Papers › Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials
Ergodicity of Langevin Dynamics and its Discretizations for Non-smooth Potentials
Lorenz Fruehwirth, Andreas Habring
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This article is concerned with sampling from Gibbs distributions π(x)∝e⁻ᵁ⁽ˣ⁾ using Markov chain Monte Carlo methods. In particular, we investigate Langevin dynamics in the continuous- and the discrete-time setting for such distributions with potentials U(x) which are strongly-convex but possibly non-differentiable. We show that the corresponding subgradient Langevin dynamics are exponentially ergodic to the target density π in the continuous setting and that certain explicit as well as semi-implicit discretizations are geometrically ergodic and approximate π for vanishing discretization step size. Moreover, we prove that the discrete schemes satisfy the law of large numbers allowing to use consecutive iterates of a Markov chain in order to compute statistics of the stationary distribution posing a significant reduction of computational complexity in practice. Numerical experiments are provided confirming the theoretical findings and showcasing the practical relevance of the proposed methods in imaging applications.
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