This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive noise, (2) the problem is ill-posed and regularization is introduced in a Bayesian framework by an a prior density, and (3) the latter is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one. This paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that this approach is particularly effective and remarkably simple. In addition, it offers a guarantee of convergence in a clearly identified situation. The results are clearly confirmed by numerical simulations.

翻译：本文针对以下情况下的反演问题展开研究：(1)观测系统由线性变换和加性噪声建模；(2)问题具有不适定性，需在贝叶斯框架中通过先验密度引入正则化；(3)该先验密度通过基于可用大规模示例集调整的扩散过程进行建模。在此背景下，后验采样问题已知具有挑战性。本文提出一种吉布斯采样算法。该研究路径尚未被深入探索，我们证明该方法具有显著的有效性与简洁性。此外，该方法在明确界定的条件下提供收敛性保证。数值模拟结果充分验证了上述结论。