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Combined Error Estimates for Local Fluctuations of SPDEs

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

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フルテキスト (公開)

1611.04629.pdf
(プレプリント), 296KB

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引用

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


引用: http://hdl.handle.net/11858/00-001M-0000-002E-73A4-2
要旨
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.