| 1. |
- Nakonechny, Alexander, et al.
(författare)
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Guaranteed estimation for inverse problems in electromagnetics and acoustics
- 2016
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Ingår i: Proceedings of the 2016 18th International Conference on Electromagnetics in Advanced Applications, ICEAA 2016. ; , s. 374-377
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Konferensbidrag (refereegranskat)abstract
- The analysis is performed and theorems are formulated concerning general form of guaranteed estimates of linear functionals from unknown data of Helmholtz transmission problems arising in electromagnetics and acoustics of inhomogeneous dielectric media.
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| 2. |
- Nakonechny, A. G., et al.
(författare)
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Aposteriori estimates in inverse problems for the Helmholtz equation
- 2017
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Ingår i: 2017 Progress In Electromagnetics Research Symposium - Fall (PIERS - FALL). - : Electromagnetics Academy. - 9781509062690 - 9781509062706 ; , s. 3526-3531
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Konferensbidrag (refereegranskat)abstract
- We consider the problem of estimation of the right-hand sides of the Helmholtz equation that models electromagnetic and acoustic wave fields when that initial data is uncertain. In the case when measurement errors depend on solutions, we construct algorithms of computation of the optimal estimates which are compatible with the measurement data. It is shown that approximate a posteriori estimates of the right-hand sides are expressed via solutions of linear algebraic equations.
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| 3. |
- Nakonechny, Alexander G., et al.
(författare)
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Guaranteed a posteriori estimates of right-hand sides in transmission problems for Helmholtz equations
- 2017
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Ingår i: 2017 XXXIIND GENERAL ASSEMBLY AND SCIENTIFIC SYMPOSIUM OF THE INTERNATIONAL UNION OF RADIO SCIENCE (URSI GASS). - : Institute of Electrical and Electronics Engineers (IEEE). - 9789082598704 - 9781509044696 - 9789082598711
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Konferensbidrag (refereegranskat)abstract
- In this work we describe a method of obtaining guaranteed a posteriori estimates of unknown right-hand sides of the Helmholtz transmission problems from indirect measurements of a solution to this problem. The obtained results can be applied in various models of electromagnetics and acoustics that describe excitation of transparent bodies by sources of different kinds.
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| 4. |
- Nakonechnyi, Oleksandr, et al.
(författare)
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Guaranteed a posteriori estimation of uncertain data in exterior Neumann problems for Helmholtz equation from inexact indirect observations of their solutions
- 2021
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Ingår i: Inverse Problems in Science and Engineering. - : Taylor & Francis. - 1741-5977 .- 1741-5985. ; 29:4, s. 525-535
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Tidskriftsartikel (refereegranskat)abstract
- We consider the problem of guaranteed estimation of unknown right-hand sides of the equations entering the statement of the exterior Neumann problems for the Helmholtz equation from indirect observations of their solutions. A method is developed for the determination of guaranteed a posteriori estimates of this right-hand sides which are compatible with measurement data. It is shown that such estimates can be expressed via solutions of a uniquely solvable system of the Helmholtz equations.
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| 5. |
- Nakonechnyi, Oleksandr, et al.
(författare)
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Guaranteed Estimation of Solutions to the Cauchy Problem When the Restrictions on Unknown Initial Data Are Not Posed
- 2021
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Ingår i: Mathematics. - : MDPI. - 2227-7390. ; 9:24
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Tidskriftsartikel (refereegranskat)abstract
- The paper deals with Cauchy problems for first-order systems of linear ordinary differential equations with unknown data. It is assumed that the right-hand sides of equations belong to certain bounded sets in the space of square-integrable vector-functions, and the information about the initial conditions is absent. From indirect noisy observations of solutions to the Cauchy problems on a finite system of points and intervals, the guaranteed mean square estimates of linear functionals on unknown solutions of the problems under consideration are obtained. Under an assumption that the statistical characteristics of noise in observations are not known exactly, it is proved that such estimates can be expressed in terms of solutions to well-defined boundary value problems for linear systems of impulsive ordinary differential equations.
