Metamodeling using extended radial basis functions: a comparative approach

The process of constructing computationally benign approximations of expensive computer simulation codes, or metamodeling, is a critical component of several large-scale multidisciplinary design optimization (MDO) approaches. Such applications typically involve complex models, such as finite elements, computational fluid dynamics, or chemical processes. The decision regarding the most appropriate metamodeling approach usually depends on the type of application. However, several newly proposed kernel-based metamodeling approaches can provide consistently accurate performance for a wide variety of applications. The authors recently proposed one such novel and effective metamodeling approach—the extended radial basis function (E-RBF) approach—and reported highly promising results. To further understand the advantages and limitations of this new approach, we compare its performance to that of the typical RBF approach, and another closely related method—kriging. Several test functions with varying problem dimensions and degrees of nonlinearity are used to compare the accuracies of the metamodels using these metamodeling approaches. We consider several performance criteria such as metamodel accuracy, effect of sampling technique, effect of sample size, effect of problem dimension, and computational complexity. The results suggest that the E-RBF approach is a potentially powerful metamodeling approach for MDO-based applications, as well as other classes of computationally intensive applications.     

Non-linear systems identification using radial basis functions

This paper investigates the identification of discrete-time non-linear systems using radial basis functions. A forward regression algorithm based on an orthogonal decomposition of the regression matrix is employed to select a suitable set of radial basis function centers from a large number of possible candidates and this provides, for the first time, fully automatic selection procedure for identifying parsimonious radial basis function models of structure-unknown non-linear systems. The relationship between neural networks and radial basis functions is discussed and the application of the algorithms to real data is included to demonstrate the effectiveness of this approach.     

An interpretive approach for interactive real-time control through a network

The CERN Super Proton Synchrotron (SPS) accelerator represents a large industrial control problem, with the additional complication that the control strategies are not fixed. Right from the beginning, it was decided to use an interpretive language for the control process. The main goal of this approach was to reduce the amount of programming effort needed for a proper control of the accelerator, by allowing the actual users — physicists, engineers and technicians — to write the control programs they needed. It very quickly appeared that this method was ideal for providing interactive tools for a control system based on a network.The requirement for interactive network control has been implemented by leaving to the user the choice of those parts of the network best suited for the execution of sections of his program. For that reason interpreter instructions have been defined, allowing the user to define where in the network a logical unit of his algorithm should be executed, and where individual items of the data should be stored.This strategy allows easy access to all the hardware through a distributed data-base. The interpreter being basically a string handler, equipment control can be packed into a module which is called from the program through a name with a standard input-output scheme. All this allows algorithm exportation, while network transfers are minimized.Such a system has been in operation since 1976. Many non-professional programmers are regularly writing or modifying ‘network procedures’ for everchanging control purposes. The simplicity and flexibility of the interpretive method has enabled many technical innovations to be installed over the years, and some of these are described.     

Automated freehand sketch segmentation using radial basis functions

Freehand sketching is widely regarded as an efficient and natural way for interaction between computers and humans. We present a robust computerized scheme to automatically segment freehand sketches into a series of components with specific geometric meaning regardless of whether these are generated online or offline. This task is a necessary first step toward sketch understanding. By exploiting the interpolation/extrapolation characteristic of radial basis functions (RBFs), a greedy algorithm consisting of forward and backward operations is proposed for finding the minimum set of segmentation points that can be used to reconstruct with high fitting accuracy freehand sketches in the form of implicit functions. To obtain segmentation points, a simple angle-based rule is used to remove “bridging” points that provide a smooth transition between consecutive sketch components. Feasibility of the proposed algorithm is demonstrated by a preliminary performance assessment study using ten computer generated drawings. These experiments show that in this dataset sensitivity of the segmentation was higher than 97.5% with a false positive (FP) rate of approximately 25%. The majority of false positive identifications are located on arc regions where a larger number of segmentation points are needed for reconstruction purposes. The primary contribution of this algorithm is that it transforms an ambiguous problem, namely, freehand sketch segmentation, into an implicit function fitting operation. Therefore, this proposed approach has several advantages, including independence of the actual sketching activity, and the ability for a satisfactory detection of the transition point between a line and an arc or between two arcs.     

