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的一项具有挑战性的任务。当分布在不同模态之间具有精确对称性时，直接在这些模态之间跳跃可以显著加速采样过程。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。我们首先通过与近似对称性相关的群轨道平均，从目标分布构造出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，我们应用多层蒙特卡洛方法。我们讨论了如何结合退火重要性采样与回火转移来实现这些步骤。与传统多层方法相比，由于参考分布与目标分布更为接近，所提方法可能更有效。文中展示了伊辛模型的数值结果，以说明所提方法的效率。