A fundamental numerical problem in many sciences is to compute integrals. These integrals can often be expressed as expectations and then approximated by sampling methods. Monte Carlo sampling is very competitive in high dimensions, but has a slow rate of convergence. One reason for this slowness is that the MC points form clusters and gaps. Quasi-Monte Carlo methods greatly reduce such clusters and gaps, and under modest smoothness demands on the integrand they can greatly improve accuracy. This can even take place in problems of surprisingly high dimension. This talk will introduce the basics of QMC and randomized QMC. It will include discrepancy and the Koksma-Hlawka inequality, some digital constructions and some randomized QMC methods that allow error estimation and sometimes bring improved accuracy.

1. QMC tutorial 1
Quasi-Monte Carlo
A tutorial introduction
Art B. Owen
Stanford University
MCQMC 2018 will be in Rennes, July 1–8 2018
SAMSI Workshop on QMC, August 2017

2. QMC tutorial 2
MC and QMC and other points
Monte Carlo (MC) and quasi-Monte Carlo (QMC) methods come down to
sampling the input space of a function.
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Top row, left to right
MC, grid, and two QMC methods
Top row, left to right
Two more QMC, sparse grid, blue noise
SAMSI Workshop on QMC, August 2017

3. QMC tutorial 3
Outline
1) What QMC is (utmost stratification)
2) Why it works (discrepancy, variation and Koksma-Hlawka)
3) How it works (constructions)
4) Randomized QMC
5) When it works best (effective dimension, tractability, weighted spaces)
(room for Bayesian thinking here)
6) (maybe) QMC ∩ MCMC
7) (maybe) QMC ∩ Particles
SAMSI Workshop on QMC, August 2017

4. QMC tutorial 4
Landmark papers in MC
Some landmark papers where Monte Carlo was applied:
• Physics Metropolis et al. (1953)
• Chemistry (reaction equations) Gillespie (1977)
• Financial valuation Boyle (1977)
• Bootstrap resampling Efron (1979)
• OR (discrete event simulation) Tocher & Owen (1960)
• Bayes (maybe 5 landmarks in early days)
• Nonsmooth optimization Kirkpatrick et al. (1983)
• Computer graphics (path tracing) Kajiya (1988)
SAMSI Workshop on QMC, August 2017

5. QMC tutorial 5
Landmark uses of QMC
• Particle transport methods in physics / medical imaging Jerome Spanier++
• Financial valuation, some early examples Paskov & Traub 1990s
• Graphical rendering Alex Keller++
(They got an Oscar!)
• Solving PDEs Frances Kuo, Christoph Schwab++, 2015
• Particle methods Chopin & Gerber (2015)
The next landmark methods
Some strong candidate areas:
• machine learning
• Bayes
• uncertainty quantification (UQ)
Clementine Prieur’s tutorial is in UQ.
SAMSI Workshop on QMC, August 2017

6. QMC tutorial 6
The next landmark
QMC methods dominate when we have to integrate a high dimensional function
that is nearly a sum of smooth functions of just a few of its input variables.
It is hard to tell a priori that this will happen. The original integrand might fail to be
smooth and yet it could be nearly a sum of smooth functions. Sloan, Kuo, Griebel
The best way is to find out is to run QMC on some example problems.
SAMSI Workshop on QMC, August 2017

7. QMC tutorial 7
MC and QMC
We estimate
µ =
[0,1]d
f(x) dx by ˆµ =
1
n
n
i=1
f(xi), xi ∈ [0, 1]d
In plain MC, the xi are IID U[0, 1]d
. In QMC they’re ’spread evenly’.
Non uniform
µ =
Ω
f(x)p(x) dx and ˆµ =
1
n
n
i=1
f(ψ(ui)), ui ∈ [0, 1]s
ψ : [0, 1]s
→ Rd
If u ∼ U[0, 1]s
then x = ψ(u) ∼ p
Many methods fit this framework. Devroye (1986)
Acceptance-rejection remains awkward.
SAMSI Workshop on QMC, August 2017

