Optimal Monte Carlo integration on closed manifolds
| Autor: | Martin Ehler, Manuel Gräf, Chris J. Oates |
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| Rok vydání: | 2017 |
| Předmět: |
Statistics and Probability
Unit sphere FOS: Computer and information sciences Pure mathematics Reproducing kernel Monte Carlo method 010103 numerical & computational mathematics 01 natural sciences Theoretical Computer Science Methodology (stat.ME) 010104 statistics & probability symbols.namesake FOS: Mathematics Mathematics - Numerical Analysis 0101 mathematics Statistics - Methodology Mathematics Simplex 65C05 65C60 Bayesian cubature Hilbert space Order (ring theory) Numerical Analysis (math.NA) Covering radius Sobolev space Computational Theory and Mathematics Cover (topology) symbols Monte Carlo integration Statistics Probability and Uncertainty |
| DOI: | 10.48550/arxiv.1707.04723 |
| Popis: | The worst case integration error in reproducing kernel Hilbert spaces of standard Monte Carlo methods with n random points decays as $$n^{-1/2}$$n-1/2. However, the re-weighting of random points, as exemplified in the Bayesian Monte Carlo method, can sometimes be used to improve the convergence order. This paper contributes general theoretical results for Sobolev spaces on closed Riemannian manifolds, where we verify that such re-weighting yields optimal approximation rates up to a logarithmic factor. We also provide numerical experiments matching the theoretical results for some Sobolev spaces on the sphere $${\mathbb {S}}^2$$S2 and on the Grassmannian manifold $${\mathcal {G}}_{2,4}$$G2,4. Our theoretical findings also cover function spaces on more general sets such as the unit ball, the cube, and the simplex. |
| Databáze: | OpenAIRE |
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