Positivity constraints in linear inverse problems--I. General theory

Abstract

Summary. A method is given which completely solves linear inverse problems with positive constraints. Equivalent solutions form a convex combination (i.e. with barycentric coefficients) of a special set of solutions, called the extremals. Conversely, the set of all linear convex combinations of the extremals coincide with the set of equivalent solutions. Thus the extremals completely define the set of solutions, exactly like the vertices of a triangle define the whole triangle. A strategy is given to obtain these extremals in a natural order, through several applications of well-known algorithms, taking into account all physical information. Similar methods are given to evaluate the range of the set of equivalent solutions, and to manage numerical uncertainties. The choice of a solution inside the set is discussed. In the Appendix B, several typical choices are given, and should meet physicists requirements in most of these problems.