Overdamped Langevin dynamics are reversible stochastic differential equations which are commonly used to sample probability measures in high-dimensional spaces, such as the ones appearing in computational statistical physics and Bayesian inference. By varying the diffusion coefficient, there are in fact infinitely many overdamped Langevin dynamics which are reversible with respect to the target probability measure at hand. This suggests to optimize the diffusion coefficient in order to increase the convergence rate of the dynamics, as measured by the spectral gap of the generator associated with the stochastic differential equation. We analytically study this problem here, obtaining in particular necessary conditions on the optimal diffusion coefficient. We also derive an explicit expression of the optimal diffusion in some appropriate homogenized limit. Numerical results, both relying on discretizations of the spectral gap problem and Monte Carlo simulations of the stochastic dynamics, demonstrate the increased quality of the sampling arising from an appropriate choice of the diffusion coefficient.

翻译：过阻尼朗之万动力学是一类可逆随机微分方程，常用于高维空间中的概率测度采样，例如计算统计物理学和贝叶斯推断中出现的测度。通过改变扩散系数，实际上存在无穷多种过阻尼朗之万动力学可逆于当前目标概率测度。这提示我们可以通过优化扩散系数来提高动力学的收敛速率，该速率由随机微分方程对应生成元的谱隙度量。本文对此问题进行了解析研究，特别获得了最优扩散系数的必要条件。我们还在适当的均匀化极限下推导出最优扩散系数的显式表达式。数值结果（包括基于谱隙问题离散化的计算和随机动力学的蒙特卡洛模拟）表明，通过恰当选择扩散系数可显著提升采样质量。