  The Langevin algorithms are frequently used to sample the posterior
distributions in Bayesian inference. In many practical problems, however, the
posterior distributions often consist of non-differentiable components, posing
challenges for the standard Langevin algorithms, as they require to evaluate
the gradient of the energy function in each iteration. To this end, a popular
remedy is to utilize the proximity operator, and as a result one needs to solve
a proximity subproblem in each iteration. The conventional practice is to solve
the subproblems accurately, which can be exceedingly expensive, as the
subproblem needs to be solved in each iteration. We propose an approximate
primal-dual fixed-point algorithm for solving the subproblem, which only seeks
an approximate solution of the subproblem and therefore reduces the
computational cost considerably. We provide theoretical analysis of the
proposed method and also demonstrate its performance with numerical examples.
