Analysis of Exchange Ratio for Exchange Monte Carlo Method Kenji Nagata, Sumio Watanabe Tokyo Institute of Technology Japan

Contents Background  Exchange Monte Carlo (EMC) method  Design of EMC method Main result  Settings  Symmetrized Kullback divergence  Average exchange ratio Discussion and Conclusion

Contents Background  Exchange Monte Carlo (EMC) method  Design of EMC method Main result  Settings  Symmetrized Kullback divergence  Average exchange ratio Discussion and Conclusion

Exchange Monte Carlo (EMC) method [Hukushima, 96] huge computational cost! Multi Canonical algorithmSimulated tempering Exchange Monte Carlo (EMC) method

EMC method is to generate the sample sequence from the following joint distribution, :temperatures Target distribution ： EMC method Sampling from the following target distribution!

EMC method Carrying out the following two updates alternately. １． [conventional MCMC sampling] Parallel sampling from each distribution by using conventional MCMC method. ２． [exchange process] The exchange of two position, and, is tried and accepted with the following probability. Hereafter, we call “ exchange ratio ”.

EMC method

EMC method : standard normal distribution

Contents Background  Exchange Monte Carlo (EMC) method  Design of EMC method Main result  Settings  Symmetrized Kullback divergence  Average exchange ratio Discussion and Conclusion

Design of EMC method How should and be set? Temperature has close relation to the exchange ratio.

Design of EMC method Acceptance ratio of exchange process. For efficient EMC method, the average exchange ratio needs to be of and nearly constant for all temperatures.

Design of EMC method : the expectation of over. For efficient EMC method, the symmetrized Kullback divergence needs to be nearly constant over the various temperatures.

Purpose We analytically clarify the symmetrized Kullback divergence and the average exchange ratio in a low temperature limit,. Average exchange ratio Symmetrized Kullback divergence Criteria for the design of EMC method We need previous EMC simulations in order to obtain these values. The accuracy of experimental values are unknown. Purpose

Contents Background  Exchange Monte Carlo (EMC) method  Design of EMC method Main result  Settings  Symmetrized Kullback divergence  Average exchange ratio Discussion and Conclusion

Settings We consider the EMC method between the following two distribution,

Main result The symmetrized Kullback divergence converges to the following value for : rational number

Main result [Watanabe,2001]

Main result The average exchange ratio converges to the following value for : rational number

Main result If,

Contents Background  Exchange Monte Carlo (EMC) method  Design of EMC method Main result  Settings  Symmetrized Kullback divergence  Average exchange ratio Discussion and Conclusion

Discussion Average exchange ratio : Symmetrized Kullback divergence : From these results, we can adjust the temperatures without carrying out the previous EMC simulations. These results can be used as criteria for checking the convergence of EMC simulations.

Discussion Average exchange ratio : geometrical progression!! Symmetrized Kullback divergence :

Discussion Condition of theorem large enough! Not applicable.Applicable.

Conclusion We analytically clarified the symmetrized Kullback divergence and the average exchange ratio in a low temperature limit. As a result, it is clarified that  The set of temperature should be set as a geometrical progression in order to make the average exchange ratio constant over the various temperatures. As the future works,  Verifying the theoretical results in this study experimentally.  Constructing the design of EMC method.