Report
Monte Carlo methods on compact complex manifolds using Bergman kernels
| العنوان: | Monte Carlo methods on compact complex manifolds using Bergman kernels |
|---|---|
| المؤلفون: | Lemoine, Thibaut, Bardenet, Rémi |
| المساهمون: | Centre de Recherche en Informatique, Signal et Automatique de Lille - UMR 9189 (CRIStAL), Centrale Lille-Université de Lille-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS), ANR-20-CHIA-0002,Baccarat,Apprentissage bayésien pour les modèles coûteux, avec applications à la biologie cellulaire(2020), European Project: ERC-2019-STG-851866,Blackjack |
| المصدر: | https://hal.science/hal-04575374 ; 2024. |
| بيانات النشر: | CCSD |
| سنة النشر: | 2024 |
| المجموعة: | LillOA (HAL Lille Open Archive, Université de Lille) |
| مصطلحات موضوعية: | Bergman kernel, complex manifolds, determinantal point processes, Monte Carlo integration, MSC Classification: 65C05, 32Q10 (primary), 60F5, 32U05 (secondary), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], [MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] |
| الوصف: | 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 $d_\R=2d$, 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. |
| نوع الوثيقة: | report |
| اللغة: | English |
| Relation: | info:eu-repo/grantAgreement//ERC-2019-STG-851866/EU/Fast Monte Carlo integration with repulsive processes/Blackjack |
| الاتاحة: | https://hal.science/hal-04575374 https://hal.science/hal-04575374v1/document https://hal.science/hal-04575374v1/file/MC_Bergman.pdf |
| Rights: | http://creativecommons.org/licenses/by/ ; info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.2226F958 |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |