Workshop on Inverse Problems: Econometry, Numerical Analysis and Optimization, Statistics, Touluse 2005
What statisticians, numericians, engineers or econometricians mean by "inverse problem" often differs.
For a statistician, an inverse problem is an estimation problem of a function which is not directly observed. The data are finite in number and contain errors, whose variance decreases with the number of observations, as they do in classical inference problems, while the unknown typically is infinite dimensional, as it is in nonparametric regression.
For numericians, the noise is more an error induced by the fact that the real data are not directly observed. But the asymptotics differ, as the regularity conditions imposed for the solution.
Finally, in econometrics the structural approach combines data observation and economic model. The parameter of interest is defined as a solution of a functional equation depending on the data distribution. Hence the operator in the underlying inverse problem is in general unknown.
Many questions arise naturally in all the different fields, which are of great both applied and theoretical interest: identifiability, consistency and optimality in various forms, iterative methods. There have been great advances in the study of inverse problems within these three communities and we think that it is time for a workshop where the different point of views could be confronted, leading to exchanges of methodologies and several improvements. For instance non linear inverse problems have been studied in numerical analysis while statistical literature on this topics is scarce. Unknown inverse operators are common in econometrics but the problem is not well studied in statistics. On the other hand, adaptive estimation and optimal rates of convergence are common in statistics but not in the other fields.
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