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Free keywords:
Mathematics, Numerical Analysis, math.NA,Mathematics, Dynamical Systems, math.DS,Mathematics, Probability, math.PR
Abstract:
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.
