Sensitivity theory for nonlinear systems. I. Nonlinear functional analysis approach

Concepts of nonlinear functional analysis are employed to investigate the mathematical foundations underlying sensitivity theory. This makes it possible not only to ascertain the limitations inherent in existing analytical approaches to sensitivity analysis, but also to rigorously formulate a considerably more general sensitivity theory for physical problems characterized by systems of nonlinear equations and by nonlinear functionals as responses. Two alternative formalisms, labeled the ’’forward sensitivity formalism’’ and the ’’adjoint sensitivity formalism,’’ are developed in order to evaluate the sensitivity of the response to variations in the system parameters. The forward sensitivity formalism is formulated in normed linear spaces, and the existence of the Gâteaux differentials of the operators appearing in the problem is shown to be both necessary and sufficient for its validity. This formalism is conceptually straightforward and can be advantageously used to assess the effects of relatively few parameter alterations on many responses. On the other hand, for problems involving many parameter alterations or a large data base and comparatively few functional-type responses, the alternative adjoint sensitivity formalism is computationally more economical. However, it is shown that this formalism can be developed only under conditions that are more restrictive than those underlying the validity of the forward sensitivity formalism. In particular, the requirement that operators acting on the state vector and on the system parameters must admit densely defined Gâteaux derivatives is shown to be of fundamental importance for the validity of this formalism. The present analysis significantly extends the scope of sensitivity theory and provides a basis for still further generalizations.