Freeform determination of a nonlinear diffusion coefficient by a reduced adjoint system

In this article we deal with the determination of a diffusion coefficient function in a quasi-linear parabolic system. The motivation comes from a metallurgy setting. The solution method is based on the output least-squares approach (ols) with minimization applying the adjoint equation. As the diffusion coefficient has an a-priori unknown form, we apply freeform determination in which the coefficient is defined by a linear interpolation. To avoid higher order of interpolation of the coefficient, as well as to enable the use of an adjoint system which is of the same complexity as the original partial differential equation, a reduced adjoint equation is used, applying a mapping strategy. This method is compared with a standard Levenberg-Marquardt approach.