Adaptive Monte Carlo Algorithms Applied to Heterogeneous Transport Problems

2009 ◽  
pp. 209-225 ◽  
Author(s):  
Katherine Bhan ◽  
Rong Kong ◽  
Jerome Spanier
Keyword(s):  
Monte Carlo ◽  
Adaptive Monte Carlo ◽  
Monte Carlo Algorithms ◽  
Transport Problems

scholarly journals Convergence rates for optimised adaptive importance samplers

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Ömer Deniz Akyildiz ◽  
Joaquín Míguez
Keyword(s):  
Monte Carlo ◽  
Rate Of Convergence ◽  
Exponential Family ◽  
Convergence Rates ◽  
Adaptation Strategy ◽  
Adaptive Monte Carlo ◽  
Asymptotic Bounds ◽  
Monte Carlo Algorithms ◽  
The Mean ◽  
Number Of Iterations

AbstractAdaptive importance samplers are adaptive Monte Carlo algorithms to estimate expectations with respect to some target distribution which adapt themselves to obtain better estimators over a sequence of iterations. Although it is straightforward to show that they have the same $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) convergence rate as standard importance samplers, where N is the number of Monte Carlo samples, the behaviour of adaptive importance samplers over the number of iterations has been left relatively unexplored. In this work, we investigate an adaptation strategy based on convex optimisation which leads to a class of adaptive importance samplers termed optimised adaptive importance samplers (OAIS). These samplers rely on the iterative minimisation of the $$\chi ^2$$ χ 2 -divergence between an exponential family proposal and the target. The analysed algorithms are closely related to the class of adaptive importance samplers which minimise the variance of the weight function. We first prove non-asymptotic error bounds for the mean squared errors (MSEs) of these algorithms, which explicitly depend on the number of iterations and the number of samples together. The non-asymptotic bounds derived in this paper imply that when the target belongs to the exponential family, the $$L_2$$ L 2 errors of the optimised samplers converge to the optimal rate of $$\mathcal {O}(1/\sqrt{N})$$ O ( 1 / N ) and the rate of convergence in the number of iterations are explicitly provided. When the target does not belong to the exponential family, the rate of convergence is the same but the asymptotic $$L_2$$ L 2 error increases by a factor $$\sqrt{\rho ^\star } > 1$$ ρ ⋆ > 1 , where $$\rho ^\star - 1$$ ρ ⋆ - 1 is the minimum $$\chi ^2$$ χ 2 -divergence between the target and an exponential family proposal.

Adaptive Monte Carlo Algorithms for Stopped Diffusion

2005 ◽  
pp. 59-88 ◽  
Author(s):  
Anna Dzougoutov ◽  
Kyoung-Sook Moon ◽  
Erik von Schwerin ◽  
Anders Szepessy ◽  
Raúl Tempone
Keyword(s):  
Monte Carlo ◽  
Adaptive Monte Carlo ◽  
Monte Carlo Algorithms

Monte Carlo Algorithms for Moments of Transition Arrays

1988 ◽  
Vol 102 ◽  
pp. 79-81
Author(s):  
A. Goldberg ◽  
S.D. Bloom
Keyword(s):  
Monte Carlo ◽  
State Vector ◽  
Basis Vector ◽  
Collective State ◽  
Dispersion Characteristics ◽  
Dipole Strength ◽  
The Third ◽  
Monte Carlo Algorithms ◽  
Third Moment ◽  
Very High

AbstractClosed expressions for the first, second, and (in some cases) the third moment of atomic transition arrays now exist. Recently a method has been developed for getting to very high moments (up to the 12th and beyond) in cases where a “collective” state-vector (i.e. a state-vector containing the entire electric dipole strength) can be created from each eigenstate in the parent configuration. Both of these approaches give exact results. Herein we describe astatistical(or Monte Carlo) approach which requires onlyonerepresentative state-vector |RV> for the entire parent manifold to get estimates of transition moments of high order. The representation is achieved through the random amplitudes associated with each basis vector making up |RV>. This also gives rise to the dispersion characterizing the method, which has been applied to a system (in the M shell) with≈250,000 lines where we have calculated up to the 5th moment. It turns out that the dispersion in the moments decreases with the size of the manifold, making its application to very big systems statistically advantageous. A discussion of the method and these dispersion characteristics will be presented.

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