)
Ian Sloan’s 80th Birthday Conference, June 18–19, 2018

2. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
They ﬁll space better than IID
They provide superior approximations to the high dimensional
integrals when the coordinates decrease in importance
We have ways to choose the sample size adaptively
2/19

3. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Integration in One Dimension
In calculus we learn that the rectangle rule satisﬁes
1
0
f(x) dx −
1
n
n−1
i=0
f
i
n

2

3n
f
2
We might have missed that
1
0
f(x) dx −
1
n
n−1
i=0
f
i
n
2 ζ(2r)
(2πn)r
f(r)
2
,
provided f, f , . . . , f(r−1) are continuous and periodic
What can be done for
[0,1)d
f(x) dx for d = 2, 6, 12, 100, 1000?
3/19

4. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
d = 6, n = 64, Grid
4/19

5. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
d = 6, n = 64, Near Orthogonal Array of Strength 3 Resembles Grid in 3-D
5/19

6. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
d = 6, n = 64, Near Orthogonal Array of Strength 2 Resembles a Grid in 2-D
6/19

7. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
d = 6, n = 64, Good Lattice Point Set Resembles a Grid in 1-D
7/19

8. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
d = 6, n = 64, The Competition: IID
8/19

9. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Sloan-Kachoyan Uniﬁcation
Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d
Grid (Strength d OA) C =
1
2
Id
Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples.
SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford
University Press, Oxford, 1994). 9/19

10. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Sloan-Kachoyan Uniﬁcation
Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d
Grid (Strength d OA) C =
1
2
Id
Strength 3 Near OA C =
1
4

1 0 0 1 2 3
0 1 0 2 3 1
0 0 1 3 1 2

Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples.
SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford
University Press, Oxford, 1994). 9/19

11. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Sloan-Kachoyan Uniﬁcation
Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d
Grid (Strength d OA) C =
1
2
Id
Strength 3 Near OA C =
1
4

1 0 0 1 2 3
0 1 0 2 3 1
0 0 1 3 1 2

Strength 2 Near OA C =
1
8
1 0 1 3 2 3
0 1 1 2 3 1
Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples.
SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford
University Press, Oxford, 1994). 9/19

12. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Sloan-Kachoyan Uniﬁcation
Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d
Grid (Strength d OA) C =
1
2
Id
Strength 3 Near OA C =
1
4

1 0 0 1 2 3
0 1 0 2 3 1
0 0 1 3 1 2

Strength 2 Near OA C =
1
8
1 0 1 3 2 3
0 1 1 2 3 1
Good Lattice Point Set C =
1
64
31 11 23 3 19 13
Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples.
SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford
University Press, Oxford, 1994). 9/19

13. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Sloan-Kachoyan Uniﬁcation
Rank s lattice sets take the form {iC ∩ [0, 1)d : i ∈ Zs}, C is s × d
Grid (Strength d OA) C =
1
2
Id
Strength 3 Near OA C =
1
4

1 0 0 1 2 3
0 1 0 2 3 1
0 0 1 3 1 2

Strength 2 Near OA C =
1
8
1 0 1 3 2 3
0 1 1 2 3 1
Good Lattice Point Set C =
1
64
31 11 23 3 19 13
How do we decide which kind of lattice is best?
Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples.
SIAM J. Numer. Anal. 24, 116–128 (1987), Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford
University Press, Oxford, 1994). 9/19

14. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
An Ian Story
My ﬁrst interaction with Ian was when he acted as editor
for my ﬁrst quasi-Monte Carlo manuscript .
Ian is always quite encouraging,
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected
printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19

15. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
An Ian Story
My ﬁrst interaction with Ian was when he acted as editor
for my ﬁrst quasi-Monte Carlo manuscript .
Ian is always quite encouraging,
and his feedback included page after page of encourage-
ment,
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected
printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19

16. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
An Ian Story
My ﬁrst interaction with Ian was when he acted as editor
for my ﬁrst quasi-Monte Carlo manuscript .
Ian is always quite encouraging,
and his feedback included page after page of encourage-
ment,
which led to a much better paper. Thank you, Ian.
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected
printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 10/19

17. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Cubature Error For Lattice Sets
f(x) =
k∈Zd
f(k)e2π

