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The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems
National Taras Shevchenko University, Kiev, Ukraine .
National Taras Shevchenko University, Kiev, Ukraine .
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electrical Engineering, Mathematics and Science, Mathematics.
2019 (English)In: Minimax Theory and its Applications, ISSN 2199-1413, Vol. 4, no 2, p. 305-328Article in journal (Refereed) Published
Abstract [en]

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. 

Place, publisher, year, edition, pages
Heldermann Verlag , 2019. Vol. 4, no 2, p. 305-328
Keywords [en]
Minimax estimation, noisy observations, inverse Helmholtz transmission problem, incomplete data, minimax mean-square estimates
National Category
Mathematics
Identifiers
URN: urn:nbn:se:hig:diva-30972ISI: 000495981800005OAI: oai:DiVA.org:hig-30972DiVA, id: diva2:1370398
Available from: 2019-11-15 Created: 2019-11-15 Last updated: 2019-11-28Bibliographically approved

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Shestopalov, Yury

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