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.
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Notes
- 1.
Random variable ξ with values in Hilbert space H is considered as a function \(\xi: \it{\Omega} \rightarrow H\) mapping random events \(E \in \mathcal{B}\) to Borel sets in H (Borel σ-algebra in H is generated by open sets in H).
- 2.
This operator exists according to the Riesz theorem.
- 3.
For vectors \({\mathbf{V}}^{(1)} = \left (V _{1}^{(1)},V _{2}^{(1)},V _{3}^{(1)}\right ),\) \({\mathbf{V}}^{(2)} = \left (V _{1}^{(2)},V _{2}^{(2)},V _{3}^{(2)}\right ) \in {\mathbb{C}}^{3}\) we set \(\left ({\mathbf{V}}^{(1)},{\mathbf{V}}^{(2)}\right )_{{\mathbb{C}}^{3}} =\sum _{ i=1}^{3}V _{i}^{(2)}\bar{V }_{i}^{(2)}.\)
- 4.
This problem is uniquely solvable since, owing to (22), the sesquilinear form \({a}^{{\ast}}(\cdot,\cdot )\) is also coercive in \(H_{0}({\rm rot},D)\).
- 5.
Here and below we denote by \(C(\bar{D}_{j})\) a class of functions continuous in the domain \(\bar{D}_{j}.\)
- 6.
By
$$\displaystyle{\mbox{ Sp}\left (\mathbf{Q}_{j}^{i}(x)\tilde{\mathbf{R}}_{r_{ 1}}^{(i)}(x,x)\right )}$$we denote the traces of matrices \(\mathbf{Q}_{j}^{i}(x)\tilde{\mathbf{R}}_{j}^{(i)}(x,x)\), i.e. the sum of diagonal elements of these matrices, i = 1, 2. 
- 7.
We use the following notation: if \(\mathbf{A}(\xi ) = [a_{ij}(\xi )]_{i,j=1}^{N}\) is a matrix depending on variable ξ that varies on measurable set Ω, then we define \(\int _{\it{\Omega} }\mathbf{A}(\xi )\,d\xi\) by the equality
$$\displaystyle{\int _{\it{\Omega} }\mathbf{A}(\xi )\,d\xi = \left [\int _{\it{\Omega} }a_{ij}(\xi )\,d\xi \right ]_{i,j=1}^{N}.}$$
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Acknowledgements
This work is supported by the Visby program of the Swedish Institute.
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Podlipenko, Y., Shestopalov, Y. (2013). Guaranteed Estimates of Functionals from Solutions and Data of Interior Maxwell Problems Under Uncertainties. In: Beilina, L., Shestopalov, Y. (eds) Inverse Problems and Large-Scale Computations. Springer Proceedings in Mathematics & Statistics, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-319-00660-4_10
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DOI: https://doi.org/10.1007/978-3-319-00660-4_10
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