Mathematical and computational framework of computational material diagnostics method based on limited indentation tests is proposed. An inversion method aims to determine elastoplastic properties from an indentation loading curve. Mathematical model, based on deformation theory, leads to quasi-static elastoplastic contact problem, given by the the monotonically increasing values alpha(i) > 0 of the indentation depth. The identification problem is formulated as an inverse problem of determining the stress-strain curve(sigma i) = sigma(i)(e(i)) from an experimentally given indentation curve P = P(alpha). The inversion method is based on the parametrization of the stress-strain curve, according to the discrete values of the indentation depth, and uses only a priori information as monotonicity of the unknown function sigma(i) = sigma(i)(e(i)). It is shown that the ill-conditionedness of the identification problems depends on the state discretization parameter Delta e(i). An algorithm of optimal selection of state discretization parameters is proposed as a new regularization scheme. Remeshing algorithm presented here allows to find the unknown contact zone on each step of indentation with high accuracy. Numerical parametric studies are performed for different materials with various elastic and plastic properties. The obtained results show high predictive capability of the proposed computational model. The presented numerical results permit also to construct relationships between various parameters chracterizing material behaviour under spherical indentation.