Monte Carlo methods on compact complex manifolds using Bergman kernels
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916247424204800 |
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| author | Lemoine, Thibaut Bardenet, Rémi |
| author_facet | Lemoine, Thibaut Bardenet, Rémi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09203 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monte Carlo methods on compact complex manifolds using Bergman kernels Lemoine, Thibaut Bardenet, Rémi Complex Variables Numerical Analysis Probability 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. |
| title | Monte Carlo methods on compact complex manifolds using Bergman kernels |
| topic | Complex Variables Numerical Analysis Probability |
| url | https://arxiv.org/abs/2405.09203 |