Source reconstruction algorithms for coupled parabolic systems from internal measurements of one scalar state

This paper is devoted to the study of source reconstruction algorithms for coupled systems of heat equations, with either constant or spatially dependent coupling terms, where internal measurements are available from a reduced number of observed states. Two classes of systems are considered. The first comprises parabolic equations with constant zero-order coupling terms (through a matrix potential term or via the diffusion matrix). The second type considers parabolic equations coupled by a matrix potential that depends on spatial variables, which leads to the analysis of a non-self-adjoint operator. In all configurations, the source is assumed to be of separate variables, the temporal part is a known scalar function, and the spatial dependence is an unknown vector field. Several numerical examples using the finite element method in 1D and 2D are presented to show the reconstruction of space-dependent sources.