Numerical simulation of inverse retrospective problems for a nonlinear heat equation

The paper proposes a method for numerically solving the inverse problem of restoring the initial condition for a nonlinear one-dimensional heat equation. The proposed method is based on the use of the parametric identification method, the implicit gradient descent method of minimizing the residual functional and the regularization method of A.N. Tikhonov. Also we developed algorithm and a software package for numerical solution. Numerous results of numerical experiments are obtained and discussed. The analysis of the behavior of decision functions w/w the use of the regularizing functional A.N. Tikhonov and the effects of the regularizing parameter is carried out. An algorithm for finding the optimal value of the regularizing parameter based on a grid search with cross-validation (K-fold) is proposed. The results of numeric experiments using the proposed numerical method showed that the error of the results obtained does not exceed the error in the experimental data.