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Numerical Analysis of Lognormal Diffusions on the Sphere
(2016)Research reports / Seminar for Applied MathematicsReport -
Isotropic Gaussian random fields on the sphere
(2013)SAM Research ReportSample regularity and fast simulation of isotropic Gaussian random fields on the sphere are for example of interest for the numerical analysis of stochastic partial differential equations and for the simulation of ice crystals or Saharan dust particles as lognormal random fields. We recall the results from SAM Report 2013-15, which include the approximation of isotropic Gaussian random fields with convergence rates as well as the regularity ...Report -
Isotropic Gaussian random fields on the sphere: regularity, fast simulation, and stochastic partial differential equations
(2013)SAM Research ReportIsotropic Gaussian random fields on the sphere are characterized by Karhunen-Loève 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 Hölder continuity and sample differentiability of the random fields is discussed. Rates of convergence of their finitely truncated Karhunen-Loève ...Report -
Covariance structure of parabolic stochastic partial differential equations
(2012)SAM Research ReportIn this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.Report -
Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations
(2011)SAM Research ReportWe analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time ...Report -
Multilevel Monte Carlo method for parabolic stochastic partial differential equations
(2011)SAM Research ReportWe analyze the convergence and complexity of multilevel Monte Carlo discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show under low regularity assumptions on the solution that the judicious combination of low order Galerkin discretizations in space and an Euler-Maruyama discretization in time yields mean square convergence of order one in space and of order 1/2 in time to ...Report