Inverse $t$-source problem and a strict positivity property for coupled subdiffusion systems
This work addresses source identification in fractional diffusion systems, which is incremental as it extends existing inverse problem methods to coupled multi-component cases with theoretical and algorithmic improvements.
The authors tackled the inverse problem of determining temporal source terms in coupled fractional diffusion systems using single-point observations, establishing Lipschitz stability with all components observed and uniqueness with single-component observation under constraints, and demonstrated their iterative regularizing ensemble Kalman method achieves accurate, robust, and scalable recovery in numerical tests.
This article is concerned with the inverse problem on determining the temporal component of the source term in a coupled system of time-fractional diffusion equations by single point observation. Under a non-degeneracy condition on the known spatial component, we establish the Lipschitz stability by observing all solution components by a series representation of the mild solution. To reduce the observation data, we prove the strict positivity of some fractional integral of the solution to the homogeneous problem by a modified Picard iteration. This, together with a coupled Duhamel's principle, lead us to the uniqueness of the inverse problem by observing any single solution component under a specific structural constraint on the unknown. Numerically, we propose an iterative regularizing ensemble Kalman method (IREKM) for the simultaneous recovery of the temporal sources. Through extensive numerical tests, we demonstrate its accuracy, robustness against noise, and scalability with respect to the number of components. Our findings highlight the essential roles of the non-degeneracy condition, measurement configuration, and fractional structural constraints in ensuring reliable reconstructions. The proposed framework provides both rigorous theoretical guarantees and a practical algorithmic approach for multi-component source identification in fractional diffusion systems.