8. QMC tutorial 8
Illustration
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Monte Carlo
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Fibonacci lattice
MC and two QMC methods in the unit square
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Hammersley sequence
MC points always have clusters and gaps. What is random is where they appear.
QMC points avoid clusters and gaps to the extent that mathematics permits.
SAMSI Workshop on QMC, August 2017

9. QMC tutorial 9
Measuring uniformity
We need a way to verify that the points xi are ‘spread out’ in [0, 1]d
.
The most fruitful way is to show that
U {x1, x2, . . . , xn}
.
= U[0, 1]d
Discrepancy
A discrepancy is a distance F − ˆFn between measures
F = U[0, 1]d
and ˆFn = U {x1, x2, . . . , xn}.
There are many discrepancies.
SAMSI Workshop on QMC, August 2017

10. QMC tutorial 10
Local discrepancy
Did the box [0, a) get it’s fair share of points?
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a
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0.6
0.70
b
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0.45
Local discrepancy at a, b
δ(a) = ˆFn([0, a)) − F([0, a)) =
13
32
− 0.6 × 0.7 = −0.01375
Star discrepancy
D∗
n = D∗
n(x1, . . . , xn) = sup
a∈[0,1)d
|δ(a)| i.e., δ ∞
For d = 1 this is Kolmogorov-Smirnov.
SAMSI Workshop on QMC, August 2017

11. QMC tutorial 11
More discrepancies
D∗
n = sup
a∈[0,1)d
ˆFn([0, a)) − F([0, a))
Dn = sup
a,b∈[0,1)d
ˆFn([a, b)) − F([a, b))
D∗
n Dn 2d
D∗
n
Lp
discrepancies
D∗p
n =
[0,1)d
|δ(a)|p
da
1/p
e.g., Warnock
Also
Wrap-around discrepancies Hickernell
Discrepancies over (triangles, rotated rectangles, balls · · · convex sets · · · ).
Beck, Chen, Schmidt, Brandolini, Travaglini, Colzani, Gigante, Cools, Pillards
Best results are only for axis-aligned hyper-rectangles. SAMSI Workshop on QMC, August 2017

12. QMC tutorial 12
QMC’s law of large numbers
1) If f is Riemann integrable on [0, 1]d
, and
2) D∗
n(x1, . . . , xn) → 0
Then
1
n
n
i=1
f(xi) →
[0,1]d
f(x) dx
How fast?
MC has the CLT.
QMC has the Koksma-Hlawka inequality.
SAMSI Workshop on QMC, August 2017

13. QMC tutorial 13
Koksma’s inequality
For d = 1 |ˆµ − µ| D∗
n(x1, . . . , xn) ×
1
0
|f (x)| dx
NB: D∗
n = δ ∞ and
1
0
|f (x)| dx is the total variation V (f).
Setup for the proof
1
0
f(x) dx = f(1) −
1
0
xf (x) dx Integration by parts
1
n
n
i=1
f(xi) = f(1) −
1
n
n
i=0
i(f(xi+1) − f(xi)
=
xi+1
xi
f (x) dx
) Summation by parts
A few more steps, use f continuous
|µ − ˆµ| = · · · =
1
0
δ(x)f (x) dx δ ∞ f 1 = D∗
n × V (f)
SAMSI Workshop on QMC, August 2017

14. QMC tutorial 14
Koksma-Hlawka theorem
1
n
n
i=1
f(xi) −
[0,1)d
f(x) dx D∗
n × VHK(f)
VHK is the total variation in the sense of Hardy (1905) and Krause (1903)
Multidimensional variation has a few surprises for us.
Puzzler
Is this a 100% confidence interval?
SAMSI Workshop on QMC, August 2017

