  The interaction between the foundation structures and the soil has been
developed for many engineering applications. For the determination of the
stress in foundation structure it is needed to determine the influence of the
stiffness of soil with respect to the displacement w of the deformable plate
(direct problem), and viceversa, how the stiffness of the foundation structure
affects the resulting subsidence (inverse problem). In this paper, we deal with
the Winkler mathematical model and propose to use an efficient Ensemble Kalman
Inversion scheme (EKI) that regularizes iteratively the ill-posedness of the
inverse problem. It is a regularizing optimizer used in Bayesian inverse
problems that samples particles in pseudo-time introducing a motion due to the
movement of these particles. The EKI algorithm converges to the solution of an
optimization problem that minimizes the objective function. In this context we
show how to reconstruct the Winkler subgrade reaction coefficient of a
rectangular thin plate loaded with an existing building by using the EKI
methodology combined by the finite difference method (FDM) to discretize the
biharmonic operator of the governing equations.
