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：多模态分布采样是科学计算中的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，直接在这些模式间跳跃能显著加速采样过程。然而，大多数实际应用中的分布并不具备精确对称性。本文考虑具有近似对称性的分布。我们首先通过沿近似对称性对应的群轨道取平均，从目标分布构造一个精确对称的参考分布。接着，通过构建参考分布与目标分布之间的连续路径，应用多层级蒙特卡洛方法。我们讨论了如何通过退火重要采样和回火转移实现这些步骤。与传统多层级方法相比，本文提出的方法更为有效，因为参考分布与目标分布更加接近。通过伊辛模型的数值结果验证了所提方法的效率。