Online modeling with tunable RBF network

In this paper, we propose a novel online modeling algorithm for nonlinear and nonstationary systems using a radial basis function (RBF) neural network with a fixed number of hidden nodes. Each of the RBF basis functions has a tunable center vector and an adjustable diagonal covariance matrix. A multi-innovation recursive least square (MRLS) algorithm is applied to update the weights of RBF online, while the modeling performance is monitored. When the modeling residual of the RBF network becomes large in spite of the weight adaptation, a node identified as insignificant is replaced with a new node, for which the tunable center vector and diagonal covariance matrix are optimized using the quantum particle swarm optimization (QPSO) algorithm. The major contribution is to combine the MRLS weight adaptation and QPSO node structure optimization in an innovative way so that it can track well the local characteristic in the nonstationary system with a very sparse model. Simulation results show that the proposed algorithm has significantly better performance than existing approaches.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.

A fast adaptive tunable RBF network for nonstationary systems

This paper describes a novel on-line learning approach for radial basis function (RBF) neural network. Based on an RBF network with individually tunable nodes and a fixed small model size, the weight vector is adjusted using the multi-innovation recursive least square algorithm on-line. When the residual error of the RBF network becomes large despite of the weight adaptation, an insignificant node with little contribution to the overall system is replaced by a new node. Structural parameters of the new node are optimized by proposed fast algorithms in order to significantly improve the modeling performance. The proposed scheme describes a novel, flexible, and fast way for on-line system identification problems. Simulation results show that the proposed approach can significantly outperform existing ones for nonstationary systems in particular.

A new adaptive multiple modelling approach for non-linear and non-stationary systems

This paper proposes a novel adaptive multiple modelling algorithm for non-linear and non-stationary systems. This simple modelling paradigm comprises K candidate sub-models which are all linear. With data available in an online fashion, the performance of all candidate sub-models are monitored based on the most recent data window, and M best sub-models are selected from the K candidates. The weight coefficients of the selected sub-model are adapted via the recursive least square (RLS) algorithm, while the coefficients of the remaining sub-models are unchanged. These M model predictions are then optimally combined to produce the multi-model output. We propose to minimise the mean square error based on a recent data window, and apply the sum to one constraint to the combination parameters, leading to a closed-form solution, so that maximal computational efficiency can be achieved. In addition, at each time step, the model prediction is chosen from either the resultant multiple model or the best sub-model, whichever is the best. Simulation results are given in comparison with some typical alternatives, including the linear RLS algorithm and a number of online non-linear approaches, in terms of modelling performance and time consumption.

The method of least squares used to invert an orbit problem

Six parameters uniquely describe the orbit of a body about the Sun. Given these parameters, it is possible to make predictions of the body's position by solving its equation of motion. The parameters cannot be directly measured, so they must be inferred indirectly by an inversion method which uses measurements of other quantities in combination with the equation of motion. Inverse techniques are valuable tools in many applications where only noisy, incomplete, and indirect observations are available for estimating parameter values. The methodology of the approach is introduced and the Kepler problem is used as a real-world example. (C) 2003 American Association of Physics Teachers.

The method of predicate least squares refinement

Parameters to be determined in a least squares refinement calculation to fit a set of observed data may sometimes usefully be `predicated' to values obtained from some independent source, such as a theoretical calculation. An algorithm for achieving this in a least squares refinement calculation is described, which leaves the operator in full control of the weight that he may wish to attach to the predicate values of the parameters.

