A major difficulty in Markov chain Monte Carlo simulation is the restriction to very small perturbations of the chain at each iteration required to maintain stability or reasonable acceptance rate. A similar problem arises in optimization, where efficient and practical preconditioning techniques (changes of variables that alleviate the restriction on perturbations) have been developed over many decades. Unfortunately, the most practical and widely used of these techniques do not have analogues in the sampling context. We develop a general framework for preconditioning the underdamped Langevin sampling technique and suggest practical preconditioning strategies based on estimating local covariance information from parallel Markov chain Monte Carlo methods. The use of collective dynamics eliminates multiplicative noise and stabilizes the dynamics, thus providing a practical approach to difficult anisotropic sampling problems in high dimensions. Numerical experiments with model problems demonstrate that dramatic potential speedups, compared to various alternative schemes, are attainable.
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