scholarly journals Optimal Monte Carlo integration on closed manifolds

2019 ◽  
Vol 29 (6) ◽  
pp. 1203-1214 ◽  
Author(s):  
Martin Ehler ◽  
Manuel Gräf ◽  
Chris. J. Oates
Keyword(s):  
Monte Carlo ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Monte Carlo Integration ◽  
Worst Case ◽  
Logarithmic Factor ◽  
Approximation Rates ◽  
Bayesian Monte Carlo ◽  
Kernel Hilbert Spaces ◽  
Theoretical Results

Abstract 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.

scholarly journals Derandomised lattice rules for high dimensional integration

2019 ◽  
Vol 60 ◽  
pp. C247-C260
Author(s):  
Y. Kazashi ◽  
F. Y. Kuo ◽  
I. H. Sloan
Keyword(s):  
Monte Carlo ◽  
Sobolev Spaces ◽  
Monte Carlo Integration ◽  
High Dimensional ◽  
Lattice Rules ◽  
Worst Case ◽  
Multiple Integration ◽  
Quasi Monte Carlo ◽  
Lattice Methods ◽  
Numer Math

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of \(1/(2N)\) where \(N\) is the number of points in the lattice. By applying the principle that there is always at least one choice as good as the average, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order \({1/N^2}\). We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging. References J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numer., 22:133–288, 2013. doi:10.1017/S0962492913000044. J. Dick, D. Nuyens, and F. Pillichshammer. Lattice rules for nonperiodic smooth integrands. Numer. Math., 126(2):259–291, 2014. doi:10.1007/s00211-013-0566-0. T. Goda, K. Suzuki, and T. Yoshiki. Lattice rules in non-periodic subspaces of sobolev spaces. Numer. Math., 141(2):399–427, 2019. doi:10.1007/s00211-018-1003-1. F. Y. Kuo. Lattice rule generating vectors. URL http://web.maths.unsw.edu.au/ fkuo/lattice/index.html. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput., 75:903–920, 2006. doi:10.1090/S0025-5718-06-01785-6. I. H. Sloan and S. Joe. Lattice methods for multiple integration. Oxford Science Publications. Clarendon Press and Oxford University Press, 1994. URL https://global.oup.com/academic/product/lattice-methods-for-multiple-integration-9780198534723. I. H. Sloan and H. Wozniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex., 14(1):1–33, 1998. doi:10.1006/jcom.1997.0463. I. H. Sloan, F. Y. Kuo, and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput., 71:1609–1641, 2002. doi:10.1090/S0025-5718-02-01420-5.

scholarly journals ESTIMATION OF VOLUME USING THE NUCLEATOR AND LATTICE POINTS

2019 ◽  
Vol 38 (2) ◽  
pp. 141
Author(s):  
Domingo Gomez-Perez ◽  
Javier Gonzalez-Villa ◽  
Florian Pausinger
Keyword(s):  
Monte Carlo ◽  
Compact Subset ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Spaces ◽  
Lattice Points ◽  
Monte Carlo Integration ◽  
Quasi Monte Carlo ◽  
Kernel Hilbert Spaces

The nucleator is a method to estimate the volume of a particle, i.e. a compact subset of ℝ3, which is widely used in Stereology. It is based on geometric sampling and known to be unbiased. However, the prediction of the variance of this estimator is non-trivial and depends on the underlying sampling scheme.We propose well established tools from quasi-Monte Carlo integration to address this problem. In particular, we show how the theory of reproducing kernel Hilbert spaces can be used for variance prediction and how the variance of estimators based on the nucleator idea can be reduced using lattice (or lattice-like) points. We illustrate and test our results on various examples.

scholarly journals Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

scholarly journals Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces

2020 ◽  
pp. 1-34
Author(s):  
Ernesto De Vito ◽  
Nicole Mücke ◽  
Lorenzo Rosasco
Keyword(s):  
Sobolev Spaces ◽  
Riemannian Manifolds ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Differential Operators ◽  
Reproducing Kernels ◽  
Reproducing Kernel Hilbert Spaces ◽  
Kernel Hilbert Spaces ◽  
Euclidean Setting ◽  
And Diffusion

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples. While connections between Sobolev spaces, differential operators and RKHS are well known in the Euclidean setting, here we present a self-contained study of analogous connections for Riemannian manifolds. By collecting a number of results in unified a way, we think our study can be useful for researchers interested in the topic.

Quasi-Monte Carlo tractability of integration problem in function spaces defined over products of balls

2019 ◽  
Vol 17 (06) ◽  
pp. 1950043
Author(s):  
Jie Zhang ◽  
Yongping Liu
Keyword(s):  
Monte Carlo ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Weighted Sobolev Spaces ◽  
Sufficient Conditions ◽  
Quasi Monte Carlo ◽  
Integration Problem ◽  
Kernel Hilbert Spaces ◽  
Necessary And Sufficient ◽  
Uniformly Bounded

Recently, quasi-Monte Carlo (QMC) rules for multivariate integration in some weighted Sobolev spaces of functions defined over unit cubes [Formula: see text], products of [Formula: see text] copies of the simplex [Formula: see text] and products of [Formula: see text] copies of the unit sphere [Formula: see text] have been well-studied in the literature. In this paper, we consider QMC tractability of integrals of functions defined over product of [Formula: see text] copies of the ball [Formula: see text]. The space is a tensor product of [Formula: see text] reproducing kernel Hilbert spaces defined by positive and uniformly bounded weights [Formula: see text] for [Formula: see text]. We obtain matching necessary and sufficient conditions in terms of this weights for various notions of QMC tractability, including strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniformly weak tractability and [Formula: see text]-weak tractability.

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