Differentiation of inverse functions in spaces without norm

Serovaiskii, S. Ya.

doi:10.1007/bf01078852

Funct Anal Its Appl

1994

DOI: 10.1007/bf01078852

|View full text |Cite

Differentiation of inverse functions in spaces without norm

S. Ya. Serovaiskii¹

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2005

Publication Types

Select...

Article1

Relationship

Self Cite0

Independent1

Authors

Journals

Cited by 1 publication

(1 citation statement)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The established results, representing a development of previously reported works [8,9] by the present author give promise that extended derivatives may successfully replace the classical derivatives in various applications. Among such applications are numerous problems that arise in nonlinear analysis (the implicit function theorem and the Lusternik theorem), in differential topology (notions of differentiable manifold, tangent space, and tangent fibration), in the Lee group theory (relations between Lee groups and Lee algebras), in the theory…”

mentioning

confidence: 70%

Calculation of functional gradients and extended differentiation of operators

Serovajsky¹

2005

Journal of Inverse and Ill-posed Problems

View full text Add to dashboard Cite

Qualitative and quantitative analysis of optimal-control problems and inverse problems in mathematical physics largely implies the calculation of functional gradients giving optimality criteria or closing errors. As a rule, this procedure is used in calculating necessary extremum conditions and in gradient methods. If a system under study is linear, then calculation of a functional gradient presents no serious difficulties. Yet, in nonlinear problems of mathematical physics there arise substantial difficulties stemming from the fact that, as the control (parameter to be identified) is put to variation, the state function of the system also undergo changes. Thus, for the functional gradient to be found, differentiability of the state function with respect to control must be established. Substantiation of this property for nonlinear infinite-dimensional systems is a point far from being trivial. Moreover, the example given below shows that, although the system of interest contains no nonsmooth expressions, the required property can be violated. This circumstance does not allow one to calculate functional gradients by standard methods and use traditional optimization methods. Nonetheless, the system under study displays a property that can be considered as a weak form of differentiability of the state function with respect to control. As a result, we arrive at the notion of extended derivative of an operator, presenting generalization of the classical types of operator derivatives. With the help of this notion, one can find a functional gradient even if its direct determination is impossible. In this way, we can enlarge the area of applicability of optimization methods. It is shown that, for extended derivatives, an analogue to the inverse function theorem can be formulated which makes it possible to give a general scheme permitting calculation of functional gradients in nonlinear problems of mathematical physics even in those cases in which the state function turns out to be a nondifferentiable function of the control variable.

show abstract

mentioning

confidence: 70%

Calculation of functional gradients and extended differentiation of operators

Serovajsky¹

2005

Journal of Inverse and Ill-posed Problems

View full text Add to dashboard Cite

show abstract

scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.

Contact Info

hi@scite.ai

10624 S. Eastern Ave., Ste. A-614

Henderson, NV 89052, USA

Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information

Made with 💙 for researchers

Part of the Research Solutions Family.

Differentiation of inverse functions in spaces without norm

Cited by 1 publication

References 1 publication

Calculation of functional gradients and extended differentiation of operators

Calculation of functional gradients and extended differentiation of operators

Contact Info

Product

Resources

About