Inverse problems = Quest for information

Abstract

Note: The 1982 paper has been rewritten, to produce a searchable PDF file. This text is essentially identical to the original. We examine the general non-linear inverse problem with a finite number of parameters. In order to permit the incorporation of any a priori information about parameters and any distribution of data (not only of gaussian type) we propose to formulate the problem not using single quantities (such as bounds, means, etc.) but using probability density functions for data and parameters. We also want our formulation to allow for the incorporation of theoretical errors, i.e. non-exact theoretical relationships between data and parameters (due to discretization, or incomplete theoretical knowledge); to do that in a natural way we propose to define general theoretical relationships also as probability density functions. We show then that the inverse problem may be formulated as a problem of combination of information: the experimental information about data, the a priori information about parameters, and the theoretical information. With this approach, the general solution of the non-linear inverse problem is unique and consistent (solving the same problem, with the same data, but with a different system of parameters does not change the solution).