QMC: THE THEORY
Variation: an intuitive measure of smoothness (or “wiggliness”):
J
𝑉 𝑓 = K |𝑓′(𝑥)|𝑑𝑥
]
•
A less intuitive multivariate generalization:
Hardy-Krause variation:
J
l
𝑉›œ 𝑓 = N
N
J
𝜕ž 𝑓
K…K
𝜕𝑥Pw … 𝜕𝑥PŸ
žRJ J£Pw ¤⋯¤PŸ £l ]
]
𝑑𝑥Pw … 𝑑𝑥PŸ
r RJ,¡¢Pw ,…PŸ
65
QMC: THE THEORY
Koksma-Hlawka theorem. Suppose 𝑓: 0,1
variation. Then
l
→ ℝ is of bounded HK-
Q
1
K
𝑓 𝑥 𝑑𝑥 − N 𝑓 𝑥P
𝑛
¨
],J
≤ 𝐷 ∗ (𝑥J , … 𝑥Q )𝑉›œ (𝑓)
PRJ
• Complicated ingredients aside, the final result is quite intuitive.
• The error bound is proportional to the non-uniformity of the
points and also to the wiggliness of the function.
• We don’t know 𝑉›œ (𝑓), but it’s a constant, which gives us the
asymptotics of the error (as a function of discrepancy).
66
MC VS QMC: THE THEORY
Asymptotics:
•
•
Koksma-Hlawka: for a given function, the QMC integration
error is proportional to the star-discrepancy of the QMC
sequence.
Commonly cited result: the star-discrepancy of the ddimensional Halton set is
Θ
•
log 𝑛
𝑛
l
Q: is this a positive or negative result? (Also when
compared to MC?)
67
MC VS QMC: THE THEORY
Asymptotics:
•
•
The QMC integration error is proportional to the stardiscrepancy of the QMC sequence.
Commonly cited result: the star-discrepancy of the ddimensional Halton set is
Θ
log 𝑛
𝑛
l
With 2 parameters, this notation is shady!
•
What is meant here is that as a function of n (with d fixed!),
the discrepancy is proportional to
•
—˜™ Q ¨
Q
.
But the proportionality constant depends (horribly!) on d.
68
QMC IN HIGH DIMENSIONS
The Halton sequence (as most other QMC sequences) becomes
visibly less impressive in high dimensions:
d=10
1
1
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
d=30
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0.2
0.4
0.6
d=5
0.8
1
0
0.2
0.4
0.6
0.8
1
d=20
69
QMC IN HIGH DIMENSIONS
•
—˜™ Q ¨
Q
tends to zero a lot faster than
J
.
Q
•
In low dimensions, QMC is great!
•
In high dimensions the dimension-free scaling of MC
catches up. QMC stops working around 15-20
dimensions.
•
Even without the hidden dimension-independent constant,
compare
—˜™ Q ¨
Q
values of n…
and
J
Q
for small d and small/moderate
70
STATISTICAL BOUNDS IN
QUASI MONTE-­CARLO
We can also get statistical bounds similar to those from MC:
• Shift every element in the the QMC sequence by the same
uniform random vector in 0,1 l .
• The discrepancy of the shifted set is approximately the same
as the original. (This is fairly intuitive.)
• The estimates of the integral arising from different shifted
sets are independent.
• The same method (as we saw in MC) for computing the
endpoints of a confidence interval applies.
71
MATLAB
Sauer has a simple implementation of the Halton set
Built-in, with more features:
• haltonset
• H = haltonset(d) initializes a d-dimensional array
• H(1:n, :) produces an n-by-d array with the first n points
• Skip, Scramble, ...
• It doesn’t implement the random shift, but it’s easy to do
manually. (HW 1.)
72
QMC/INTEGRATION OUTLOOK
There are many other (better) QMC sequences and grids than just
Halton. Popular ones include:
1. Sobolev sequence. Complicated, won’t discuss.
Available in Matlab: sobolset.
2. Integration lattices. Easy to explain, surprisingly
complicated theory.
l
𝐿l = 𝑣 = N ℎ¡ 𝑣¡ such that ℎ¡ ∈ ℤ for all 𝑗 = 1, … , 𝑑 ¡RJ
•
linearly indep. vectors (generators) chosen to satisfy ℤj ⊆ 𝐿j
Not available in Matlab, but there is a good open source
implementation: Lattice Builder
(https://github.com/umontreal-simul/latbuilder).
73
QMC/INTEGRATION OUTLOOK
What happened to Gaussian quadrature, polynomial exactness
and all that cool stuff we learned in 427?
3. Gaussian quadrature: some theory exists. Very difficult
to characterize the domains for which Gaussian
quadrature formulas exist. (Remember, in the univariate
setting there was a unique formula for each degree.)
4. Sparse grids. …
74
SPARSE GRIDS
Sparse grids.
• Exact for multivariate
polynomials up to a
chosen total degree. (Full
grids are exact for every
polynomial of a given
degree in each variable.)
• A combination of lower
order grids.
• As a result, they use way
fewer points.
• But they also have some
negative weights!
75