紧辛流形上的蒙特卡洛方法
Monte Carlo methods on compact symplectic manifolds
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中文总结 AI 辅助
该研究针对预量子化紧辛流形上的积分问题，采用Bochner-Schrödinger算子谱投影关联的行列式点过程作为求积节点构建无偏蒙特卡洛估计器，证明其满足中心极限定理且均方误差衰减速率达最优，扩展了紧复流形上的相关前期成果。
中文摘要 AI 辅助
我们在预量子化紧辛流形上，针对任意$C^1$函数关于光滑黎曼体积形式的积分，构建了一种无偏蒙特卡洛估计器，其求积节点采用与Bochner-Schrödinger算子的适当谱投影相关联的行列式点过程。我们证明该估计器满足中心极限定理，且均方误差的衰减速率达到了Bakhvalov在欧几里得空间中研究的最优最坏情况速率。这些结果扩展了Lemoine和Bardenet关于紧复流形上蒙特卡洛方法的前期成果。
英文摘要
We build an unbiased Monte Carlo estimator of the integral of any $C^1$ function on a prequantized compact symplectic manifold against a smooth Riemannian volume form, taking for quadrature nodes the determinantal point process associated with an appropriate spectral projection of the Bochner-Schrödinger operator. We show that the estimator satisfies a central limit theorem, and the decay rate of the mean squared error reaches the optimal worst-case rate investigated by Bakhvalov in Euclidean spaces. These results extend previous results of Lemoine and Bardenet on Monte Carlo methods on compact complex manifolds.
