A General Method for the Solution of Inverse Problems in Transport Phenomena

The typical inverse problems in transport phenomena are given by partial differential equations with unknown boundary conditions, which are to be estimated from measurements corresponding to solutions of the PDEs or of their gradients. The resulting problem is an ill-posed problem, which can't be solved unless it is adequately regularized, because arbitrarily small errors on the data can give rise to very large deviations in the reconstruction of the unknown boundary conditions. In other words the estimated solution does not depend continuously on the data. The method proposed in this paper is the generalization of methods already applied to a number of problems (diffusion, heat transfer, percolation). The unknown boundary condition is replaced by a piecewise constant (or linear) functions with unknown coefficients. This approximation makes it possible to solve the resulting equation analytically and to estimate the coefficients by comparing theoretical and experimental values by means of linear least squares methods. The ill-posedness of this class of problems (resulting in singular matrices of the least squares problems) can be tackled using the well-known method proposed by Tikhonov. The value of the parameter contained in Tikhonov's method is estimated from the statistical noise on data by means of the Morozov's discrepancy principle. If additional information on the structure of the solution is available, specialised algorithms can be developed on a case-by-case basis. The presence of additional unknown non-linear parameters in the partial differential equations (such as diffusion coefficients, conductivities or adsorption coefficients of plant roots) leads to a particular model known as separable least squares. This procedure makes use of the conditional optimality principle for carrying out the overall identification problem in terms of the non-linear parameters only.