A new refinement indicator for adaptive parameterization: Application to the estimation of the diffusion coefficient in an elliptic problem

We propose a new refinement indicator (NRI) for adaptive parameterization to determine the diffusion coefficient in an elliptic equation in two-dimensional space. The diffusion coefficient is assumed to be a piecewise constant space function. The unknowns are both the parameter values and the zonation. Refinement indicators are used to localize parameter discontinuities in order to construct iteratively the zonation (parameterization). The refinement indicator is obtained usually by using the first-order effect on the objective function of removing degrees of freedom for a current set of parameters. In this work, in order to reduce the computation costs, we propose a new refinement indicator based on the second-order effect on the objective function. This new refinement indicator depends on the objective function, and its first and second derivatives with respect to the parameter constraints. Numerical experiments show the high efficiency of the new refinement indicator compared to the standard one.