Monte Carlo Simulation methods are widely used to solve complex engineering and scientific problems. The principle of Monte Carlo Simulation is averaging over a large number of random samples to approximate desired quantities. However, the convergence rate is 1/N, where N is the number of samples. In this slides, we introduce the concept of Superefficient Monte Carlo simulations, which can achieve 1/N^2 convergence rate. We discuss both the theoretical and practical aspects of the Superefficient Monte Carlo simulation methodology.

1. Super Efficient Monte Carlo
Simulation
Cheng-An Yang
Advisor: Prof. Yao

2. Monte Carlo Simulation
• We want to evaluate
• Define
• Approximate I by the N-sample average:
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Average
Xi
B(Xi)

3. Convergence Rate
• By LLN, sample average converges almost surely.
• Approximation error decays like 1/N.
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4. Convergence Rate of Super-Efficient
MC Simulation
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5. • Initial state
Chaotic Dynamical System
• Evolution of the state is
governed by a mapping
T such that
• Arbitrarily close initial
states grow apart
exponentially.

6. • State space:
• Mapping: pth order Chebyshev polynomial
Ex. Chebyshev dynamical system
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7. Chaotic Monte Carlo Simulation
Time average Ensemble average
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Chaotic sequence
Average
Xi
B(Xi)
Birkhoff theorem

8. Super-efficient MC Simulation
• First introduced by Umeno in 1999.
• Rewrite the error variance as
• We say the Chaotic MC simulation is Super-
Efficient (SE). 8

9. When is MC simulation Super-Efficient?
• Super-efficiency and Lebesgue spectrum:
9
…
…
0 1 2 ……

10. Necessary and Sufficient Condition for SE
• If the dynamical system has a Lebesgue
spectrum, then the chaotic MC simulation is
super-efficient if and only if
Generalized Fourier Series Expansion of B:
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11. When is MC simulation Super-Efficient?
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……
0 …
…
1 2
• If all the row sum equals to zero, then the
chaotic MC simulation is SE.

12. Approximate SEMC
• Key observation: adding zero-mean terms will
not affect the integral; but can improve
dynamical correlation:
• In practice we need to approximate dλ by the
finite sum
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Mean = 0

13. ASE Algorithm
• Apply chaotic MC on the modified integrand
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Compensator F(x)
Average
Xi
B(Xi)
F(Xi)

14. Ex. ASE
• Consider the integrand on (-1,1):
• Using Chebyshev dynamical system with order 2.
• Choose Λ = {1,3,5,7,9}, approximate B up to 5
terms for each λ in Λ.
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15. Ex. ASE
10
3
10
4
10
5
10
6
10
7
10
-10
10
-8
10
-6
10
-4
N
sN
2
Conventional
SE
n = 100
n = 1000
n = 10000
n = 100000
• The more samples we spent on estimating dλ,
the better the convergence rate is.
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for some ζ.
• The error variance:
• Effective convergence
rate: 1/Nα, 1≤ α ≤2.

16. Progressive ASE
• Idea: estimate dλ along the way.
• Leads to the Progressive ASE (PASE) variant.
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Average
Xi
B(Xi)
F(Xi)

17. Ex. PASE
• Using the previous integrand
10
3
10
4
10
5
10
6
10
7
10
-10
10
-8
10
-6
10
-4
N
sN
2
Conventional
SE
n = 100
n = 1000
n = 10000
n = 100000
PASE
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a(DecayExponent)
103 104 105 106
1
1.2
1.4
1.6
1.8
2
N (samples)
Conventional
SE
PASE
n = 100
n = 1000
n = 10000
n = 100000

18. Summary and Future Works
• Some Chaotic Monte Carlo simulations are
super-efficient:
• SE can be characterized by Lebesgue spectrum.
• Proposed a PASE algorithm:
• Find efficient ways to generalize SEMC to high
dimensional integrands.
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19. • Denote the time average
• Autocorrelation:
Efficiency of Chaotic MC
• After some math, the error variance is given by
Statistical Dynamical
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20. Ex. Chebyshev Dynamical System
• Recall the Chebyshev dynamical system with
order p has the mapping
• By the semi-group property of Tp, the (λ, j)-th
basis function is given by
• F = {0,1,2,…}, Λ = relative prime number to p.
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