研究目的
To recover electromagnetic ?elds at the past time by noisy measurement data at the present time using the Tikhonov regularization method to cope with the ill-posedness of the governing backward nonlinear Maxwell’s equations.
研究成果
The study validates the variational source condition for the Tikhonov regularization of ill-posed backward nonlinear Maxwell’s equations, leading to power-type convergence rates under appropriate regularity assumptions. The results contribute to the understanding of convergence rates for PDE-constrained optimization problems.
研究不足
The study is theoretical and focuses on mathematical analysis without experimental validation. The convergence rate of the Tikhonov regularization method can be arbitrarily slow without additional conditions.
1：Experimental Design and Method Selection:
The study employs the Tikhonov regularization method for the inverse problem governed by nonlinear evolutionary Maxwell’s equations. The methodology includes the use of semigroup theory for convergence analysis and adjoint calculus for deriving optimality conditions.
2：Sample Selection and Data Sources:
The study focuses on recovering the initial value of the mild solution for the Maxwell system from given measurement at final time with noise data.
3：List of Experimental Equipment and Materials:
The study involves mathematical modeling and analysis without specific experimental equipment.
4：Experimental Procedures and Operational Workflow:
The process involves formulating the inverse problem, applying Tikhonov regularization, and analyzing convergence rates under a variational source condition.
5：Data Analysis Methods:
The analysis includes spectral theory, complex interpolation theory, and fractional Sobolev spaces to validate the proposed variational source condition.
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