Yuri Podlipenko | Searcharxiv

arXiv2018

The minimax approach to the estimation of solutions to first order linear systems of ordinary differential periodic equations with inexact data

We consider first-order linear systems of ordinary differential equations with periodic coefficients. Supposing that right-hand sides of equations are not known and subjected to some quadratic restrictions, we obtain optimal, in certain sense, estimates of solutions to above-mentioned problems from indirect noisy observations of these solutions on a finite system of points and intervals.

math.CA↗

arXiv2014

Mixed variational approach to finding guaranteed estimates from solutions and right-hand sides of the second-order linear elliptic equations under incomplete data

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 look 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. We study also the problem of guaranteed estimation of right-hand sides of mixed variational equations.

math.AP↗