Pryshlyak V. Garanteed estimation of solutions to boundary value problems for the Helmholtz equation under uncertainties.

Thesis for the Degree of Candidate of Sciences in Physics and Mathematics in speciality 01.05.04 -- system analysis and theory of optimal decisions. Taras Shevchenko National University, Kyiv, 2009. For the first time we consider the statement of the problem of minimax estimation of the parameters of external BVPs for the Helmholtz equation in arbitrary unbounded domains with finite boundary that arise in the mathematical theory of wave diffraction. For the systems described by such BVPs, we obtain representations for minimax estimates of the values of functionals from the observed solutions and right-hand sides that enter the problem statement; quadratic restrictions are imposed on unknown deterministic data and second moment of observation noise. We also obtain representations for the estimation errors. The representations are obtained in terms of the solutions to certain uniquely solvable systems of integro-differential and integral equations in bounded domains. When the unknown solutions of external BVPs for the Helmholtz equation are observed on a system of surfaces, we reduce the problem of finding minimax estimates to solving some integral equation systems on multi-connected surfaces (or contours), the latter being a union of the obstacle boundary and the surfaces on which the observations are made. We prove the unique solvability of the obtained integral equations. The estimation techniques elaborated in this work are of big importance for the development of the theory of inverse acoustic and electromagnetic wave scattering by bounded obstacles. Key words: minimax estimates, observations, systems of integro-differential equations, boundary value problems for the Helmholtz equation