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，例如计算统计物理和贝叶斯推断中出现的情形。通过改变扩散系数，实际上存在无穷多个针对目标概率测度可逆的过阻尼朗之万动力学。这提示我们优化扩散系数以提高动力学的收敛速率，该速率由随机微分方程生成元的光谱间隙衡量。我们在此对这一问题进行解析研究，特别得到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了最优扩散的显式表达式。数值结果——既包括基于光谱间隙问题离散化的方法，也包括随机动力学蒙特卡洛模拟——表明，通过适当选择扩散系数可提高采样质量。