
Minimax Theory and its Applications 01 (2016), No. 2, 197244 Copyright Heldermann Verlag 2016 Mixed Variational Approach to Finding Guaranteed Estimates for Solutions and RightHand Sides of the SecondOrder Linear Elliptic Equations under Incomplete Data Yuri Podlipenko National Taras Shevchenko University, Kiev, Ukraine yourip@mail.ru Yury Shestopalov University of Gävle, Gävle, Sweden yuyshv@hig.se 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 righthand 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 righthand sides of mixed variational equations. [ Fulltextpdf (363 KB)] for subscribers only. 