Tractability of the Quasi-Monte Carlo Quadrature with Halton Points for Elliptic Pdes with Random Diffusion
نویسندگان
چکیده
This article is dedicated to the computation of the moments of the solution to stochastic partial differential equations with log-normal distributed diffusion coefficient by the Quasi-Monte Carlo method. Our main result is the polynomial tractability for the QuasiMonte Carlo method based on the Halton sequence. As a by-product, we obtain also the strong tractability of stochastic partial differential equations with uniformly elliptic diffusion coefficient by the Quasi-Monte Carlo method. Numerical experiments are given to validate the theoretical findings.
منابع مشابه
Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in R (d = 1, 2, 3), with diffusion coefficient a(x, ω) given as a lognormal random field, i.e., a(x, ω) = exp(Z(x, ω)) where x is the spatial variable and Z(x, ·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded ...
متن کاملA qMC-spectral method for elliptic PDEs with random coefficients on the unit sphere
We present a quasi-Monte Carlo spectral method for a class of elliptic partial differential equations (PDEs) with random coefficients defined on the unit sphere. The random coefficients are parametrised by the Karhunen-Loève expansion, while the exact solution is approximated by the spherical harmonics. The expectation of the solution is approximated by a quasi-Monte Carlo integration rule. A m...
متن کاملFast QMC Matrix-Vector Multiplication
Quasi-Monte Carlo (QMC) rules 1/N ∑N−1 n=0 f(ynA) can be used to approximate integrals of the form ∫ [0,1]s f(yA) dy, where A is a matrix and y is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications fr...
متن کاملA Constructive Approach to Strong Tractability Using Quasi-Monte Carlo Algorithms
We prove in a constructive way that multivariate integration in appropriate weighted Sobolev classes is strongly tractable and the e-exponent of strong tractability is 1 (which is the best-possible value) under a stronger assumption than Sloan and Wo! zniakowski’s assumption. We show that quasi-Monte Carlo algorithms based on the Sobol sequence and Halton sequence achieve the convergence order ...
متن کاملThe error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension
Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013