Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y_n is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Y_n to Y in terms of the error |E[Y - Y_n]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y] - E_N[Y_n]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |E_N[Y - Y_n]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y - Y_n]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations.Comment: 16 pages, 5 figures; formulated Section 2 independently of SPDEs, shortened Section 3, added example of geometric Brownian motion in Section

Isotropic Gaussian random fields on the sphere: Regularity, fast simulation and stochastic partial differential equations

Isotropic Gaussian random fields on the sphere are characterized by Karhunen-Lo\`{e}ve expansions with respect to the spherical harmonic functions and the angular power spectrum. The smoothness of the covariance is connected to the decay of the angular power spectrum and the relation to sample H\"{o}lder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Lo\`{e}ve expansions in terms of the covariance spectrum are established, and algorithmic aspects of fast sample generation via fast Fourier transforms on the sphere are indicated. The relevance of the results on sample regularity for isotropic Gaussian random fields and the corresponding lognormal random fields on the sphere for several models from environmental sciences is indicated. Finally, the stochastic heat equation on the sphere driven by additive, isotropic Wiener noise is considered, and strong convergence rates for spectral discretizations based on the spherical harmonic functions are proven.Comment: Published at http://dx.doi.org/10.1214/14-AAP1067 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Fast simulation of Gaussian random fields

Fast Fourier transforms are used to develop algorithms for the fast generation of correlated Gaussian random fields on d-dimensional rectangular regions. The complexities of the algorithms are derived, simulation results and error analysis are presented.Comment: 15 pages, 8 figures. Typos corrected in Algorithm 3, Remark (4), Algorithm 4, Remark (5), and Algorithm 5, Remark (5

Kolmogorov-Chentsov theorem and differentiability of random fields on manifolds

A version of the Kolmogorov-Chentsov theorem on sample differentiability and H\"older continuity of random fields on domains of cone type is proved, and the result is generalized to manifolds.Comment: 8 pages. Potential Analysis, February 201

Numerical analysis of lognormal diffusions on the sphere

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. H\"older regularity in $L^p$ sense for isotropic Gaussian random fields is obtained and related to the regularity of the driving lognormal coefficients. This yields regularity in $L^p$ sense of the solution to the diffusion problem in Sobolev spaces. Convergence rate estimates of multilevel Monte Carlo Finite and Spectral Element discretizations of these problems are then deduced. Specifically, a convergence analysis is provided with convergence rate estimates in terms of the number of Monte Carlo samples of the solution to the considered diffusion equation and in terms of the total number of degrees of freedom of the spatial discretization, and with bounds for the total work required by the algorithm in the case of Finite Element discretizations. The obtained convergence rates are solely in terms of the decay of the angular power spectrum of the (logarithm) of the diffusion coefficient. Numerical examples confirm the presented theory.Comment: 35 pages, 1 figure; rewritten Sections 2 and 3, added numerical experiment

Mean-square stability analysis of approximations of stochastic differential equations in infinite dimensions

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory.Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of results; corrected typo

Erratum: Fast simulation of Gaussian random fields[Monte Carlo Methods Appl. 17 (2011), 195-214]

In the paper "Fast simulation of Gaussian random fields”, a typo occurred. Instead of it should read in Algorithm3.1, Remarkd). For convenience of the reader we reproduce below the complete corrected algorithm

Galerkin--Chebyshev approximation of Gaussian random fields on compact Riemannian manifolds

A new numerical approximation method for a class of Gaussian random fields on compact Riemannian manifolds is introduced. This class of random fields is characterized by the Laplace--Beltrami operator on the manifold. A Galerkin approximation is combined with a polynomial approximation using Chebyshev series. This so-called Galerkin--Chebyshev approximation scheme yields efficient and generic sampling algorithms for Gaussian random fields on manifolds. Strong and weak orders of convergence for the Galerkin approximation and strong convergence orders for the Galerkin--Chebyshev approximation are shown and confirmed through numerical experiments.Comment: Version submitted to a peer-reviewed journal. Changes: fixed residual typos, new outline. 33 pages, 5 figure

Numerical approximation and simulation of the stochastic wave equation on the sphere

Solutions to the stochastic wave equation on the unit sphere are approximated by spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided and shown to depend only on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schr\uf6dinger equation on the unit sphere

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived inL 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler-Maruyama approximation. Finally, simulations complete the pape