The Multi-Dimensional Refinement Indicators Algorithm for Optimal Parameterization

The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori information by reducing the number of unknowns according to the physics of the problem. The refinement indicators algorithm provides a fruitful adaptive parameterization technique that parsimoniously opens the degrees of freedom in an iterative way. We present a new general form of the refinement indicators algorithm that is applicable to the estimation of multi-dimensional parameters in any PDE. In the linear case, we state the relationship between the refinement indicator and the decrease of the usual least-squares data misfit objective function. We give numerical results in the simple case of the identity model, and this application reveals the refinement indicators algorithm as an image segmentation technique.