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Minimax Theory and its Applications 03 (2018), No. 1, [final page numbers not yet available]
Copyright Heldermann Verlag 2018



Minimax State Estimates for Abstract Neumann Problems

Sergiy Zhuk
IBM Research, Dublin, Ireland
sergiy.zhuk@ie.ibm.com

Olexander Nakonechnii
T. Shevchenko University, Kyiv, Ukraine
nakonechniy@unicyb.kiev.ua



We present analytic expressions of minimax (worst-case) estimates for the solution of linear abstract Neumann problems in Hilbert space with uncertain (not necessarily bounded!) inputs and boundary conditions given incomplete observations with stochastic noise. The latter is assumed to have uncertain but bounded correlation operator. It is demonstrated that the minimax estimate is asymptotically exact under mild assumptions on observations and bounding sets. A relationship between the proposed estimates and a robust pseudo-inversion of compact operators is revealed. This relationship is demonstrated on an academic numerical example: homogeneous Neumann problem for Poisson equation in two spatial dimensions.

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