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...

Inverse heat conduction problems, which are one of the most important groups of problems, are often ill-posed and complicated problems, and their optimization process has lots of local extrema. This paper provides a novel computational procedure based on finite differences method and league championship algorithm to solve a one-dimensional inverse heat conduction problem. At the beginning, we u...

Successful employment of numerical techniques for the forward and inverse electrocardiographic (ECG) problems requires the ability to both quantify and minimize approximation errors introduced as part of the discretization process. Conventional finite element discretization and refinement strategies effective for the forward problem may become inappropriate for the inverse problem because of it...

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...