Sampling from multimodal distributions is a challenging task in scientific computing. When a distribution has an exact symmetry between the modes, direct jumps among them can accelerate the samplings significantly. However, the distributions from most applications do not have exact symmetries. This paper considers the distributions with approximate symmetries. We first construct an exactly symmetric reference distribution from the target one by averaging over the group orbit associated with the approximate symmetry. Next, we can apply the multilevel Monte Carlo methods by constructing a continuation path between the reference and target distributions. We discuss how to implement these steps with annealed importance sampling and tempered transitions. Compared with traditional multilevel methods, the proposed approach can be more effective since the reference and target distributions are much closer. Numerical results of the Ising models are presented to illustrate the efficiency of the proposed method.

翻译：从多模态分布中采样是科学计算中一项具有挑战性的任务。当分布的模式间存在精确对称性时，直接跳跃于各模式之间可显著加速采样过程。然而，大多数应用场景中的分布并不具备精确对称性。本文针对具有近似对称性的分布展开研究。我们首先通过在近似对称性对应的群轨道上进行平均，从目标分布构造出精确对称的参考分布。随后，通过构建参考分布与目标分布之间的连续路径，应用多层蒙特卡洛方法。我们讨论了如何结合退火重要采样和温度跃迁技术实现上述步骤。相较于传统多层方法，由于参考分布与目标分布更为接近，所提方法可取得更高的效率。最后给出伊辛模型的数值结果以验证该方法的有效性。