Since the classical iterative methods for solving nonlinear ill-posed problems are locally convergent, this paper constructs a robust and widely convergent method for identifying parameter based on homotopy algorithm, and investigates this method’s convergence in the light of Lyapunov theory. Furthermore, we consider 1-D elliptic type equation to testify that the homotopy regularization can ide...

Following a discussion of the relation of these problems to applications , intended to clarify the considerations which must be handled in order to obtain genuinely useful results, we consider techniques for determining optimal approximationss and consequent optimal error bounds for certain classes of ill-posed problems with appropriate a priori information.

Many works have shown that strong connections relate learning from examples to regularization techniques for ill-posed inverse problems. Nevertheless by now there was no formal evidence neither that learning from examples could be seen as an inverse problem nor that theoretical results in learning theory could be independently derived using tools from regularization theory. In this paper we pro...

GMRES is one of the most popular iterative methods for the solution of large linear systems of equations. However, GMRES generally does not perform well when applied to the solution of linear systems of equations that arise from the discretization of linear ill-posed problems with error-contaminated data represented by the right-hand side. Such linear systems are commonly referred to as linear ...

We consider large scale ill-conditioned linear systems arising from discretization of ill-posed problems. Regularization is imposed through an (assumed known) upper bound constraint on the solution. An iterative scheme, requiring the computation of the smallest eigenvalue and corresponding eigenvector, is used to determine the proper level of regularization. In this paper we consider several co...

Convergent methodology for ill-posed problems is typically equivalent to application of an operator dependent on a single parameter derived from the noise level and the data (a regularization parameter or terminal iteration number). In the context of a given problem discretized for purposes of numerical analysis, these methods can be viewed as resulting from imposed prior constraints bearing th...

Multilevel methods are popular for the solution of well-posed problems, such as certain boundary value problems for partial differential equations and Fredholm integral equations of the second kind. However, little is known about the behavior of multilevel methods when applied to the solution of linear ill-posed problems, such as Fredholm integral equations of the first kind, with a right-hand ...

This paper is devoted to the numerical analysis of ill-posed problems of evolution equations in Banach spaces using certain classes of stochastic one step methods. The linear stability properties of these methods are studied. Regularisation is given by the choice of the regularisation parameter as = p n ; where n is the stepsize and provides the convergence on smooth initial data. The case of t...

A ‘correct’ interpretation of the computational complexity of an ill-posed problem is formulated as a cost/effectiveness balance for the use of available data to obtain adequate solutions for an application. This composition with an application, is seen as the real problem, leading to the conclusion that some apparently ill-posed problems are, in context, really well-posed with a reasonable ass...

Let A be a linear, closed, densely defined unbounded operator in a Hilbert space. Assume that A is not boundedly invertible. If Eq. (1) Au = f is solvable, and ‖fδ − f ‖ δ, then the following results are provided: Problem Fδ(u) := ‖Au− fδ‖2 + α‖u‖2 has a unique global minimizer uα,δ for any fδ , uα,δ = A∗(AA∗ + αI)−1fδ . There is a function α = α(δ), limδ→0 α(δ)= 0 such that limδ→0 ‖uα(δ),δ − y...