Simultaneous estimation of spatially dependent diffusion coefficient and source term in a nonlinear 1D diffusion problem

This work deals with the use of the conjugate gradient method in conjunction with an adjoint problem formulation for the simultaneous estimation of the spatially varying diffusion coefficient and of the source term distribution in a one-dimensional nonlinear diffusion problem. In the present approach, no a priori assumption is required regarding the functional form of the unknowns. This work can be physically associated with the detection of material non-homogeneities, such as inclusions, obstacles or cracks, in heat conduction, groundwater flow and tomography problems. Three versions of the conjugate gradient method are compared for the solution of the present inverse problem, by using simulated measurements containing random errors in the inverse analysis. Different functional forms, including those containing sharp corners and discontinuities, are used to generate the simulated measurements and to address the accuracy of the present solution approach.