15. QMC tutorial 15
Rates of convergence
It is possible to get D∗
n = O
log(n)d−1
n
.
Then
|ˆµ − µ| = o(n−1+
) vs Op(n−1/2
) for MC
What about those logs?
Maybe log(n)d−1
/n 1/
√
n
Low effective dimension (later) counters them
As do some randomizations (later)
Roth (1954)
D∗
n = o
log(n)(d−1)/2
n
is unattainable
Gap between log(n)(d−1)/2
and log(n)d−1
subject to continued work.
E.g., Lacey, Bilyk
SAMSI Workshop on QMC, August 2017

16. QMC tutorial 16
Tight and loose bounds
They are not mutually exclusive.
Koksma-Hlawka is tight
|ˆµ − µ| (1 − )D∗
n(x1, . . . , xn) × VHK(f) fails for some f
KH holds as an equality for a worst case function, e.g., f
.
= ±δ.
It even holds if an adversary sees your xi before picking f.
Koksma-Hlawka is also very loose
It can greatly over-estimate actual error. Usually δ and f are dissimilar.
ˆµ − µ = − δ, f
Just like Chebychev’s inequality
It is also tight and very loose. E.g., Pr |N(0, 1)| 10 0.01 is loose.
Yes: 1.5 × 10−23
10−2
SAMSI Workshop on QMC, August 2017

17. QMC tutorial 17
Variation
Multidimensional Hardy-Krause variation has surprises for us. O (2005)
f(x1, x2) =



1, x1 + x2 1/2
0, else
VHK(f) = ∞ on [0, 1]2
VHK(f ) < ∞, for some f with f − f 1 <
Cusps
For general a ∈ Rd
,
f = max(aT
x, 0) =⇒ VHK(f) = ∞ for d 3
f = max(aT
x, 0)2
=⇒ VHK(f) = ∞ for d 4
QMC-friendly discontinuities
Axis parallel discontinuities may have VHK < ∞.
Used by e.g., X. Wang, I. Sloan, Z. He SAMSI Workshop on QMC, August 2017

18. QMC tutorial 18
Extensibility
For d = 1, the equispaced points xi = (i − 1/2)/n have D∗
n =
1
2n
Best possible.
q q q q q
0 1
But where do we put the n+1’st point?
We cannot get D∗
n = O(1/n) along a sequence x1, x2, . . . .
Extensible sequences
Take first n points of x1, x2, x3, . . . , xn, xn+1, xn+2, . . . .
Then we can get D∗
n = O((log n)d
/n).
No known extensible constructions get O((log n)d−1
/n).
SAMSI Workshop on QMC, August 2017

19. QMC tutorial 19
van der Corput
qqn=1
qqqn=2
qq qqn=3
qq qqqn=4
qq qq qqn=5
qq qq qqqn=6
qq qq qq qqn=7
qq qq qq qqqn=8
qq qq qq qq qqn=9
qq qq qq qq qqqn=10
qq qq qq qq qq qqn=11
qq qq qq qq qq qqqn=12
qq qq qq qq qq qq qqn=13
qq qq qq qq qq qq qqqn=14
qq qq qq qq qq qq qq qqn=15
qq qq qq qq qq qq qq qqqn=16
qq qq qq qq qq qq qq qq qqn=17
SAMSI Workshop on QMC, August 2017

20. QMC tutorial 20
van der Corput
i φ2(i)
1 1 0.1 1/2 0.5
2 10 0.01 1/4 0.25
3 11 0.11 3/4 0.75
4 100 0.001 1/8 0.125
5 101 0.101 5/8 0.625
6 110 0.011 3/8 0.375
7 111 0.111 7/8 0.875
8 1000 0.0001 1/16 0.0625
9 1001 0.1001 9/16 0.5625
Take xi = φ2(i). Extensible with D∗
n = O(log(n)/n).
Commonly xi = φ2(i − 1) starts at x1 = 0. SAMSI Workshop on QMC, August 2017

21. QMC tutorial 21
Halton sequences
The van der Corput trick works for any base. Use bases 2, 3, 5, 7, . . .
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72 Halton points
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864 random points
Halton sequence in the unit square
Via base b digital expansions
i =
K
k=0
bk
aik → φb(i) ≡
K
k=0
b−1−k
aik
xi = (φ2(i), φ3(i), . . . , φp(i)) SAMSI Workshop on QMC, August 2017