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| 6. |
- Nakonechnyi, Oleksandr, et al.
(författare)
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The Minimax Estimation Method for a Class of Inverse Helmholtz Transmission Problems
- 2019
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Ingår i: Minimax Theory and its Applications. - : Heldermann Verlag. - 2199-1413. ; 4:2, s. 305-328
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Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 7. |
- Podlipenko, Yury, et al.
(författare)
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Guaranteed Estimates of Functionals from Solutions and Data of Interior Maxwell Problems Under Uncertainties
- 2013
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Ingår i: Springer Proceedings in Mathematics & statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 52, s. 135-167
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Tidskriftsartikel (refereegranskat)abstract
- We are looking for linear with respect to observations optimal estimates of solutions and right-hand sides of Maxwell 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 variational equations and prove that Galerkin approximations of the obtained variational equations converge to their exact solutions. © Springer International Publishing Switzerland 2013.
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| 8. |
- Podlipenko, Yury, et al.
(författare)
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Guaranteed Estimates of Linear Continuous Functionals of Solutions and Right-hand Sides of the Helmholtz Equation in the Domains with Infinite Boundaries under Uncertainties
- 2013
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Ingår i: PIERS 2013 STOCKHOLM. - 9781934142264 ; , s. 65-69
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Konferensbidrag (refereegranskat)abstract
- We consider the construction of guaranteed estimates of linear continuous function als of the unknown solutions and right-hand sides of the Helmholtz equation; the boundary value problems under study are associated with the wave diffraction by a bounded body D situated in a domain Omega is an element of R-n, n = 2, 3, whose boundary partial derivative Omega stretches to infinity (e.g., a wedge or a layer) and Green's function Phi(k) (x, y), (x, y is an element of Omega, x not equal y) corresponding to wave number k with k > 0 and boundary condition (I)k (x, y)vertical bar y is an element of Omega = 0 is known [4]. Here, for a function u(y) defined in (Omega) over bar Bu(y)vertical bar(y is an element of partial derivative Omega) + beta partial derivative u(y)/partial derivative y vertical bar(y is an element of partial derivative Q), alpha, beta = 0, 1, alpha + beta = 1, v is outward normal to aft We assume that right-hand sides of the equations entering the problem statement are not known; the only available information is that they belong to a bounded set of the space of square-integrable functions. In order to solve these estimation problems we need additional data: observations in the form of certain linear transformations of the solution distorted by noise. The latter are realizations of the random fields with the unknown second moment functions belonging to a given bounded set in the appropriate functional space. The approach set forth in and developed in this study allows us to obtain optimal estimates of the unknown solution or righthand sides of the equations and linear functionals, i.e., estimates sought in the class of functionals linear with respect to observations for which the maximal mean-square estimation error taken over all elements belonging to the aforementioned sets takes minimal value. Such estimates are called minimax or guaranteed estimates. We obtain representations for these estimates and estimation errors in terms of solutions to certain integro-differential or integral equations in bounded subdomains of domain Omega \ D.
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| 9. |
- Podlipenko, Y., et al.
(författare)
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Guaranteed estimation of solutions to transmission problems for Helmholtz equation with uncertain data from their indirect noisy observations
- 2016
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Ingår i: 2016 URSI International Symposium on Electromagnetic Theory (EMTS). - : IEEE conference proceedings. - 9781509025022 - 9781509025039 ; , s. 93-95
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Konferensbidrag (refereegranskat)abstract
- We investigate the estimation problems of linear functionals from solutions to transmission problems for Helmholtz equation with inexact data. The right-hand sides of equations entering the statements of transmission problems and the statistical characteristics of observations errors are supposed to be unknown and belonging to the certain sets. It is shown that the linear mean square estimates of the above-mentioned functionals and estimation errors are expressed via solutions to the systems of transmission problems of the special type.
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| 10. |
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