Monte Carlo methods on compact complex manifolds using Bergman kernels
<|reference_start|>Monte Carlo methods on compact complex manifolds using Bergman kernels: In this paper, we propose a new randomized method for numerical integration on a compact complex manifold with respect to a continuous volume form. Taking for quadrature nodes a suitable determinantal point process, we build an unbiased Monte Carlo estimator of the integral of any Lipschitz function, and show that the estimator satisfies a central limit theorem, with a faster rate than under independent sampling. In particular, seeing a complex manifold of dimension $d$ as a real manifold of dimension $d_{\mathbb{R}}=2d$, the mean squared error for $N$ quadrature nodes decays as $N^{-1-2/d_{\mathbb{R}}}$; this is faster than previous DPP-based quadratures and reaches the optimal worst-case rate investigated by [Bakhvalov 1965] in Euclidean spaces. The determinantal point process we use is characterized by its kernel, which is the Bergman kernel of a holomorphic Hermitian line bundle, and we strongly build upon the work of Berman that led to the central limit theorem in [Berman, 2018].We provide numerical illustrations for the Riemann sphere.<|reference_end|>
@article{lemoine2024monte,
    title={Monte Carlo methods on compact complex manifolds using Bergman kernels},
    author={Thibaut Lemoine (CRIStAL), R'emi Bardenet (TAO, CRIStAL)},
    journal={arXiv preprint arXiv:2405.09203},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09203},
    primaryClass={math.CV cs.NA math.NA math.PR}
}