Benutzerhandbuch Datenschutzhinweis Impressum Kontakt





Combined Error Estimates for Local Fluctuations of SPDEs


Kürschner,  Patrick
Computational Methods in Systems and Control Theory, Max Planck Institute for Dynamics of Complex Technical Systems, Max Planck Society;

Externe Ressourcen
Es sind keine Externen Ressourcen verfügbar
Volltexte (frei zugänglich)

(Preprint), 296KB

Ergänzendes Material (frei zugänglich)
Es sind keine frei zugänglichen Ergänzenden Materialien verfügbar

Kuehn, C., & Kürschner, P. (in preparation). Combined Error Estimates for Local Fluctuations of SPDEs.

In this work, we study the numerical approximation of local fluctuations of certain classes of parabolic stochastic partial differential equations (SPDEs). Our focus is on effects for small spatially-correlated noise on a time scale before large deviation effects have occurred. In particular, we are interested in the local directions of the noise described by a covariance operator. We introduce a new strategy and prove a Combined ERror EStimate (CERES) for the four main errors: the spatial discretization error, the local linearization error, the local relaxation error to steady state, and the approximation error via an iterative low-rank matrix algorithm. In summary, we obtain one CERES describing, apart from modelling of the original equations and standard round-off, all the sources of error for a local fluctuation analysis of an SPDE in one estimate. To prove our results, we rely on a combination of methods from optimal Galerkin approximation of SPDEs, covariance moment estimates, analytical techniques for Lyapunov equations, iterative numerical schemes for low-rank solution of Lyapunov equations, and working with related spectral norms for different classes of operators.