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Minimax Theory and its Applications 01 (2016), No. 2, 197--244
Copyright Heldermann Verlag 2016

Mixed Variational Approach to Finding Guaranteed Estimates for Solutions and Right-Hand Sides of the Second-Order Linear Elliptic Equations under Incomplete Data

Yuri Podlipenko
National Taras Shevchenko University, Kiev, Ukraine

Yury Shestopalov
University of Gävle, Gävle, Sweden

We investigate the problem of guaranteed estimation of values of linear continuous functionals defined on solutions to mixed variational equations generated by linear elliptic problems from indirect noisy observations of these solutions. We assume that right-hand sides of the equations, as well as the second moments of noises in observations are not known; the only available information is that they belong to given bounded sets in the appropriate functional spaces. We are looking for linear with respect to observations optimal estimates of solutions of aforementioned equations called minimax or guaranteed estimates. We develop constructive methods for finding these estimates and estimation errors which are expressed in terms of solutions to special mixed variational equations and prove that Galerkin approximations of the obtained variational equations converge to their exact solutions. Finally we study the problem of guaranteed estimation of right-hand sides of mixed variational equations.

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