Overview
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ABSTRACT
The construction of a criterion to be minimized, which is the basis for solving any inverse problem, is introduced. The case of inversion of measurements using a linear model is dealt with. The study of its possibly ‘ill-posed’ character, the ordinary least square estimator and its variance-covariance matrix are detailed. The different techniques for inverting measurements using a non-linear model are then detailed. Examples of inversion in steady or unsteady conduction are presented, starting from simulated measurements that integrate some noise. Explosion of estimations, in the case of a bad conditioning of the sensitivity matrix is highlighted.
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Read the articleAUTHORS
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Denis MAILLET: Professor Emeritus, University of Lorraine (UL) - Laboratoire d'Énergétique et de Mécanique Théorique et Appliquée (LEMTA) – CNRS and UL
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Yvon JARNY: Professor Emeritus, University of Nantes - Laboratoire de Thermique et Énergie de Nantes (LTeN) – UMR CNRS 6607 Nantes
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Daniel PETIT: Professor Emeritus, École Nationale Supérieure de Mécanique et d'Aérotechnique (ISAE-ENSMA) - Institut P' UPR CNRS 3346 Département Fluides, Thermique, Combustion – Poitiers
INTRODUCTION
In the article
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
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
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KEYWORDS
least squares | estimation | condition number | covariance matrix
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Physics of energy
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Inverse problems in thermal diffusion
Bibliography
Websites
French Thermal Society (see proceedings of Metti 5-2011 and Metti 6-2015 schools) http://www.sft.asso.fr
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