Computer Modelling in Tomography and Ill-Posed Problems

Regularization of Monotone Variational Inequalities with Mosco Approximations of the Constraint Sets

“…Since the operator A k is s-w-demiclosed, the set A k y k is weakly closed in X * and, thus, v k ∈ A k y k . Taking lim inf as n → ∞ on both sides of (18) we obtain that v k satisfies (17). Observe that…”

“…Proof. According to A2(ii), one can reproduce without essential modifications the reasoning which proves the claim (17) in order to show that for each k ∈ N, there exists ψ k ∈ Ax k such that…”

“…Recall that an operator A: X → 2 X * is called strongly-weakly demiclosed (s-w-demiclosed for short) on a subset M of Dom A if for any pair of sequences {u k } k∈N ⊂ M and {ξ k } k∈N ⊂ X * such that ξ k ∈ Au k for all k ∈ N, u k → u and ξ k ξ we have that ξ ∈ Au. The problem of finding solutions for variational inequalities in the form (1) occurs in various fields and is often ill-posed in the sense that either the set S(A, f, ) is not a singleton and/or small perturbations of the problem data A, f and may lead to significant distortions of this set (see works of Tikhonov and Arsenin [18,33], Kinderlehrer and Stampacchia [15], Liskovets [20], Kaplan and Tichatschke [14], Engl, Hanke and Neubauer [12], Bonnans and Shapiro [8], Lavrent'ev, Zerkal and Trofimov [17]). Ill-posedness of the problem creates difficulties in applications where the problem data are given by sequences of approximations: even if the approximative data A k , f k and k are close to the original data A, f and , respectively, the solution set S(A k , f k , k ) of the variational inequality in which the original data are replaced by their respective approximations may happen to be empty or far from the solution set S(A, f, ) of the variational inequality (1).…”

An Application of Abel's Method to the Inverse Radon Transform

“…There is a well-developed theory surrounding the computation of the inverse Radon transform which has found many fruitful applications (see monographs [2], [3], [4] [5], [6]). One such family of numerical methods finds approximations of a function by expanding it into a series in terms of functions whose inverse Radon transforms are known ( [2], Ch.…”

Suppression of ring artefacts when tomographing anisotropically attenuating samples

“…Image reconstruction is always an unstable problem, i.e. small errors in input data lead to significant variations in the solution, see examples by Lavrent'ev et al (2001) and Natterer & Wü bbeling (2007). Also from the theory of ill-posed problems it is known that without a priori information about properties of a sample and noise levels in input data you may achieve a solution which varies significantly from the exact one.…”