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Super Efficient Monte Carlo Simulation
Cheng-An YangAdvisor: Prof. Yao
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Monte Carlo Simulation
• We want to evaluate
• Define
• Approximate I by the N-sample average:
AverageXi B(Xi)
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Convergence Rate
• By LLN, sample average converges almost surely.
• Approximation error decays like 1/N.
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Convergence Rate of Super-Efficient MC Simulation
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• Initial state
Chaotic Dynamical System
• Evolution of the state is governed by a mapping T such that
• Arbitrarily close initial states grow apart exponentially.
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• State space:• Mapping: pth order Chebyshev polynomial
Ex. Chebyshev dynamical system
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Chaotic Monte Carlo Simulation
Time average Ensemble average
Chaotic sequence
AverageXi B(Xi)
Birkhoff theorem
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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).
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When is MC simulation Super-Efficient?
• Super-efficiency and Lebesgue spectrum:
…
… …
0 1 2 ……
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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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When is MC simulation Super-Efficient?
…
… …
0 …
…
1 2
• If all the row sum equals to zero, then the chaotic MC simulation is SE.
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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
Mean = 0
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ASE Algorithm
• Apply chaotic MC on the modified integrand
Compensator F(x)
AverageXi
B(Xi)
F(Xi)
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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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Ex. ASE
103
104
105
106
107
10-10
10-8
10-6
10-4
N
sN2
Conventional
SE
n = 100n = 1000n = 10000n = 100000
• The more samples we spent on estimating dλ, the better the convergence rate is.
for some ζ.
• The error variance:
• Effective convergence rate: 1/Nα, 1≤ α ≤2.
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Progressive ASE
• Idea: estimate dλ along the way.• Leads to the Progressive ASE (PASE) variant.
AverageXi
B(Xi)
F(Xi)
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Ex. PASE
• Using the previous integrand
103
104
105
106
107
10-10
10-8
10-6
10-4
N
sN2
Conventional
SE
n = 100n = 1000
n = 10000n = 100000
PASE
a (
Dec
ay E
xpon
ent)
103 104 105 1061
1.2
1.4
1.6
1.8
2
N (samples)
Conventional
SE
PASE
n = 100n = 1000n = 10000
n = 100000
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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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• Denote the time average• Autocorrelation:
Efficiency of Chaotic MC
• After some math, the error variance is given by
Statistical Dynamical
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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.