Fixed Point Iteration Method Pdf Free

Download File ===> https://bltlly.com/2t8oMk

Inverse problems consist of recovering a signal from a collection of noisy measurements. These problems can often be cast as feasibility problems. This work--by Howard Heaton, Sam Wu Fung, Aviv Gibali, Wotao Yin--fuses data-driven regularization and convex feasibility in a theoretically sound manner. This is accomplished using feasibility-based fixed point networks (F-FPNs) which define a collection of nonexpansive operators. Fixed point iteration is used to compute fixed points of these operators, and weights of the operators are tuned so that the fixed points closely represent available data. Numerical examples demonstrate performance increases by F-FPNs when compared to standard TV-based recovery methods for CT reconstruction.

In a wide range of mathematical, computational, economical, modeling and engineering problems, the existence of a solution to a theoretical or real world problem is equivalent to the existence of a fixed point for a suitable map or operator. Fixed points are therefore of paramount importance in many areas of mathematics, sciences and engineering.

The theory itself is a beautiful mixture of analysis (pure and applied), topology and geometry. Over the last 60 years or so, the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in such diverse fields as biology, chemistry, physics, engineering, game theory and economics.

In numerous cases finding the exact solution is not possible; hence it is necessary to develop appropriate algorithms to approximate the requested result. This is strongly related to control and optimization problems arising in the different sciences and in engineering problems. Many situations in the study of nonlinear equations, calculus of variations, partial differential equations, optimal control and inverse problems can be formulated in terms of fixed point problems or optimization.

The aim of this journal is to report new fixed point results, methods and algorithms as well as their applications in which the indispensability of the fixed point results is highlighted or is the common substrate. It will cover topics such as

A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function.

In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. Specifically, given a function f {\displaystyle f} with the same domain and codomain, a point x 0 {\displaystyle x_{0}} in the domain of f {\displaystyle f} , the fixed-point iteration is

According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.[1]

In order theory, the least fixed point of a function from a partially ordered set (poset) to itself is the fixed point which is less than each other fixed point, according to the order of the poset. A function need not have a least fixed point, but if it does then the least fixed point is unique.

In combinatory logic for computer science, a fixed-point combinator is a higher-order function fix {\displaystyle {\textsf {fix}}} that returns a fixed point of its argument function, if one exists. Formally, if the function f has one or more fixed points, then

In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog.

A fixed-point theorem is a result saying that at least one fixed point exists, under some general condition.[6] Some authors claim that results of this kind are amongst the most generally useful in mathematics.[7]

In particular, in the light of our observations from Sect. 2, the algebraic preconditioner arises as the discretisation of the Sobolev preconditioner; or said differently, the preconditioned algebraic gradient method (27) matches, up to the damping parameter, the discretisation of the unified iteration scheme (6). We remark that this is not a new insight, but is already known in the literature: for linear problems, this and many more observations are well presented in [33], and in the context of nonlinear problems we refer to [24] and [9, Ch. 8]. 2b1af7f3a8