22. QMC tutorial 22
Digital nets
Halton sequences are balanced if n is a multiple of 2a
and 3b
and 5c
. . .
Digital nets use just one base b =⇒ balance all margins equally.
Elementary intervals
Some elementary intervals in base 5
SAMSI Workshop on QMC, August 2017

23. QMC tutorial 23
Digital nets
E =
s
j=1
aj
bkj
aj + 1
bkj
, 0 aj < bkj
(0, m, s)-net
n = bm
points in [0, 1)s
. If vol(E) = 1/n then E has one of the n points.
e.g. Faure (1982) points, prime base b s
(t, m, s)-net
If E deserves bt
points it gets bt
points. Integer t 0.
e.g. Sobol’ (1967) points base 2
Smaller t is better (but a construction might not exist).
minT project
Sch¨urer & Schmid give bounds on t given b, m and s
Monographs
Niederreiter (1992) Dick & Pillichshammer (2010)
SAMSI Workshop on QMC, August 2017

24. QMC tutorial 24
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A (0,4,2) net
Two digital nets in base 5
The (0, 4, 2)-net is a bivariate margin of a (0, 4, 5)-net.
The parent net has 54
= 625 points in [0, 1)5
.
It balances 43,750 elementary intervals.
Think of 43,750 control variates for 625 obs.
We should remove that diagonal striping artifact (later).
SAMSI Workshop on QMC, August 2017

25. QMC tutorial 25
Digital net constructions
Write i =
K
k=0 aikbk
For xi1 ∈ [0, 1] take xi1 =
K
k=0 xi1kb1−k
where
xi1 ≡







xi10
xi11
...
xi1K







=







C
(1)
11 C
(1)
12 . . . C
(1)
1K
C
(1)
21 C
(1)
22 . . . C
(1)
2K
...
...
...
...
C
(1)
K1 C
(1)
K2 . . . C
(1)
KK














ai0
ai1
...
aiK







mod b
Generally xij = C(j)
ai mod b for i = 0, . . . , bm
− 1 and j = 1, . . . , s.
Good C(j)
give small t.
See Dick & PIllichshammer (2010), Niederreiter (1991)
Computational cost
About the same as a Tausworth random number generator.
Base b = 2 offers some advantages.
SAMSI Workshop on QMC, August 2017

26. QMC tutorial 26
Extensible nets
Nets can be extended to larger sample sizes.
(t, s)-sequence in base b
Infinite sequence of (t, m, s)-nets.
x1, . . . , xbm xbm+1, . . . , x2bm · · · xkbm+1, . . . , x(k+1)bm · · ·
(t, m, s)−net
1st
(t, m, s)−net
2nd
· · · (t, m, s)−net
b’th
(t,m+1,s)−net
· · ·
And recursively for all m t.
Examples
Sobol’ b = 2 Faure t = 0 Niederreiter & Xing b = 2 (mostly)
SAMSI Workshop on QMC, August 2017

27. QMC tutorial 27
Lattices
The other main family of QMC points. An extensive literature, e.g., Sloan & Joe,
Kuo, Nuyens, Dick, Cools, Hickernell, Lemieux, L’Ecuyer· · · See L’Ecuyer’s
tutorial.
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z = (1,41)
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z = (1,233)
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z = (1,253)
Some lattice rules for n=377
Computation like congruential generators
xi =
i
n
,
Z2i
n
,
Z3i
n
, . . . ,
Zdi
n
(mod 1) Zj ∈ N
choose Z = (1, Z2, Z3, . . . , Zd) wisely
SAMSI Workshop on QMC, August 2017

28. QMC tutorial 28
QMC error estimation
|ˆµ − µ| D∗
n × VHK(f)
Not a 100% confidence interval
• D∗
n is hard to compute
• VHK harder to get than µ
• VHK = ∞ is common, e.g., f(x1, x2) = 1x1+x2 1
Koksma-Hlawka gives |ˆµ − µ| ∞
(maybe we already knew)
Also
Koksma-Hlawka is worst case. It can be very conservative.
Recent work
GAIL project of Hickernell++ allows user specified error tolerance .
SAMSI Workshop on QMC, August 2017

