  We study the problem of approximate sampling from non-log-concave
distributions, e.g., Gaussian mixtures, which is often challenging even in low
dimensions due to their multimodality. We focus on performing this task via
Markov chain Monte Carlo (MCMC) methods derived from discretizations of the
overdamped Langevin diffusions, which are commonly known as Langevin Monte
Carlo algorithms. Furthermore, we are also interested in two nonsmooth cases
for which a large class of proximal MCMC methods have been developed: (i) a
nonsmooth prior is considered with a Gaussian mixture likelihood; (ii) a
Laplacian mixture distribution. Such nonsmooth and non-log-concave sampling
tasks arise from a wide range of applications to Bayesian inference and imaging
inverse problems such as image deconvolution. We perform numerical simulations
to compare the performance of most commonly used Langevin Monte Carlo
algorithms.
