Ill-posed sets of linear equations typically arise when discretizing certain types of integral transforms. A well known example is **image reconstruction**, which can be modelled using the Radon transform. After expanding the solution into a finite series of basis functions a large, sparse and ill-conditioned **linear system** arises. We consider the solution of such systems. In particular we study a new class of iteration methods named DROP (for Diagonal Relaxed Orthogonal Projections) constructed for solving both linear equations and linear inequalities. This class can also be viewed, when applied to linear equations, as a generalized Landweber iteration. The method is compared with other iteration methods using test data from a medical application and from electron microscopy. Our theoretical analysis include convergence proofs of the fully-simultaneous DROP algorithm for linear equations without consistency assumptions, and of block-iterative algorithms both for linear equations and linear inequalities, for the consistent case.When applying an iterative solver to an ill-posed set of linear equations the error typically initially decreases but after some iterations (depending on the amount of noise in the data, and the degree of **ill-posedness**) it starts to increase. This phenomena is called **semi-convergence**…

*Contents*

1 Introduction

1 Semi-convergence behavior of Landweber iteration

2 Projection Algorithms

3 Stopping rules

2 Summary of papers

References

Author: Nikazad, Touraj

Source: Linköping University

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