Effect of method of application of a fibrolytic enzyme product on digestive processes and milk production in Holstein-Friesian cows

Four multiparous cows with cannulas in the rumen and proximal duodenum were used in early lactation in a 4 x 4 Latin square experiment to investigate the effect of method of application of a fibrolytic enzyme product on digestive processes and milk production. The cows were given ad libitum a total mixed ration (TMR) composed of 57% (dry matter basis) forage (3:1 corn silage:grass silage) and 43% concentrates. The TMR contained (g/kg dry matter): 274 neutral detergent fiber, 295 starch, 180 crude protein. Treatments were TMR alone or TMR with the enzyme product added (2 kg/1000 kg TMR dry matter) either sprayed on the TMR 1 h before the morning feed (TMR-E), sprayed only on the concentrate the day before feeding (Concs-E), or infused into the rumen for 14 h/d (Rumen-E). There Was no significant effect on either feed intake or milk yield but both were highest on TMR-E. Rumen digestibility of dry matter, organic matter, and starch was unaffected by the enzyme. Digestibility of NDF was lowest on TMR-E in the rumen but highest postruminally. Total Tract digestibility was highest on TMR-E for dry matter, organic matter, and starch but treatment differences were nonsignificant for neutral detergent fiber: Corn silage stover retention time in the rumen was reduced by all enzyme treatments but postruminal transit time vas increased so the decline in total tract retention. time with enzymes was not significant. It is suggested that the tendency for enzymes to reduce particle retention time in the rumen may, by reducing the time available for fibrolysis to occur, at least partly explain the variability in the reported responses to enzyme treatment.

Adaptive least mean square CDMA detection with Gram–Schmidt pre-processing

The Gram-Schmidt (GS) orthogonalisation procedure has been used to improve the convergence speed of least mean square (LMS) adaptive code-division multiple-access (CDMA) detectors. However, this algorithm updates two sets of parameters, namely the GS transform coefficients and the tap weights, simultaneously. Because of the additional adaptation noise introduced by the former, it is impossible to achieve the same performance as the ideal orthogonalised LMS filter, unlike the result implied in an earlier paper. The authors provide a lower bound on the minimum achievable mean squared error (MSE) as a function of the forgetting factor λ used in finding the GS transform coefficients, and propose a variable-λ algorithm to balance the conflicting requirements of good tracking and low misadjustment.

A frequency-independent boundary element method for scattering by two-dimensional screens and apertures

We propose and analyse a hybrid numerical–asymptotic hp boundary element method (BEM) for time-harmonic scattering of an incident plane wave by an arbitrary collinear array of sound-soft two-dimensional screens. Our method uses an approximation space enriched with oscillatory basis functions, chosen to capture the high-frequency asymptotics of the solution. We provide a rigorous frequency-explicit error analysis which proves that the method converges exponentially as the number of degrees of freedom N increases, and that to achieve any desired accuracy it is sufficient to increase N in proportion to the square of the logarithm of the frequency as the frequency increases (standard BEMs require N to increase at least linearly with frequency to retain accuracy). Our numerical results suggest that fixed accuracy can in fact be achieved at arbitrarily high frequencies with a frequency-independent computational cost, when the oscillatory integrals required for implementation are computed using Filon quadrature. We also show how our method can be applied to the complementary ‘breakwater’ problem of propagation through an aperture in an infinite sound-hard screen.

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

A Galerkin boundary element method for high frequency scattering by convex polygons

In this paper we consider the problem of time-harmonic acoustic scattering in two dimensions by convex polygons. Standard boundary or finite element methods for acoustic scattering problems have a computational cost that grows at least linearly as a function of the frequency of the incident wave. Here we present a novel Galerkin boundary element method, which uses an approximation space consisting of the products of plane waves with piecewise polynomials supported on a graded mesh, with smaller elements closer to the corners of the polygon. We prove that the best approximation from the approximation space requires a number of degrees of freedom to achieve a prescribed level of accuracy that grows only logarithmically as a function of the frequency. Numerical results demonstrate the same logarithmic dependence on the frequency for the Galerkin method solution. Our boundary element method is a discretization of a well-known second kind combined-layer-potential integral equation. We provide a proof that this equation and its adjoint are well-posed and equivalent to the boundary value problem in a Sobolev space setting for general Lipschitz domains.