Fast Finite Difference Approximation for Identifying Parameters in a Two-dimensional Space-fractional Nonlocal Model with Variable Diffusivity Coefficients

In this paper, we consider an inverse problem for identifying the fractional derivative indices in a two-dimensional space-fractional nonlocal model based on a generalization of the two-sided Riemann--Liouville formulation with variable diffusivity coefficients. First, we derive an implicit difference method (IDM) for the direct problem and the stability and convergence of the IDM are discussed. Second, for the implementation of the IDM, we develop a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) that is superior in computational performance to Gaussian elimination and attains the same accuracy. Third, we utilize the Levenberg--Marquardt (L-M) regularization technique combined with the Armijo rule (the popular inexact line search condition) to solve the modified nonlinear least squares model associated with the parameter identification. Finally, we carry out numerical tests to verify the accuracy and efficiency of the IDM. Numerical investigations are performed with both accurate data and noisy data to check the effectiveness of the L-M regularization method. The convergence behavior of the L-M for the inverse problem involving the space-fractional diffusion model is shown graphically.