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Minimax Theory and its Applications 03 (2018), No. 1, 001--021
Copyright Heldermann Verlag 2018

Minimax State Estimates for Abstract Neumann Problems

Sergiy Zhuk
IBM Research, Dublin, Ireland

Olexander Nakonechnii
T. Shevchenko University, Kyiv, Ukraine

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 the observation operator and the 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.

Keywords: Minimax, filtering, linear operator equations, Neumann problems, pseudo-inversion, differentiation.

MSC: 93C05, 93C20, 93C25, 93C41, 93E10, 93B03, 93B07

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