29. QMC tutorial 29
Randomized QMC
1) Make xi ∼ U[0, 1)d
individually,
2) keeping D∗
n(x1, . . . , xn) = O(n−1+
) collectively.
R independent replicates
ˆµ =
1
R
R
r=1
ˆµr
Var(ˆµ) =
1
R(R − 1)
R
r=1
(ˆµr − ˆµ)2
If VHK(f) < ∞ then
E((ˆµ − µ)2
) = O(n−2+
)
Random shift Cranley & Patterson (1976)
Scrambled nets O (1995,1997,1998)
Survey in L’Ecuyer & Lemieux (2005)
SAMSI Workshop on QMC, August 2017

30. QMC tutorial 30
Rotation modulo 1
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After
Cranley−Patterson rotation
Shift the points by u ∼ U[0, 1)s
with wraparound:
xi → xi + u (mod 1).
Commonly used on lattice rules.
Can also be used with nets.
At least it removes x1 = 0. SAMSI Workshop on QMC, August 2017

31. QMC tutorial 31
Digit scrambling
1) Chop space into b slabs. Shuffle.
2) Repeat within each of b slabs.
3) Then within b2
sub-slabs.
4) Ad infinitum b3
, b4
, . . .
5) And the same for all s coordinates.
Each xi ∼ U[0, 1)s
and x1, . . . , xn still a net (a.s.). O (1995)
SAMSI Workshop on QMC, August 2017

32. QMC tutorial 32
Digit scrambling
Mathematically it is a permutation operation on the base b digits of
xij =
K
k=1
aijk b−k
→ ˜xij =
K
k=1
˜aijk b−k
The mapping aijk → ˜aijk = πjk(aijk) preserves the net property.
In the previous ‘nested’ scramble πjk also depends on aij1, . . . , aij,k−1
Simpler scrambles
Random linear permutations
a → g + h × a mod p (prime p = b)
g ∼ U{0, 1, . . . , b − 1}, h ∼ U{1, 2, . . . , b − 1}
Matousek (1998) has nested linear permutations
Digital shift
aijk → aijk + gij (mod p)
For b = 2 xi → xi = xi ⊕ u bitwise XOR
SAMSI Workshop on QMC, August 2017

33. QMC tutorial 33
Example scrambles
Two components of the first 530 points of a Faure (0, 53)-net in base 53.
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Nested uniform
Randomized Faure points
The digital shift is much like a Cranley-Patterson rotation.
It uses just one random u for all points: xi = xi ⊕ u.
Random linear Matousek (1998) and nested uniform O (1995) have the same
Var(ˆµ).
SAMSI Workshop on QMC, August 2017

34. QMC tutorial 34
Unscrambled Faure
First n = 112
= 121 points of Faure (0, 11)-net in [0, 1]11
.
−2 −1 0 1 2
−1.0−0.50.00.51.0
qqqqqqqqqqqq
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2x2 ++ x3 −− x6 ++ x8 −− 2x10 −− x11
x3−−x6−−x8++x11
−1.0 −0.5 0.0 0.5 1.0−1.0−0.50.00.51.0
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x4 ++ x6 −− x10 −− x11
x1−−x2−−x8++x9
Two projections of 121 Faure points
Unscrambled points are very structured.
SAMSI Workshop on QMC, August 2017

35. QMC tutorial 35
Sobol’ projections
Using code from Bratley & Fox (1988).
Implementations differ based on choices of ‘direction numbers’.
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x26 versus x31
Three projections of 1024 Sobol points
At some higher sample size the empty blocks in x26 vs x31 fill in.
Expect empty blocks in higher order margins to last longer than those in low
dimensional margins.
SAMSI Workshop on QMC, August 2017

