Identification of transmissivity coefficients by mollification techniques. Part I: One-dimensional elliptic and parabolic problems

Abstract

We discuss the numerical identification of the transmissivity coefficient in the one-dimensional linear and nonlinear elliptic and parabolic partial differential equations. This is a quite general problem arising—for example—in porous media and related to groundwater modeling and reservoir simulation. This problem belongs to the class field of inverse problems. It is well-known that it is ill-posed because small errors in the data might produce large errors in the computed solution. We combine the mollification techniques and finite differences to address the ill-posedness of this parameter estimation problem. Several numerical experiments support the stability and convergence of the algorithms.