−1k·x for f(k) =
[0,1)d
f(x)e−2π

−1k·x
If {xi}n−1
i=0
is a lattice set, B2(x) = x2 − x + 1/6 is the quadratic Bernoulli polynomial, and
w(k1, . . . , kd) :=
kj=0
kj , e.g., w(0, −3, 5, 0) = 15, ∂uf :=
[0,1)u
∂uf
∂xu
dxu ∀u ∈ {1, . . . , d}
then via Fourier or reproducing kernel Hilbert space analysis
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi)

−1 +
1
n
n−1
i=0
d
j=1
[1 + 2π2B2(xĳ)] f(k)w(k)
k 2
Korobov
−1 +
1
n
n−1
i=0
d
j=1
1 +
1
2
B2(xĳ) ∂uf 2 u 2
Sobolev
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected
printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 11/19

18. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Cubature Error For Lattice Sets
f(x) =
k∈Zd
f(k)e2π

−1k·x for f(k) =
[0,1)d
f(x)e−2π

−1k·x
If {xi}n−1
i=0
is a lattice set, B2(x) = x2 − x + 1/6 is the quadratic Bernoulli polynomial, and
w(k1, . . . , kd) :=
kj=0
kj , e.g., w(0, −3, 5, 0) = 15, ∂uf :=
[0,1)u
∂uf
∂xu
dxu ∀u ∈ {1, . . . , d}
then via Fourier or reproducing kernel Hilbert space analysis
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
2
B2(xĳ)
O(n−1+δ)
∂uf 2
γ|u|
u 2
Should γ = 1 or 2π or ...?
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer. Anal. 33. corrected
printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016 (1996). 12/19

19. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
The Choice of Lattice Points Depends on γ
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
2
B2(xĳ)
∂uf 2
γ|u|
u 2
Korobov γ = 2π
Sobolev γ = 1
13/19

20. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
2
B2(xĳ)
∂uf 2
γ|u|
u 2
14/19

21. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Sloan-Woźniakowski Discovery of Coordinate Weights
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
j
2
B2(xĳ)
∂uf 2
j∈u
γj
u 2
Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Eﬃcient for High Dimensional
Integrals? J. Complexity 14, 1–33 (1998). 14/19

22. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Sloan-Woźniakowski Discovery of Coordinate Weights
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
j
2
B2(xĳ)
∂uf 2
j∈u
γj
u 2
An idea that launched 1000 papers
Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Eﬃcient for High Dimensional
Integrals? J. Complexity 14, 1–33 (1998). 14/19

23. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
A Few of Those 1000 Papers
Thank you, Ian, for our fruitful collaboration.
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and
Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On
Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo
Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. &
Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911
(2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1
Approximation
over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. &
Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and
Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19

24. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
A Few of Those 1000 Papers
Thank you, Ian, for our fruitful collaboration.
Ian’s friend: “Ian collaborates with everybody!”
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and
Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On
Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo
Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. &
Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911
(2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1
Approximation
over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. &
Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and
Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19

25. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
A Few of Those 1000 Papers
Thank you, Ian, for our fruitful collaboration.
Ian’s friend: “Ian collaborates with everybody!”
Ian’s UNSW colleague: “Ian does the work of ten.”
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over Bounded and
Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. On
Tractability of Weighted Integration for Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo
Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71, H., F. J., Sloan, I. H. &
Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate Integration. Math. Comp. 73, 1903–1911
(2004), H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for Weighted L1
Approximation
over Bounded or Unbounded Regions. SIAM J. Numer. Anal. 43, 1003–1020 (2005), H., F. J., Sloan, I. H. &
Wasilkowski, G. W. The Strong Tractability of Multivariate Integration Using Lattice Rules. in Monte Carlo and
Quasi-Monte Carlo Methods 2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273. 15/19

26. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
In Practice, How Large Should n Be?
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
j
2
B2(xĳ)
∂uf 2
j∈u
γj
u 2
ε
H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for Cubature. in Contemporary
Computational Mathematics — a celebration of the 80th birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. &
Woźniakowski, H.) (Springer-Verlag, 2018), 597–619. doi:10.1007/978-3-319-72456-0, Jagadeeswaran, R. &
H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. 16/19

27. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
In Practice, How Large Should n Be?
[0,1]d
f(x) −
1
n
n−1
i=0
f(xi) −1 +
1
n
n−1
i=0
d
j=1
1 +
γ2
j
2
B2(xĳ)
∂uf 2
j∈u
γj
u 2
ε
Look at the discrete Fourier coeﬃcients (O(n log n) additional cost) and infer the decay rate of
true coeﬃcients assuming that the integrand lies in a cone C of nice functions.
H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for Cubature. in Contemporary
Computational Mathematics — a celebration of the 80th birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. &
Woźniakowski, H.) (Springer-Verlag, 2018), 597–619. doi:10.1007/978-3-319-72456-0, Jagadeeswaran, R. &
H., F. J. Automatic Bayesian Cubature. in preparation. 2018+. 16/19

28. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Option Pricing
fair price =
Rd
e−rT max

1
d
d
j=1
Sj − K, 0

e−zTz/2
(2π)d/2
dz ≈ \$13.12
Sj = S0e(r−σ2/2)jT/d+σxj = stock price at time jT/d,
x = Az, AAT = Σ = min(i, j)T/d
d
i,j=1
, T = 1/4, d = 13 here
Error Median Worst 10% Worst 10%
Tolerance Method Error Accuracy n Time (s)
1E−2 IID diﬀ 2E−3 100% 6.1E7 33.000
1E−2 Sh. Latt. PCA 1E−3 100% 1.6E4 0.041
1E−2 Bayes. Latt. PCA 2E−3 99% 1.6E4 0.051
Choi, S.-C. T., Ding, Y., H., F. J., Jiang, L., Ll . A. Jiménez Rugama, Li, D., Jagadeeswaran, R., Tong, X.,
Zhang, K., et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2). MATLAB software.
2013–2017. http://gailgithub.github.io/GAIL_Dev/. 17/19

29. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
They ﬁll space better than IID
They provide superior approximations to the high dimensional
integrals when the coordinates decrease in importance
We have ways to choose the sample size adaptively
18/19

30. Thank you
Slides available on SpeakerDeck at
speakerdeck.com/fjhickernell/ian-sloan-80th-birthday

31. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
Sloan, I. H. & Kachoyan, P. J. Lattice Methods for Multiple Integration: Theory, Error
Analysis and Examples. SIAM J. Numer. Anal. 24, 116–128 (1987).
Sloan, I. H. & Joe, S. Lattice Methods for Multiple Integration. (Oxford University Press,
Oxford, 1994).
H., F. J. Quadrature Error Bounds with Applications to Lattice Rules. SIAM J. Numer.
Anal. 33. corrected printing of Sections 3-6 in ibid., 34 (1997), 853–866, 1995–2016
(1996).
Sloan, I. H. & Woźniakowski, H. When Are Quasi-Monte Carlo Algorithms Eﬃcient for
High Dimensional Integrals? J. Complexity 14, 1–33 (1998).
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration over
Bounded and Unbounded Regions in Rs. Math. Comp. 73, 1885–1901 (2004).
19/19

32. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Tractability of Weighted Integration for
Certain Banach Spaces of Functions. in Monte Carlo and Quasi-Monte Carlo Methods
2002 (ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 51–71.
H., F. J., Sloan, I. H. & Wasilkowski, G. W. On Strong Tractability of Weighted Multivariate
Integration. Math. Comp. 73, 1903–1911 (2004).
H., F. J., Sloan, I. H. & Wasilkowski, G. W. A Piece-Wise Constant Algorithm for
Weighted L1
Approximation over Bounded or Unbounded Regions. SIAM J. Numer. Anal.
43, 1003–1020 (2005).
H., F. J., Sloan, I. H. & Wasilkowski, G. W. The Strong Tractability of Multivariate
Integration Using Lattice Rules. in Monte Carlo and Quasi-Monte Carlo Methods 2002
(ed Niederreiter, H.) (Springer-Verlag, Berlin, 2004), 259–273.
H., F. J., Jiménez Rugama, Ll. A. & Li, D. Adaptive Quasi-Monte Carlo Methods for
Cubature. in Contemporary Computational Mathematics — a celebration of the 80th
19/19

33. Points in a Cube Error Analysis Coordinate Weights When to Stop Summary References
birthday of Ian Sloan (eds Dick, J., Kuo, F. Y. & Woźniakowski, H.) (Springer-Verlag,
2018), 597–619. doi:10.1007/978-3-319-72456-0.
Jagadeeswaran, R. & H., F. J. Automatic Bayesian Cubature. in preparation. 2018+.
Choi, S.-C. T. et al. GAIL: Guaranteed Automatic Integration Library (Versions 1.0–2.2).
MATLAB software. 2013–2017. http://gailgithub.github.io/GAIL_Dev/.
(ed Niederreiter, H.) Monte Carlo and Quasi-Monte Carlo Methods 2002.
(Springer-Verlag, Berlin, 2004).
19/19