Monte Carlo integration is a powerful tool for scientific and statistical computation, but faces significant challenges when the integrand is a multi-modal distribution, even when the mode locations are known. This work introduces novel Monte Carlo sampling and integration estimation strategies for the multi-modal context by leveraging a generalized version of the stochastic Warp-U transformation Wang et al. [2022]. We propose two flexible classes of Warp-U transformations, one based on a general location-scale-skew mixture model and a second using neural ordinary differential equations. We develop an efficient sampling strategy called Warp-U sampling, which applies a Warp-U transformation to map a multi-modal density into a uni-modal one, then inverts the transformation with injected stochasticity. In high dimensions, our approach relies on information about the mode locations, but requires minimal tuning and demonstrates better mixing properties than conventional methods with identical mode information. To improve normalizing constant estimation once samples are obtained, we propose a stochastic Warp-U bridge sampling estimator, which we demonstrate has higher asymptotic precision per CPU second compared to the original approach proposed by Wang et al. [2022]. We also establish the ergodicity of our sampling algorithm. The effectiveness and current limitations of our methods are illustrated through simulation studies and an application to exoplanet detection.

翻译：蒙特卡洛积分是科学与统计计算的有力工具，但在被积函数为多模态分布时面临显著挑战，即使已知模态位置。本研究通过利用Wang等人[2022]提出的随机Warp-U变换的广义版本，针对多模态场景提出了新颖的蒙特卡洛采样与积分估计策略。我们提出了两类灵活的Warp-U变换：第一类基于广义位置-尺度-偏度混合模型，第二类采用神经常微分方程。我们开发了一种称为Warp-U采样的高效采样策略，该策略通过Warp-U变换将多模态密度映射为单模态密度，随后注入随机性进行逆变换。在高维情形中，我们的方法依赖于模态位置信息，但仅需极少参数调整，并展现出比具有相同模态信息的传统方法更好的混合特性。为在获得样本后改进归一化常数估计，我们提出了一种随机Warp-U桥式采样估计器，经证明其单位CPU时间内的渐近精度高于Wang等人[2022]提出的原始方法。我们还证明了采样算法的遍历性。通过模拟研究和系外行星探测的实际应用，展示了所提方法的有效性与当前局限性。