  Monte Carlo integration is fundamental in scientific and statistical
computation, but requires reliable samples from the target distribution, which
poses a substantial challenge in the case of multi-modal distributions.
Existing methods often involve time-consuming tuning, and typically lack
tailored estimators for efficient use of the samples. This paper adapts the
Warp-U transformation [Wang et al., 2022] to form multi-modal sampling strategy
called Warp-U sampling. It constructs a stochastic map to transport a
multi-modal density into a uni-modal one, and subsequently inverts the
transport but with new stochasticity injected. For efficient use of the samples
for normalising constant estimation, we propose (i) an unbiased estimation
scheme based coupled chains, where the Warp-U sampling is used to reduce the
coupling time; and (ii) a stochastic Warp-U bridge sampling estimator, which
improves its deterministic counterpart given in Wang et al. [2022]. Our overall
approach requires less tuning and is easier to apply than common alternatives.
Theoretically, we establish the ergodicity of our sampling algorithm and that
our stochastic Warp-U bridge sampling estimator has greater (asymptotic)
precision per CPU second compared to the Warp-U bridge estimator of Wang et al.
[2022] under practical conditions. The advantages and current limitations of
our approach are demonstrated through simulation studies and an application to
exoplanet detection.
