An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

Estimating Particle Shape and Orientation Using Volume Tensors

We develop statistical procedures for estimating shape and orientation of arbitrary three-dimensional particles. We focus on the case where particles cannot be observed directly, but only via sections. Volume tensors are used for describing particle shape and orientation, and we derive stereological estimators of the tensors. These estimators are combined to provide consistent estimators of the moments of the so-called particle cover density. The covariance structure associated with the particle cover density depends on the orientation and shape of the particles. For instance, if the distribution of the typical particle is invariant under rotations, then the covariance matrix is proportional to the identity matrix. We develop a non-parametric test for such isotropy. A flexible Lévy-based particle model is proposed, which may be analysed using a generalized method of moments in which the volume tensors enter. The developed methods are used to study the cell organization in the human brain cortex.

Associating optical measurements and estimating orbits of geocentric objects through population-based meta-heuristic methods

Currently several thousands of objects are being tracked in the MEO and GEO regions through optical means. The problem faced in this framework is that of Multiple Target Tracking (MTT). In this context both, the correct associations among the observations and the orbits of the objects have to be determined. The complexity of the MTT problem is defined by its dimension S. The number S corresponds to the number of fences involved in the problem. Each fence consists of a set of observations where each observation belongs to a different object. The S ≥ 3 MTT problem is an NP-hard combinatorial optimization problem. There are two general ways to solve this. One way is to seek the optimum solution, this can be achieved by applying a branch-and- bound algorithm. When using these algorithms the problem has to be greatly simplified to keep the computational cost at a reasonable level. Another option is to approximate the solution by using meta-heuristic methods. These methods aim to efficiently explore the different possible combinations so that a reasonable result can be obtained with a reasonable computational effort. To this end several population-based meta-heuristic methods are implemented and tested on simulated optical measurements. With the advent of improved sensors and a heightened interest in the problem of space debris, it is expected that the number of tracked objects will grow by an order of magnitude in the near future. This research aims to provide a method that can treat the correlation and orbit determination problems simultaneously, and is able to efficiently process large data sets with minimal manual intervention.

Estimating the benefit of a second bone anchored hearing implant in unilaterally implanted users with a testband.

Conclusion Using a second bone anchored hearing implant (BAHI) mounted on a testband in unilaterally implanted BAHI users to test its potential advantage pre-operatively under-estimates the advantage of two BAHIs placed on two implants. Objectives To investigate how well speech understanding with a second BAHI mounted on a testband approaches the benefit of bilaterally implanted BAHIs. Method Prospective study with 16 BAHI users. Eight were implanted unilaterally (group A) and eight were implanted bilaterally (group B). Aided speech understanding was measured. Speech was presented from the front and noise came either from the left, right, or from the front in two conditions for group A (with one BAHI, and with two BAHIs, where the second device was mounted on a testband) and in three conditions for group B (same two conditions as group A, and in addition with both BAHIs mounted on implants). Results Speech understanding in noise improved with the additional device for noise from the side of the first BAHI (+0.7 to +2.1 dB) and decreased for noise from the other side (-1.8 dB to -3.9 dB). Improvements were highest (+2.1 dB, p = 0.016) and disadvantages were smallest (-1.8 dB, p = 0.047) with both BAHIs mounted on implants. Testbands yielded smaller advantages and higher disadvantages of the additional BAHI (average difference = -0.9 dB).