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：本文研究从 $\mathbb{R}^d$ 上具有形式 $\pi(x) \propto e^{-U(x)}$ 密度（其中 $U(x)=F(x)+G(Kx)$，$K$ 为线性算子，$G$ 不可微）的概率分布 $\pi$ 中采样的问题。提出两种不同方法，均采用关于 $G\circ K$ 的次梯度步骤，但根据 $F$ 的正则性，可分别实现关于 $F$ 的显式或隐式梯度步骤。针对两种方法，均提供了非渐近收敛性证明，且当 $F$ 更正则时收敛结果得到改进。此外，通过简单二维示例进行数值实验以展示收敛速率，并针对贝叶斯成像实例验证了所提方法在高维数据中的实际可行性。