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The estimation of distributed parameters in a partial differential equation (PDE) from measures of the solution of the PDE may lead to underdetermination problems. The choice of a parameterization is a frequently used 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 distributed multidimensional 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 .

The multidimensional refinement indicators algorithm for optimal parameterization