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A nonparametric problem for stochastic PDEs

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Stochastic partial differential equations (SPDEs) are models for random processes evolving in time and space. They can be used to describe various phenomena, for example in neuroscience, oceanography, biophysics and finance. The practical implementation of SPDE models, however, hinges on a good understanding of the underlying information structure (in time and space) and on the development of robust statistical methods. In this talk, I will discuss the basic ideas for this analysis and explain what difficulties arise from the spatial structure. Based on recent results in (1), the focus will be on nonparametric inference for spatially localized measurements and the challenging journey to a suitable Bernstein-von-Mises theorem in the context of Bayesian statistics.

(1) R. Altmeyer, M. Reiss (2020). Nonparametric estimation for linear SPD Es from local measurements. Annals of Applied Probability, to appear. arXiv preprint arXiv:1903.06984.

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