Abstract
What is an ill-posed inverse problem? The answer to this question is the main objective of the present paper and the pre-requisite to follow the material requires only elementary calculus. The first mathematical formulation of an inverse problem, due to N. H. Abel, together with the fundamental work by Jacques Hadamard, are explored at the beginning of the paper. A prototype system is used to consider the regularization concept. Three numerical methods, the Tikhonov regularization, the decomposition into singular values and the Hopfield neural networks, applied to remove the singularity are examined. General aspects of the ill-posed inverse problems in chemistry with emphasis in thermodynamics and a set of general rules for other areas of science are also analyzed.
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