Mild Differentiability Conditions For Newton Method In Banach Spaces Frontiers

The Newton method is a widely used optimization algorithm in various fields, including mathematics, computer science, and engineering. It is particularly effective in solving nonlinear equations and finding critical points of differentiable functions. However, the standard Newton method assumes full differentiability, which restricts its applicability to certain classes of functions. This article explores the concept of mild differentiability conditions for the Newton method in Banach spaces frontiers, opening up new possibilities for optimization problems in these spaces.

The Newton method is based on the idea of linearizing a function in the vicinity of its current approximation and solving the resulting linear equation. This linearization process relies heavily on the differentiability of the function. In Banach spaces, the concept of differentiability is more general than in the finite-dimensional case, allowing for more versatile applications.

In the classical Banach space theory, the concept of Gateaux differentiability was introduced as a natural extension of the concept of differentiability in Euclidean spaces. However, this notion is restrictive for the Newton method as it requires the existence of directional derivatives in all directions. The mild differentiability concept provides a more relaxed requirement.

Mild Differentiability Conditions for Newton's Method in Banach Spaces (Frontiers in Mathematics)

by Eckehard W. Mielke(1st ed. 2020 Edition, Kindle Edition)

4.6 out of 5

Language	:	English
File size	:	6085 KB
Screen Reader	:	Supported
Print length	:	192 pages

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Mild Differentiability Conditions

In Banach spaces, a function is considered to be mildly differentiable if it satisfies certain conditions that allow for the application of linearization techniques similar to the Newton method. These conditions are less stringent than the standard differentiability conditions and allow for a broader class of functions to be optimized using the Newton method.

One of the key conditions for mild differentiability is the existence of the Fréchet derivative. The Fréchet derivative provides a linear approximation of a function, allowing for an iterative approach to finding critical points. Additionally, the function must satisfy a Lipschitz condition on its Fréchet derivative, ensuring that the linearization process is well-behaved.

Another important condition is the existence of a bounded inverse of the Fréchet derivative. This condition guarantees the convergence of the Newton method and provides stability in finding critical points. Without this condition, the Newton method may fail to converge or yield inaccurate results.

Applications in Banach Spaces

The extension of the Newton method to Banach spaces with mild differentiability conditions opens up new possibilities for optimization problems in various fields. One example is the optimization of nonlinear partial differential equations (PDEs).

Nonlinear PDEs arise in many scientific and engineering applications, and their numerical solution is often challenging. The Newton method, with its ability to provide rapid convergence, is a natural choice for solving these equations. By extending the Newton method to Banach spaces, mild differentiability conditions allow for the optimization of nonlinear PDEs on a broader class of functions, leading to more accurate and efficient solutions.

Another application area is optimization problems in functional analysis. Functional analysis deals with the study of spaces of functions and operators, and optimization problems in this field often require specialized techniques. The Newton method, with mild differentiability conditions, offers a powerful tool for solving optimization problems in functional analysis, enabling researchers to tackle complex problems more effectively.

The Newton method is a widely used optimization algorithm that has proven effective in various fields. By extending its applicability to Banach spaces through mild differentiability conditions, new frontiers are opened up in optimization problems. The concept of mild differentiability allows a broader class of functions to be optimized, leading to more accurate and efficient solutions.

Whether it's nonlinear PDEs or optimization problems in functional analysis, the Newton method with mild differentiability conditions provides a valuable framework for tackling complex problems in these areas. As researchers continue to explore the boundaries of optimization in Banach spaces, the development of new techniques and algorithms based on mild differentiability conditions will undoubtedly play a significant role in advancing the field.

Mild Differentiability Conditions for Newton's Method in Banach Spaces (Frontiers in Mathematics)

by Eckehard W. Mielke(1st ed. 2020 Edition, Kindle Edition)

4.6 out of 5

Language	:	English
File size	:	6085 KB
Screen Reader	:	Supported
Print length	:	192 pages

FREE

In this book the authors use a technique based on recurrence relations to study the convergence of the Newton method under mild differentiability conditions on the first derivative of the operator involved. The authors’ technique relies on the construction of a scalar sequence, not majorizing, that satisfies a system of recurrence relations, and guarantees the convergence of the method. The application is user-friendly and has certain advantages over Kantorovich’s majorant principle. First, it allows generalizations to be made of the results obtained under conditions of Newton-Kantorovich type and, second, it improves the results obtained through majorizing sequences. In addition, the authors extend the application of Newton’s method in Banach spaces from the modification of the domain of starting points. As a result, the scope of Kantorovich’s theory for Newton’s method is substantially broadened. Moreover, this technique can be applied to any iterative method.

This book is chiefly intended for researchers and (postgraduate) students working on nonlinear equations, as well as scientists in general with an interest in numerical analysis.