This paper is concerned with sampling from probability distributions $\pi$ on $\mathbb{R}^d$ admitting a density of the form $\pi(x) \propto e^{-U(x)}$, where $U(x)=F(x)+G(Kx)$ with $K$ being a linear operator and $G$ being non-differentiable. Two different methods are proposed, both employing a subgradient step with respect to $G\circ K$, but, depending on the regularity of $F$, either an explicit or an implicit gradient step with respect to $F$ can be implemented. For both methods, non-asymptotic convergence proofs are provided, with improved convergence results for more regular $F$. Further, numerical experiments are conducted for simple 2D examples, illustrating the convergence rates, and for examples of Bayesian imaging, showing the practical feasibility of the proposed methods for high dimensional data.

翻译：本文关注从概率分布$\pi$在$\mathbb{R}^d$上的采样问题，该分布具有密度形式$\pi(x) \propto e^{-U(x)}$，其中$U(x)=F(x)+G(Kx)$，且$K$为线性算子，$G$不可微。本文提出了两种不同方法，两者均采用关于$G\circ K$的次梯度步骤，但根据$F$的正则性，可分别实现针对$F$的显式或隐式梯度步骤。两种方法均给出了非渐近收敛性证明，且对于更正则的$F$获得了改进的收敛结果。此外，通过简单二维示例的数值实验验证了收敛速率，并通过贝叶斯成像示例展示了所提方法在高维数据中的实际可行性。