Nonlinear Dynamic Inverse Solution of the Diffusion Problem Based on Krylov Subspace Methods with Spatiotemporal Constraints ^†

In this work, we propose a nonlinear dynamic inverse solution to the diffusion problem based on Krylov Subspace Methods with spatiotemporal constraints. The proposed approach is applied by considering, as a forward problem, a 1D diffusion problem with a nonlinear diffusion model. The dynamic inverse problem solution is obtained by considering a cost function with spatiotemporal constraints, where the Krylov subspace method named the Generalized Minimal Residual method is applied by considering a linearized diffusion model and spatiotemporal constraints. In addition, a Jacobian-based preconditioner is used to improve the convergence of the inverse solution. The proposed approach is evaluated under noise conditions by considering the reconstruction error and the relative residual error. It can be seen that the performance of the proposed approach is better when used with the preconditioner for the nonlinear diffusion model under noise conditions in comparison with the system without the preconditioner.