Analysis and Optimization of Certain Parallel Monte Carlo Methods in the Low Temperature Limit

Dupuis, Paul; Wu, Guo-Jhen

doi:10.1137/21m1402029

Cited by 2 publications

(2 citation statements)

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“…The growth constant in T ε will then depend on these parameters, which will then be chosen to (either directly or indirectly, depending on the criteria used) reduce the size of T ε . For a compelling example we refer to [6], which shows how for a system with fixed well depths a scheme known as infinite swapping can be designed so that given any a > 0 one can design a scheme so that an interval of length e a/ε suffices.…”

Section: March 2 2020mentioning

confidence: 99%

“…Our interest in proving large deviations estimates when ε → 0 and T ε → ∞ is because one might find it easier to extract information in this double limit, analogous to the simplified approximation to µ ε just mentioned. These results will be applied in [6] to analyze and optimize a Monte Carlo method known as infinite swapping [5,10] when the noise is small, which happens to be both common in applications and also the setting in which the Monte Carlo method will have the greatest difficulty. We expect that the general set of results will be useful for other purposes as well.…”

Section: Introductionmentioning

confidence: 99%

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Large deviation properties of the empirical measure of a metastable small noise diffusion

Dupuis

2020

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The aim of this paper is to develop tractable large deviation approximations for the empirical measure of a small noise diffusion. The starting point is the Freidlin-Wentzell theory, which shows how to approximate via a large deviation principle the invariant distribution of such a diffusion. The rate function of the invariant measure is formulated in terms of quasipotentials, quantities that measure the difficulty of a transition from the neighborhood of one metastable set to another. The theory provides an intuitive and useful approximation for the invariant measure, and along the way many useful related results (e.g., transition rates between metastable states) are also developed.With the specific goal of design of Monte Carlo schemes in mind, we prove large deviation limits for integrals with respect to the empirical measure, where the process is considered over a time interval whose length grows as the noise decreases to zero. In particular, we show how the first and second moments of these integrals can be expressed in terms of quasipotentials. When the dynamics of the process depend on parameters, these approximations can be used for algorithm design, and applications of this sort will appear elsewhere. The use of a small noise limit is well motivated, since in this limit good sampling of the state space becomes most challenging. The proof exploits a regenerative structure, and a number of new techniques are needed to turn large deviation estimates over a regenerative cycle into estimates for the empirical measure and its moments.

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Section: March 2 2020mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%