A new approach to the reconstruction of images from Radon projections

A new approach is proposed for reconstruction of images from Radon projections. Based on Fourier expansions in orthogonal polynomials of two and three variables, instead of Fourier transforms, the approach provides a new algorithm for the computed tomography. The convergence of the algorithm is established under mild assumptions.Comment: 28 pages, accepted by Adv. in Applied Mat

The Kink Phenomenon in Fejér and Clenshaw-Curtis Quadrature

The Fejér and Clenshaw-Curtis rules for numerical integration exhibit a curious phenomenon when applied to certain analytic functions. When N, (the number of points in the integration rule) increases, the error does not decay to zero evenly but does so in two distinct stages. For N less than a critical value, the error behaves like $O(\varrho^{-2N})$, where $\varrho$ is a constant greater than 1. For these values of N the accuracy of both the Fejér and Clenshaw-Curtis rules is almost indistinguishable from that of the more celebrated Gauss-Legendre quadrature rule. For larger N, however, the error decreases at the rate $O(\varrho^{-N})$, i.e., only half as fast as before. Convergence curves typically display a kink where the convergence rate cuts in half. In this paper we derive explicit as well as asymptotic error formulas that provide a complete description of this phenomenon.\ud \ud This work was supported by the Royal Society of the UK and the National Research Foundation of South Africa under the South Africa-UK Science Network Scheme. The first author also acknowledges grant FA2005032300018 of the NRF

Efficient solution of parabolic equations by Krylov approximation methods

Numerical techniques for solving parabolic equations by the method of lines is addressed. The main motivation for the proposed approach is the possibility of exploiting a high degree of parallelism in a simple manner. The basic idea of the method is to approximate the action of the evolution operator on a given state vector by means of a projection process onto a Krylov subspace. Thus, the resulting approximation consists of applying an evolution operator of a very small dimension to a known vector which is, in turn, computed accurately by exploiting well-known rational approximations to the exponential. Because the rational approximation is only applied to a small matrix, the only operations required with the original large matrix are matrix-by-vector multiplications, and as a result the algorithm can easily be parallelized and vectorized. Some relevant approximation and stability issues are discussed. We present some numerical experiments with the method and compare its performance with a few explicit and implicit algorithms

Propagation of epistemic uncertainty in queueing models with unreliable server using chaos expansions

In this paper, we develop a numerical approach based on Chaos expansions to analyze the sensitivity and the propagation of epistemic uncertainty through a queueing systems with breakdowns. Here, the quantity of interest is the stationary distribution of the model, which is a function of uncertain parameters. Polynomial chaos provide an efficient alternative to more traditional Monte Carlo simulations for modelling the propagation of uncertainty arising from those parameters. Furthermore, Polynomial chaos expansion affords a natural framework for computing Sobol' indices. Such indices give reliable information on the relative importance of each uncertain entry parameters. Numerical results show the benefit of using Polynomial Chaos over standard Monte-Carlo simulations, when considering statistical moments and Sobol' indices as output quantities