This paper presents an inverse problem in heat conduction, namely the determination of thicknesses of three materials of known heat capacities and thermal conductivities inside a rod of given length subjected to periodic heat flows from measurements of temperatures at both ends. The unknowns are, therefore, the positions of the two interior frontiers between the three materials. Classically, they can be obtained by minimizing the least-squares, non-linear criterion, between the measured and calculated temperatures. Nevertheless, we show that the global minimum providing the solution is close to three local minima that act as traps for a descent algorithm. After providing theoretical justification of the complex temperature method, a method based in this case on the periodicity of boundary fluxes, we suggest a new criterion allowing the global characterization or not of an a priori local minimizer to be tested. It is a criterion of topological nature based on the identification of a singularity.

2015)Parametric identification of a heating mobile source in a three-dimensional geometry, Inverse Problems in Science and Engineering, 23:1, 93-111,The resolution of an inverse problem of heat conduction in a three-dimensional plate using an iterative regularization method based on Alifanov's iterative regularization method is investigated. Considering temperature observation on the upper face centre of a small thin steel plate, the time dependent strength of a plane heat source has to be identified. Two configurations are studied. For the first one, the heat source is fixed on the lower face centre. For the second one, the heat source is mobile and the trajectory is assumed to be known. For both situations, robustness of the approach is stated considering noisy measurements. IntroductionFor several decades, resolution of inverse problems in thermal sciences is a key-goal in numerous engineering applications. Related literature is quite wide since applications deal with different geometries and configurations for many identification purposes like initial conditions [1], boundary exchange coefficient [2], thermal parameters[3] or heat source characterization. In this last framework, several applications are investigated in recent references: control of welding processes using nonintrusive observations [4,5]; prediction of the thermal effect of High Energy Laser (HEL) weapons, in order to predict the potential damage on a target [6] and estimation of temperature in human tissues submitted to a laser occurrence.[7] During the two past decades, several studies have been investigated from one-dimensional geometry [8] to two-dimensional geometries (see [9][10][11]). It is well known that such inverse problems are ill-posed. [12,13] Several approaches can be implemented to deal with parametric identification: singular value decomposition [13], Tikhonov regularization [14] and function specification.[15] The Alifanov's iterative regularization method has been successfully implemented in [16]. However, three-dimensional configurations are scarcely investigated due to important CPU time required for identification purposes (in [17][18][19][20][21][22] parametric identification in a thermal context is investigated). Nowadays, heat flux identification still remains a lively research topic (see recent references [23][24][25][26]).

Inverse three-dimensional heat conduction problems devoted to heating source localization are ill posed. Identification can be performed using an iterative regularization method based on the conjugate gradient algorithm. Such a method is usually implemented off-line, taking into account observations (temperature measurements, for example). However, in a practical context, if the source has to be located as fast as possible (e.g., for diagnosis), the observation horizon has to be reduced. To this end, several configurations are detailed and effects of noisy observations are investigated.

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