Guaranteed estimation of solutions to Helmholtz transmission problems with uncertain data from their indirect noisy observations

We investigate the guaranteed estimation problem of linear functionals from solutions to transmission problems for the Helmholtz equation with inexact data. The right-hand sides of equations entering the statements of transmission problems and the statistical characteristics of observation errors are supposed to be unknown and belonging to certain sets. It is shown that the optimal 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. The results and techniques can be applied in the analysis and estimation of solution to forward and inverse electromagnetic and acoustic problems with uncertain data that arise in mathematical models of the wave diffraction on transparent bodies. Plain Language Summary We propose and develop novel efficient mathematical technique to perform guaranteed estimation of solutions to a wide family of problems with uncertain data arising in electromagnetics and acoustics. The estimation is based on observations involving unknown errors. We introduce the notion of optimalminimax estimate that minimizes maximal mean square estimation error calculated for the "worst" implementation of uncertainties and perturbations. The optimal linear mean square estimates of the estimation errors are expressed via solutions to the systems of transmission problems that can be solved numerically using available codes and programs. Beyond purely theoretical interest, the obtained results can be applied in various models that describe wave diffraction on transparent bodies, for automatized measurement in data processing systems, and interpretation of electromagnetic and acoustic observations of different types.