The differential evolution algorithm is applied to solve the optimization problem to reconstruct the production function (inverse problem) for the spatial Solow mathematical model using additional measurements of the gross domestic product for the fixed points. Since the inverse problem is ill-posed the regularized differential evolution is applied. For getting the optimized solution of the inverse problem the differential evolution algorithm is paralleled to 32 kernels. Numerical results for different technological levels and errors in measured data are presented and discussed.

The study of the clinical and epidemiological features of tuberculosis combined with HIV infection (TB + HIV) is one of the priorities in the prevention of infectious diseases and is necessary to improve the quality of medical care for patients. So this article is devoted to the mathematical model of epidemiology, its’ investigation and analysis. The previous works showed what identifiability analysis is and considered methods of performing them, such as orthogonal method, eigenvalue method and etc., for more precise clarification of model parameters. However, the choice of solving the inverse problem to restore unknown parameters is playing a huge role. So here was showed the combination of two numerical algorithms, as stochastic method of simulating annealing to determine the region of the global minimum and gradient method to determine the inverse problem in a region, of solving the inverse problem that will help to create effective treatment plan for the elimination and treatment of the disease. Abstract . The study of the clinical and epidemiological features of tuberculosis combined with HIV infection (TB + HIV) is one of the priorities in the prevention of infectious diseases and is necessary to improve the quality of medical care for patients. So this article is devoted to the mathematical model of epidemiology, its’ investigation and analysis. The previous works showed what identifiability analysis is and considered methods of performing them, such as orthogonal method, eigenvalue method and etc., for more precise clarification of model parameters. However, the choice of solving the inverse problem to restore unknown parameters is playing a huge role. So here was showed the combination of two numerical algorithms, as stochastic method of simulating annealing to determine the region of the global minimum and gradient method to determine the inverse problem in a region, of solving the inverse problem that will help to create effective treatment plan for the elimination and treatment of the disease.

In this research, we consider the inverse problem for the wave equation under an unknown initial condition. A generalized solution to the direct problem was formulated, its correctness was established, and the stability assessment was obtained. The inverse problem was reduced to an optimization problem, where the objective function was minimized using gradient methods, including the accelerated Nesterov algorithm. The conjugate problem was constructed, and the functional gradient was computed, while the existence of the Frechet derivative was proved. For the first time, the quaternion Fourier transform (QFT) was applied to the numerical solution of a direct problem, making it possible to analyze multidimensional wave processes more efficiently. A computational experiment was carried out, which demonstrated that if there is insufficient additional information, the restoration of the initial condition is incomplete. The introduction of the second boundary condition makes it possible to significantly improve the accuracy and stability of the solution. The results confirm the importance of an integrated approach and the availability of sufficient a priori information when solving inverse problems.

This study addresses the inverse problem of parameter identification in mathematical models of tumor growth under limited and noisy experimental data. Three classical growth models—logistic, Richards, and Gompertz—are investigated in the context of structural and practical identifiability. It is demonstrated that, despite structural identifiability, parameter estimation remains highly unstable due to the ill-posed nature of the inverse problem. A comparative analysis of the Levenberg–Marquardt method and a genetic algorithm shows that improvements in optimization strategies alone do not resolve this instability and may lead to overfitting. To overcome this limitation, a Tikhonov regularization framework is introduced for the Gompertz model, ensuring stable and physically interpretable parameter estimates. The regularized formulation provides a balance between data fidelity and parameter stability, resulting in improved agreement with experimental observations. To further validate the identified parameters, a reaction–diffusion partial differential equation model is employed. Numerical simulations demonstrate that regularized parameters lead to significantly different spatial tumor morphologies, including more compact structures with sharper interfaces, highlighting the impact of inverse problem regularization on forward model predictions. The results confirm that the primary limitation in tumor growth modeling lies in the ill-posedness of the inverse problem rather than in the choice of optimization algorithm. The proposed framework provides a robust approach for parameter identification and improves the reliability of predictive tumor growth models.