36. QMC tutorial 36
Scrambled net properties
Using σ2
= (f(x) − µ)2
dx
If Then N.B.
f ∈ L2
Var(ˆµ) = o(1/n) even if VHK(f) = ∞
f ∈ L2
Var(ˆµ) Γt,b,s σ2
/n for t = 0, Γ exp(1)
.
= 2.718
∂1,2,...,s
f ∈ L2
Var(ˆµ) = O(log(n)s−1
/n3
) O (1997,2008)
Γ < ∞ rules out (log n)s−1
catastrophe at finite n.
Loh (2003) has a CLT for t = 0 (and fully scrambled points).
Geometrically
Scrambling Faure breaks up the diagonal striping of the nets.
Scrambling Sobol’ points moves the full / empty blocks around.
Improved rate
RMSE is O(n−1/2
) better than QMC rate (cancellation).
Holds for nested uniform and nested linear scrambles.
SAMSI Workshop on QMC, August 2017

37. QMC tutorial 37
Scrambling vs shifting
Consider n = 2m
points in [0, 1).
QMC
van der Corput points (i − 1)/n for i = 1, . . . , n.
• · · · · · · · · · |• · · · · · · · · ·|• · · · · · · · · ·|• · · · · · · · · ·|
Shift
Shift all points by U ∼ U(0, 1) with wraparound.
Get one point in each [(i − 1)/n, i/n)
· · · • · · · · · · |· · · • · · · · · ·|· · · • · · · · · ·|· · · • · · · · · ·|
Scramble
Get a stratified sample, independent xi ∼ U[(i − 1)/n, i/n)
· · · • · · · · · · |· · · · · · · · • ·|· · · · · • · · · ·|· · • · · · · · · ·|
Random errors cancel yielding an O(n−1/2
) improvement.
SAMSI Workshop on QMC, August 2017

38. QMC tutorial 38
Higher order nets
Results from Dick, Baldeaux
Start with a net zi ∈ [0, 1)2s
dimensions.
‘Interleave’ digits of two variables to make a new one:
zi,2j = 0.g1g2g3 · · ·
zi,2j+1 = 0.h1h2h3 · · ·
−→ xi,j = 0.g1h1g2h2 · · · .
Error is O(n−2+
) under increased smoothness: ∂2s
∂x2
1...∂x2
d
f
Scrambling gets RMSE O(n−2−1/2+
)
Even higher
Start with ks dimensions interleave down to s.
Get O(n−k+
) and O(n−k−1/2+
) (under still higher smoothness)
Upshot
Very promising. Cost: many inputs and much smoothness.
Starting to be used in PDEs. Tutorial of Frances Kuo.
SAMSI Workshop on QMC, August 2017

39. QMC tutorial 39
The curse of dimension
Curse of dimension: larger d makes integration harder.
Cr
M = f : [0, 1]d
→ R
j
∂αj
∂x
αj
j
f M,
j
αj = r, αj 0
Bahkvalov I:
For any x1, . . . , xn ∈ [0, 1]d
there is f ∈ Cr
M with |ˆµn − µ| kn−r/d
Ordinary QMC like r = d
Bahkvalov II:
Random points can’t beat RMSE O(n−r/d−1/2
)
Ordinary MC like r = 0
SAMSI Workshop on QMC, August 2017

40. QMC tutorial 40
What if we beat those rates?
Sometimes we get high accuracy for large d.
It does not mean we beat the curse of dimensionality.
Bahkvalov never promised universal failure.
Only the existence of hard cases.
We may have just had an easy, non-worst case function.
Two kinds of easy
• Truncation: only the first s d components of x matter
• Superposition: the components only matter “s at a time”
Either way
f might not be “fully d-dimensional”.
SAMSI Workshop on QMC, August 2017

