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Minimax Theory and its Applications 04 (2019), No. 2, 305--328
Copyright Heldermann Verlag 2019

The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems

Oleksandr Nakonechnyi
T. Shevchenko University, Kyiv, Ukraine

Yuri Podlipenko
National Taras Shevchenko University, Kiev, Ukraine

Yury Shestopalov
University of Gävle, Gävle, Sweden

We present complete mathematical statements and perform detailed investigations of the minimax estimation problems of unknown data for the Helmholtz transmission problems from indirect noisy observations of their solutions. We construct optimal, in certain sense, estimates, which are called minimax mean-square estimates, of the values of linear functionals from unknown data. It is established that when unknown data and correlation functions of errors in observations belong to special sets, the minimax mean square estimates are expressed via solutions to certain transmission problems for systems of Helmholtz equations. We prove that these systems are uniquely solvable. Several possible generalizations of the techniques and results are proposed including applications to the problems with incomplete data and pointwise observations.

Keywords: Minimax estimation, noisy observations, inverse Helmholtz transmission problem, incomplete data, minimax mean-square estimates.

MSC: 49K20, 49N30, 49N45, 60H30, 78M50

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