Recovering Sparse Initial Conditions in Nonlocal Diffusion

This work addresses the inverse problem of recovering sparse initial data in spatial fractional parabolic equations. The associated solution operator for initial-boundary value problem is continuous and compact, implying severe ill-posedness. To overcome this, an $ℓ_1$-regularized formulation is considered. The misfit functional is shown to be differentiable and strictly convex, and the well-posedness of the regularized problem is established, including existence, uniqueness, stability, and convergence. A numerical algorithm is proposed and implemented, incorporating Nesterov’s accelerated algorithm to enhance efficiency. Numerical experiments in both one and two spatial dimensions confirm the feasibility and accuracy of the proposed approach.