Regularization of the discrete source problem in the nonlinear diffusive-logistic equation

TL;DR

This work tackles the inverse/source problem for the nonlinear diffusion-logistic equation in a discrete spatial setting, where the process is driven by an instantaneous source $\varphi(x)\delta(t)$ and observations are available at fixed times. The authors cast the recovery as minimizing the A.N. Taikhonov functional $T(q)$ and solve it via a global tensor optimization (TT) method, augmented by a priori information about the source; the model and data are handled under noisy conditions. Numerical experiments show that regularization improves source reconstruction in the presence of noise, with the TT method effectively recovering a 6-parameter initial density but facing data-limitation for 14 parameters; the choice of regularization parameter $\alpha$ critically influences accuracy, with $\alpha \approx 10^{-5}$ often providing a good balance. The results motivate hybrid approaches combining global TT with gradient-based methods to enhance robustness and scalability in higher-dimensional parameter spaces.

Abstract

A numerical algorithm for regularization of the solution of the source problem for the diffusion-logistic model based on information about the process at fixed moments of time of integral type has been developed. The peculiarity of the problem under study is the discrete formulation in space and impossibility to apply classical algorithms for its numerical solution. The regularization of the problem is based on the application of A.N. Tikhonov's approach and a priori information about the source of the process. The problem was formulated in a variational formulation and solved by the global tensor optimization method. It is shown that in the case of noisy data regularization improves the accuracy of the reconstructed source.

Figures (3)

• Figure 1: The recovered initial density function of active users $\varphi(x)$ at $\delta = 10$%. The continuous black line with circles represents the exact solution of the inverse problem, the blue line with crosses represents the initial approximation $q^0$.
• Figure 2: Reconstructed initial density function of active users $\varphi(x)$ at $\delta = 10$ and $q^0 = 0$. The continuous black line with circles represents the exact solution of the inverse problem, the purple dashed line with triangles represents the solution obtained by the TT method with regularization parameter $\alpha = 10^{-4}$, the blue-green (teal) dashed line with stars represents the solution obtained by the TT method with regularization parameter $\alpha = 10^{-7}$, and the red squares represent additional points
• Figure 3: Reconstructed initial density function of active users $\varphi(x)$ at $\delta = 0$ for the case $d = 14$. (a) comparison of the reconstruction results for the cases $d = 6$ and $d = 14$ at $\alpha = 0$; (b) reconstruction results of the source function for different values of the regularization parameter $\alpha$.