Narrow Results

In feedforward networks trained by the classic backpropagation algorithm, as introduced by Rumelhart et al., the weights are modified according to the method of steepest descent. The goal of this weight modification is to minimise the error in network-outputs for a given training set.

Basing upon Jacobs’ work, we point out drawbacks of steepest descent and suggest improvements on it. These yield a feedforward network, which adjusts its weights according to a (globally convergent) parallel coordinate descent method.

Then we combine this parallel coordinate descent with a hybrid learning rule from -rule and momentum version. For adjusting the parameters of this rule we use a Sugeno/Tagaki fuzzy controller. We only need four rules for the controlling process. Because of using the Sugeno/Takagi controller, no defuzzification has to be performed and the controller works faster than the familiar one.

We conclude that this algorithm is very suitable for fast training and global convergence.

Time-dependent wave motion coupled with bottom disturbances and poroelastic plate in a viscous incompressible fluid of finite depth is studied here. The problem is framed as an initial value problem in terms of potential and stream functions by assuming linear water wave theory. Three objectives are met in this work: first, the behaviour of the root of the dispersion relation is analyzed for plane progressive waves; second, the effect of viscosity on the asymptotic form of the plate deflection is studied. Third, the effect of plate porosity and other crucial parameters on fluid velocity are examined through various contour plots. Using Laplace and Fourier transform techniques, the expression for plate deflection is derived in terms of a highly oscillatory infinite integral and then evaluated asymptotically by using the steepest descent method. Different numerical analyses are performed by varying physical parameters such as viscosity, porosity, flexural rigidity and frequency of the wave motion. It is observed that, with an increase in the value of viscosity, the values of the real and imaginary parts of the complex root of the dispersion relation increase for frequency above a cut-off value; the maximum amplitude of the plate deflection decreases, and the vertical velocity decreases.

Gradient-type algorithms can be linked to algorithms for constructing optimal experimental designs for linear regression models. The asymptotic rate of convergence of these algorithms can be expressed through the asymptotic behaviour of an experimental design construction procedure. One well known gradient-type algorithm is the method of Steepest Descent. Here a generalised version of the Steepest Descent algorithm, with a relaxation coefficient is considered and the rate of convergence of this algorithm is investigated.

The paper is based on a course given in 2007 at an ICTP school in Alexandria, Egypt. It aims at introducing young scientists to methods to calculate asymptotic developpments of singular integrals.