The comparison model method (CMM) is applied to the identification of spatially varying thermal conductivity in a one‐dimensional domain. This method deals with the discretized steady‐state heat equation written at the nodes of a lattice, a lattice which models a stack of plane parallel layers. The required data are temperature gradient and heat source (or sink) values. The unknowns of this inverse problem are not nodal values but internode thermal conductivities, which appear in the node heat balance equation. The conductivities, e.g., the solutions to the inverse problem obtained by the CMM, are a one‐parameter family. In order to achieve uniqueness (which coincides with identifiability in this case), a suitable value of this parameter must be found. To this end we consider (1) parameterization, i.e., introducing equality constraints between the unknown coefficients, (2) the use of two data sets at least in a subdomain, and (3) self‐identifiability. Each of these items formally translates the available a priori information about the system, e.g., geophysical properties of the layers. Under the assumption that an admissible solution exists, we evaluate the effects on the solution of two types of data noise: additive noise affecting temperature and a kind of multiplicative noise affecting heat‐source terms. In the numerical examples we provide, parameterization is combined with the CMM in order to obtain the unique solution to a test inverse problem, the geophysical data of which come from a well drilled across Tertiary layers in central Italy. Finally, we consider data perturbed by pseudorandom noise. More precisely, we add noise to temperature gradients and obtain stability estimates which correctly predict the numerical results. In particular, if the layers where the parameterization constraint applies contain a strong source term (e.g., due to flowing water), the solution is relatively insensitive to noise. Also, for multiplicative noise affecting heat‐source terms, theoretical stability predictions are confirmed numerically. The main advantage of the CMM is its simple algebraic formulation. Its implementation in the field by means of a pocket calculator allows both a consistency check on the data being collected and estimates of the unknown values of conductivity.
Stability estimates for an inverse problem for the Schrödinger equation at negative energy in two dimensions
