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Combined error estimates for local fluctuations of SPDEs

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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 five main errors: the spatial discretization error, the local linearization error, the noise truncation 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 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.

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Acknowledgements

CK would like to thank the VolkswagenStiftung for support via a Lichtenberg Professorship. Furthermore, CK would like to thank Daniele Castellano for interesting discussions regarding moment equations. The work on this article was done while PK was affiliated with the Max Planck Institute for Dynamics of Complex Technical Systems in Magdeburg, Germany. We also thank two anonymous referees for their helpful comments.

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Correspondence to Patrick Kürschner.

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Communicated by: Ivan Oseledets

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Kuehn, C., Kürschner, P. Combined error estimates for local fluctuations of SPDEs. Adv Comput Math 46, 11 (2020). https://doi.org/10.1007/s10444-020-09766-2

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Keywords

  • Stochastic partial differential equation
  • Stochastic dynamics
  • Combined error estimates
  • Optimal regularity
  • Lyapunov equation
  • Low-rank approximation
  • Local fluctuations

Mathematics subject classification (2010)

  • 65C30
  • 65M99
  • 60H15
  • 35R60
  • 37M99
  • 15A24