Monte Carlo methods on compact complex manifolds using Bergman kernels

TL;DR

This paper introduces a novel Monte Carlo integration method on compact complex manifolds using Bergman kernels and determinantal point processes, achieving faster convergence rates than previous methods.

Contribution

It develops an unbiased Monte Carlo estimator with a central limit theorem for integration on complex manifolds, utilizing Bergman kernels for quadrature nodes, and demonstrates improved convergence rates.

Findings

Mean squared error decays as N^{-1-2/d_R}

Faster convergence than previous DPP-based quadratures

Numerical validation on the Riemann sphere

Abstract

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.