A New Quasi-Monte Carlo Algorithm for Numerical Integration of Smooth Functions
نویسندگان
چکیده
Bachvalov proved that the optimal order of convergence of a Monte Carlo method for numerical integration of functions with bounded kth order derivatives is O ( N− k s − 2 ) , where s is the dimension. We construct a new Monte Carlo algorithm with such rate of convergence, which adapts to the variations of the sub-integral function and gains substantially in accuracy, when a low-discrepancy sequence is used instead of pseudo-random numbers. Theoretical estimates of the worst-case error of the method are obtained. Experimental results, showing the excellent parallelization properties of the algorithm and its applicability to problems of moderately high dimension, are also presented.
منابع مشابه
Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order
We define a Walsh space which contains all functions whose partial mixed derivatives up to order δ ≥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain reproducing kernel Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital (t, α, s)-sequences achieve the ...
متن کاملNumerical Integration of Multivariate
In the present paper we study quasi-Monte Carlo methods to integrate functions representable by generalized Haar series in high dimensions. Using (t; m; s)-nets to calculate the quasi-Monte Carlo approximation, we get best possible estimates of the integration error for practically relevant classes of functions. The local structure of the Haar functions yields interesting new aspects in proofs ...
متن کاملScrambled geometric net integration over general product spaces
Quasi-Monte Carlo (QMC) sampling has been developed for integration over [0, 1] where it has superior accuracy to Monte Carlo (MC) for integrands of bounded variation. Scrambled net quadrature gives allows replication based error estimation for QMC with at least the same accuracy and for smooth enough integrands even better accuracy than plain QMC. Integration over triangles, spheres, disks and...
متن کاملAdaptive integration and approximation over hyper-rectangular regions with applications to basket option pricing
We describe an adaptive algorithm to compute piecewise sparse polynomial approximations and the integral of a multivariate function over hyper-rectangular regions in medium dimensions. The key ingredient is a quasi-Monte Carlo quadrature rule which can handle the numerical integration of both very regular and less regular functions. Numerical tests are performed on functions taken from Genz pac...
متن کاملQuasi - Monte Carlo Methods in Computer Graphics
The problem of global illumination in computer graphics is described by a Fredholm integral equation of the second kind. Due to the complexity of this equation, Monte Carlo methods provide an efficient tool for the estimation of the solution. A new approach, using quasi-Monte Carlo integration, is introduced and compared to Monte Carlo integration. We discuss some theoretical aspects and give n...
متن کامل