Inverse problems in heat diffusion . Construction and solution of the least squares problem

In the article [BE 8 265] , we addressed the problem of model construction and the resulting study of sensitivities, i.e. we put the direct problem into shape.

In this article, we show how an inverse problem can be solved by reducing it to an optimization problem. For this purpose, an object function (also called a criterion) is introduced, relating to the deviation between model and measurement and depending on the unknown parameters. Parameter estimation then consists in adopting, as solutions, the values that minimize this function. Note that, in practice, this minimization must remain compatible with the measurement noise level of the sensors used.

Here, we analyze the mathematical form of this minimization problem, which forms the basis of the inverse approach. The actual measurement inversions, taking into account the difficulties associated with real measurement (signal-to-noise ratio, poor sensitivities, etc.), are dealt with in the article [BE 8 267] .

In addition, based on the stochastic properties of noise, the relative standard deviations of the estimated parameters can also be calculated. The methods differ depending on whether the model is linear or non-linear with respect to the parameters to be estimated.

The symbols and notations used in this article are given in Table 1 of [BE 8 265] . It should be noted that only the pdf version of this article provides fully relevant notation, as the electronic version does not clearly distinguish between the different fats of the symbols.