Numerical analysis of lognormal diffusions on the sphere

Abstract

Numerical solutions of stationary diffusion equations on the unit sphere with isotropic lognormal diffusion coefficients are considered. Hölder 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.

35 pages, 1 figure; rewritten Sections 2 and 3, added numerical experiments

References in corpus (1)

• A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes

Cited by in corpus (5)

• Numerical approximation and simulation of the stochastic wave equation on the sphere
• Surface finite element approximation of spherical Whittle--Matérn Gaussian random fields
• Hölder Conditions of Local Times and Exact Moduli of non-differentiability for Spherical Gaussian fields
• Evolving surface finite element methods for random advection-diffusion equations
• Strong Local Nondeterminism and Exact Modulus of Continuity for Spherical Gaussian Fields