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Random walks for estimating densities of states and the volume of convex bodies in high dimensional spaces

Augustin Chevallier 1
1 ABS - Algorithms, Biology, Structure
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This manuscript introduces new random walks for the computation of densities of states, a central problem in statistical physics, and the computation of the volume of polytopes. First, we focus on the Wang-Landau (WL) algorithm, a recently developed stochastic algorithm computing the density of states of a physical system. We propose an efficient random walk that uses geometrical information to circumvent the following inherent difficulties: avoiding overstepping strata, toning down concentration phenomena in high-dimensional spaces, and accommodating multidimensional distribution. Experiments on various models stress the importance of these improvements to make Wang-Landau effective in challenging cases. These improvements make it possible to compute density of states for regions of the phase space of small biomolecules. Second, we study Hamiltonian Monte Carlo (HMC) with reflections on the boundary of a domain, providing an enhanced alternative to Hit-and-run (HAR) to sample a target distribution in a bounded domain. We provide a convergence bound, paving the way to more precise mixing time analysis, and a robust implementation based on multi-precision arithmetic -- a mandatory ingredient. We compare Hamiltonian Monte Carlo with Hit and Run within the polytope volume algorithm by Cousins and Vempala. The tests, conducted up to dimension 50, show that the HMC with reflections random walk outperforms HAR. Finally, using Wang-Landau requires specifying the system handled, providing a suitable random walk to explore the definition domain, and possibly accommodate different variants of Wang-Landau. We present the first generic (C++) implementation providing all such ingredients.
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Augustin Chevallier. Random walks for estimating densities of states and the volume of convex bodies in high dimensional spaces. Data Structures and Algorithms [cs.DS]. Université Côte d'Azur, 2019. English. ⟨NNT : 2019AZUR4021⟩. ⟨tel-02490412⟩

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