Adaptive Regularisation Method for Solving Nonlinear Fredholm Integral Equations of the First Kind

This paper presented a comprehensive approach to the construction of a robust regularisation technique for solving the nonlinear Fredholm integral equation of the first kind, a class of problems frequently encountered in such areas of signal processing, inverse imaging, and control theory. The purpose of the study was to develop an efficient and reasonable procedure to regularise this type of equation, which improves the accuracy of solutions in conditions where standard methods are ineffective due to noise or nonlinear distortion. The study proposed a modification of Tikhonov’s method that uses nonlinear functionals that reflect the specific structure of the original problem. Furthermore, an algorithmic strategy for selecting the normative parameter was implemented, factoring in the a priori knowledge of the expected smoothness of the solution. This enabled the development of an efficient technique that adapts to diverse types of problems and provides stable performance even under challenging conditions. Numerous experiments were conducted on both synthetic and real datasets to verify the effectiveness of the method. The findings showed that the proposed approach considerably improves the decision accuracy and convergence rate compared to standard regulatory methods, even in the presence of strong noise in the data. The comparative analysis confirmed that the new method has advantages in terms of computational efficiency and ability to adapt to diverse types of kernels and functional settings. Furthermore, experimental results demonstrated a marked reduction of errors in the recovered functions as well as a stable convergence rate, even for high dimensional problems. The proposed scheme can automatically adapt to the different nature of noise and nonlinear distortion, which makes it a versatile tool for use in many applications that require high accuracy and efficiency in solving nonlinear integral equations.