41. QMC tutorial 41
Dimensional decomposition
For u = {j1, j2, . . . , jr} ⊂ 1:d ≡ {1, 2, . . . , d} let
xu = (xj1 , . . . , xjr )
xi,u = (xij1
, . . . , xijr
)
Via ANOVA or other method, write
f(x) =
u⊆1:d
fu(xu)
Then
ˆµ − µ =
u⊆1:d
1
n
n
i=1
fu(xi,u) − fu(xu) dxu
|ˆµ − µ|
u⊆1:d
D∗
n(x1,u, . . . , xn,u) × fu
Often D∗
n(xi,u) D∗
n(xi) for small |u|. If also fu is small for large u,
then all the terms are small.
SAMSI Workshop on QMC, August 2017

42. QMC tutorial 42
Studying the good cases
Two main tools to describe it
• Weighted spaces and tractability
• ANOVA and effective dimension
Implications
Neither causes the curse to be lifted. They describe the happy circumstance
where the curse did not apply.
Both leave important gaps described below. I’ll raise as an open problem later.
SAMSI Workshop on QMC, August 2017

43. QMC tutorial 43
Weighted spaces
Hickernell (1996), Sloan & Wozniakowski (1998),
Dick, Kuo, Novak, Wasilkowski, many more
See tutorial by Hickernell
∂u
≡
j∈u
∂
∂xj
assume ∂1:d
f exists
Inner product, weights γu > 0
f 2
γ =
u⊆1:d
1
γu [0,1]u [0,1]−u
∂u
f(x) dx−u
2
dxu
Function ball Bγ,C = {f | f γ C}
Small γu =⇒ only small ∂u
f in ball.
Product weights
γu = j∈u γj where γj decrease rapidly with j.
Now f ∈ Bγ,C implies ∂u
f small when |u| large.
Many more choices: Dick, Kuo, Sloan (2013)
SAMSI Workshop on QMC, August 2017

44. QMC tutorial 44
ANOVA and effective dimension
Caflisch, Morokoff & O (1997)
ANOVA: f(x) = u⊆1:d fu(x)
Often f is dominated by its low order interactions.
Then RQMC may make a huge improvement.
Let σ2
u = Var(fu) variance component
Truncation dim. s d
u⊆1:s
σ2
u 0.99
u⊆1:d
σ2
u
Superposition dim. s d
|u| s
σ2
u 0.99
u⊆1:d
σ2
u
Mean dimension
u |u|σ2
u
u σ2
u
Liu & O (2006) Easier to estimate via Sobol’ indices.
SAMSI Workshop on QMC, August 2017

45. QMC tutorial 45
Open problem
• ANOVA captures magnitude of the low dimensional parts but not their
smoothness. Even when it verifies that f is dominated by low dimensional
parts it does not assure small |ˆµ − µ|.
• Weighted space models assure accurate estimation at least asymptotically.
However, it is not easy to decide which weighted space to use.
• Given a weighted space, there are algorithms to tune QMC points for it.
• ANOVA approaches may support an approach to choosing good weights for a
given problem, building on Sobol’ indices Sobol’++, Saltelli++, Prieur++,
Kucherenko++ or active subspaces Constantine++
The problems are about how to combine the approaches and when / whether a
resulting adaptive algorithm will be effective.
SAMSI Workshop on QMC, August 2017

46. QMC tutorial 46
Description vs engineering
Effective dimension and weighted spaces both describe settings where f can be
integrated well.
Some authors try to engineer changes in f to make it more suited to QMC.
E.g., cram importance into first few components of x.
MC vs QMC
MC places lots of effort on variance reduction.
For QMC we gain by reducing effective dimension.
Or Hardy-Krause variation (but there asymptotics are slow).
E.g., when turning u ∼ U[0, 1]d
into z ∼ N(0, Σ)
The choice of Σ1/2
affects QMC performance.
Caflisch, Morokoff & O (1997)
Acworth, Broadie & Glasserman (1998)
Imai & Tan (2014)
Best Σ can depend strongly on f. Papageorgiou (2002) SAMSI Workshop on QMC, August 2017

