THE STABILITY OF FINITE-DIMENSIONAL INVERSE PROBLEMS

In this paper the stability of inverse problems is discussed. It is ta ken into account that in inverse problems the structure of the solutio n space is usually completely different from the structure of the data space so that the definition of stability is not trivial. We solve th is problem by assuming that under experimental circumstances both the model and the data can be characterized by a finite number of paramete rs. In the formal definition that we present, we first compare distanc es in the data space and distances in the model under variations of th ese parameters. Second, a normalization is introduced to ensure that q uantities in the solution space can be compared directly with quantiti es in the data space. We note that it is impossible to obtain an objec tive solution of stability due to the freedom one has in the choice of the norm in the solution space and in the data space. This definition is used to examine the stability of linear inverse problems as well a s for Marchenko equation and inverse problems associated with transfer -matrix methods. For the Marchenko equation it is shown that the insta bility arises from the nonlinearity of the inverse problem.