Paul P. B. Eggermont

Disentangling conformational states of macromolecules in 3D-EM through likelihood optimization

Although three-dimensional electron microscopy (3D-EM) permits structural characterization of macromolecular assemblies in distinct functional states, the inability to classify projections from structurally heterogeneous samples has severely limited its application. We present a maximum likelihood–based classification method that does not depend on prior knowledge about the structural variability, and demonstrate its effectiveness for two macromolecular assemblies with different types of conformational variability: the Escherichia coli ribosome and Simian virus 40 (SV40) large T-antigen.

Iterative algorithms for large partitioned linear systems, with applications to image reconstruction

Abstract We present a unifying framework for a wide class of iterative methods in numerical linear algebra. In particular, the class of algorithms contains Kaczmarzs and Richardsons methods for the regularized weighted least squares problem with weighted norm. The convergence theory for this class of algorithms yields as corollaries the usual convergence conditions for Kaczmarzs and Richardsons methods. The algorithms in the class may be characterized as being group-iterative, and incorporate relaxation matrices, as opposed to a single relaxation parameter. We show that some well-known iterative methods of image reconstruction fall into the class of algorithms under consideration, and are thus covered by the convergence theory. We also describe a novel application to truly three-dimensional image reconstruction.

Strong underrelaxation in Kaczmarz's method for inconsistent systems

SummaryWe investigate the behavior of Kaczmarzs method with relaxation for inconsistent systems. We show that when the relaxation parameter goes to zero, the limits of the cyclic subsequences generated by the method approach a weighted least squares solution of the system. This point minimizes the sum of the squares of the Euclidean distances to the hyperplanes of the system. If the starting point is chosen properly, then the limits approach the minimum norm weighted least squares solution. The proof is given for a block-Kaczmarz method.

Multiplicative iterative algorithms for convex programming

Abstract We study multiplicative iterative algorithms for the minimization of a differentiable, convex function defined on the positive orthant of R N . If the objective function has compact level sets and has a locally Lipschitz continuous gradient, then these algorithms converge to a solution of the minimization problem. Moreover, the convergence is nearly monotone in the sense of Kullback-Leibler divergence. Special cases of these algorithms have been applied in position emission tomography and are formally related to the EM algorithm for position emission tomography.

Best asymptotic normality of the kernel density entropy estimator for smooth densities

In the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel. Under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed (i.i.d.) random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n/sup -1/2/. An essential part in the proof is obtained by exhibiting almost sure bounds for the Kullback-Leibler divergence between the kernel density estimator and its expected value. The basic technical tools are Doobs submartingale inequality and convexity (Jensens inequality).

On the quadrature error in operational quadrature methods for convolutions

SummaryWe analyze the quadrature error associated with operational quadrature methods for convolution equations. The assumptions are that the convolution kernel is inL1 and that its Laplace transform is analytic and bounded in an obtuse sector of the complex plane. Under these circumstances the Laplace transform has a slow variation property which admits a Fourier analysis of the quadrature error. We provide generalLp error estimates assuming suitable smoothness conditions on the function under convolution.

On Galerkin methods for Abel-type integral equations

Galerkin methods are studied for the numerical solution of Abel-type integral equations. In order to maintain the causality/triangularity of the resulting system of linear equations, spline functions of maximal defect are used, i.e.,

Stability and robustness of collocation methods for Abel-type integral equations

C^{ - 1}

Lp estimates of boundary integral equations for some nonlinear boundary value problems

piecewise polynomials. Asymptotically optimal error estimates in the supremum norm are proved. Both robustness and accuracy of the Galerkin methods are compared with those of collocation methods. Some numerical experiments using some “cheap” quadrature schemes that work quite well under certain circumstances only are shown, and some generally applicable quadrature methods are reported.

Improving the accuracy of collocation solutions of volterra integral equations of the first kind by local interpolation

SummaryWe study stability aspects of collocation methods for Abel-type integral equations of the first kind using piecewise polynomials. These collocation methods may be formulated as projection methods. Stability is defined as the boundedness of the sequence of projectors in their natural setting. Robustness is essentially the optimal asymptotic insensitivity to perturbations in the data. We show that stability and robustness are equivalent for the above collocation methods. This allows us to obtain optimal error estimates for some methods that are well-known to be robust. We also present numerical results for some methods which appear to be robust.