47. QMC tutorial 47
Sampling Brownian motion
Feynman-Kac/Brownian bridge
First few variables define a ‘skeleton’. The rest fill in.
0.0 0.2 0.4 0.6 0.8 1.0
−1.0−0.50.00.5
Time t
B(t)
q
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q
Brownian bridge construction of Brownian motion
See also Mike Giles++ on multi-level MC. SAMSI Workshop on QMC, August 2017

48. QMC tutorial 48
QMC ∩ MCMC
• Replace IID ui inputs to MCMC by something better.
• Completely uniformly distributed (CUD).
Like entire period of a (small) RNG.
• Very early refs:
Chentsov (1967)
Finite state space. Uses ’inversion.’
Beautiful coupling argument. (Resembles coupling from the past.)
Sobol’ (1974)
Has n × ∞ points xij ∈ [0, 1]
Samples from a row until a return to start state, then goes to next row
Gets rate O(1/n) · · · if transition probabilities are a/2b
for integers a, b
SAMSI Workshop on QMC, August 2017

49. QMC tutorial 49
QMC ∩ MCMC
For MCMC we follow one particle.
1) Step xi ← ψ(xi−1, vi) for vi ∈ (0, 1)d
.
2) For v1 = (u1, . . . , ud), v2 = (ud+1, . . . , u2d) etc.
n steps require u1, . . . , und ∈ (0, 1)
3) MC-MC uses ui ∼ U(0, 1)
4) MC-QMC uses ui CUD
Reasons for caution
1) We’re using 1 point in [0, 1]nd
with n → ∞
2) Our xi won’t be Markovian
SAMSI Workshop on QMC, August 2017

50. QMC tutorial 50
MCMC ≈ QMCT
Method Rows Columns
QMC n points d variables 1 d n → ∞
MCMC r replicates n steps 1 r n → ∞
QMC MCMC
QMC based on equidistribution MCMC based on ergodicity
SAMSI Workshop on QMC, August 2017

51. QMC tutorial 51
Some results
1) Metropolis on CUD points consistent in finite state space O & Tribble (2005)
2) Weakly CUD random points ok Tribble & O (2008)
3) Consistency in continuous problems (Gibbs and Metropolis)
Chen, Dick & O (2010)
4) Chen (2011) rate O(n−1+
) for some smooth ARMAs
5) Mackey & Gorham find best results are for µ = E(xj).
SAMSI Workshop on QMC, August 2017

52. QMC tutorial 52
Sequential QMC
JRSS-B discussion paper by Chopin & Gerber (2015)
N particles in Rd
for T time steps
Advance xnt to xn,t+1 = φ(xnt, un,t+1) each u ∈ [0, 1)S
.
Do them all with Ut+1 ∈ [0, 1)N×S
. Usually IID.
For QMC
We want rows of Ut+1 to be ‘balanced’ wrt position of xt
Easy if d = S = 1 L´ecot & Tuffin (2004)
Take Ut+1 ∈ [0, 1)N×2
Sort first column along xn,t. Update points with remaining column.
What to do if d > 1?
Alignment is tricky because Rd
is not ordered.
Related work by L’Ecuyer, L´ecot, L’Archevˆeque-Gaudet on array-RQMC.
d-dimensional alignments. Huge variance reductions. Limited theory.
SAMSI Workshop on QMC, August 2017

53. QMC tutorial 53
Using Hilbert curves
These are space-filling curves as m → ∞:
m= 1 m= 2 m= 3 m= 4
Chopin & Gerber thread a curve through xn,t ∈ Rd
Align particles via first column of RQMC points Ut+1 ∈ [0, 1)N×(1+S)
.
Use remaining S columns to advance them.
Results
Squared error is o(N−1
).
Good empirical results for modest d
NB:
I left out lots of details about particle weighting.
SAMSI Workshop on QMC, August 2017

54. QMC tutorial 54
Thanks
• NSF support over the years, most recently:
DMS-1407397 and DMS-1521145.
• Co-leaders: Hickernell, Kuo, L’Ecuyer
• SAMSI, especially Ilse Ipsen
• Sue McDonald
SAMSI Workshop on QMC, August 2017