Multidimensional inverse heat conduction problems solution via Lagrange theory and model size reduction techniques

This article is concerned with inverse heat conduction problems (IHCP) involving large-scale linear systems (i.e. multidimensional systems) with unknown sources or boundary conditions. The inverse problem is stated as an optimisation problem with a regularised quadratic objective functional, and it is solved using Lagrange theory. We demonstrate that the IHCP solution is obtained by simple time integration of a state-variable model: the inverse model , which is analytically derived from the so called direct and adjoint problems. In addition, we show that the number of differential equations in the inverse model can be strongly reduced without a significant loss of precision. The inversion is then carried out in three main steps: calculation of the full-order inverse model, size reduction, and time integration of the resulting reduced-order inverse model. A numerical example of heat sources identification in a 2D diffusion problem shows the performances and the accuracy of the proposed approach.