We describe and analyze some Monte Carlo methods for manifolds in Euclidean
space defined by equality and inequality constraints. First, we give an MCMC
sampler for probability distributions defined by un-normalized densities on
such manifolds. The sampler uses a specific orthogonal projection to the
surface that requires only information about the tangent space to the manifold,
obtainable from first derivatives of the constraint functions, hence avoiding
the need for curvature information or second derivatives. Second, we use the
sampler to develop a multi-stage algorithm to compute integrals over such
manifolds. We provide single-run error estimates that avoid the need for
multiple independent runs. Computational experiments on various test problems
show that the algorithms and error estimates work in practice. The method is
applied to compute the entropies of different sticky hard sphere systems. These
predict the temperature or interaction energy at which loops of hard sticky
spheres become preferable to chains.

针对欧几里得空间中相等和不等约束定义的流形，我们描述并分析了一些蒙特卡洛方法，其中提出了一个采用正交投影进行采样的 MCMC 采样器，来计算该流形上定义的非归一化概率分布。我们使用这个采样器来开发一个多阶段算法，并提供单次运行误差估计，以计算这种流形上的积分。计算实验表明算法和误差估计在实践中成立。该方法应用于计算不同粘性硬球体系的熵，从而预测硬粘球链环变得优于链时的温度或相互作用能量。