Blind equalization using a predictive radial basis function neural network

In this paper, we propose a novel blind equalization approach based on radial basis function (RBF) neural networks. By exploiting the short-term predictability of the system input, a RBF neural net is used to predict the inverse filter output. It is shown here that when the prediction error of the RBF neural net is minimized, the coefficients of the inverse system are identical to those of the unknown system. To enhance the identification performance in noisy environments, the improved least square (ILS) method based on the concept of orthogonal distance to reduce the estimation bias caused by additive measurement noise is proposed here to perform the training. The convergence rate of the ILS learning is analyzed, and the asymptotic mean square error (MSE) of the proposed predictive RBF identification method is derived theoretically. Monte Carlo simulations show that the proposed method is effective for blind system identification. The new blind technique is then applied to two practical applications: equalization of real-life radar sea clutter collected at the east coast of Canada and deconvolution of real speech signals. In both cases, the proposed blind equalization technique is found to perform satisfactory even when the channel effects and measurement noise are strong.     

Shape-adaptive radial basis functions

Radial basis functions for discrimination and regression have been used with some success in a wide variety of applications. Here, we investigate the optimal choice for the form of the basis functions and present an iterative strategy for obtaining the function in a regression context using a conjugate gradient-based algorithm together with a nonparametric smoother. This is developed in a discrimination framework using the concept of optimal scaling. Results are presented for a range of simulated and real data sets.     

An improved radial basis function network for visual autonomousroad following

We have developed a radial basis function network (RBFN) for visual autonomous road following. Preliminary testing of the RBFN was done using a driving simulator, and the RBFN was then installed on an actual vehicle at Carnegie Mellon University for testing in an outdoor road-following application. In our first attempts, the RBFN had some success, but it experienced some significant problems such as jittery control and driving failure. Several improvements have been made to the original RBFN architecture to overcome these problems in simulation and more importantly in actual road following, and the improvements are described in this paper.     

Approximation of insurance liability contracts using radial basis functions

ABSTRACT

We introduce the Option Interpolation Model (OIM) for accurate approximation of embedded option values in insurance liabilities. Accurate approximation is required for ex-ante risk management applications. The OIM is based on interpolation with radial basis functions, which can interpolate scattered data, and does not suffer from the curse of dimensionality. To reduce computation time we present an inversion method to determine the interpolation function weights. The robustness, accuracy and efficiency of the OIM are investigated in several numerical experiments. We show that the OIM results in highly accurate approximations.     

Scale-based clustering using the radial basis function network

This paper shows how scale-based clustering can be done using the radial basis function network (RBFN), with the RBF width as the scale parameter and a dummy target as the desired output. The technique suggests the "right" scale at which the given data set should be clustered, thereby providing a solution to the problem of determining the number of RBF units and the widths required to get a good network solution. The network compares favorably with other standard techniques on benchmark clustering examples. Properties that are required of non-Gaussian basis functions, if they are to serve in alternative clustering networks, are identified. This work, on the whole, points out an important role played by the width parameter in RBFN, when observed over several scales, and provides a fundamental link to the scale space theory developed in computational vision.     

Mesh deformation using radial basis functions for gradient-based aerodynamic shape optimization

Gradient-based aerodynamic shape optimization using computational fluid dynamics (CFD), and time dependent problems in aeroelasticity, that is, coupled calculations between computational structural mechanics (CSM) and CFD, require repeated deformations of the CFD mesh.An interpolation scheme, based on radial basis functions (RBF), is devised in order to propagate the deformations from the boundaries to the interior of the CFD mesh. This method can lower the computational costs due to the deformation of the mesh, in comparison with the usual Laplace smoothing. Moreover, the algorithm is independent of the mesh connectivities. Therefore, structured and unstructured meshes are equally treated as well as hybrid meshes.The application of this interpolation scheme in problems of aerodynamic shape optimization is also carefully investigated. When the optimization is executed by a gradient-based algorithm the cost function is differentiated with respect to the design parameters in order to obtain the gradient. The gradient is most efficiently and accurately calculated by solving a certain adjoint equation derived from the discretized flow equations. The calculation of the gradient, which is detailed in this presentation, involves the Jacobian matrix of the mesh deformation.Finally, we present the results of an optimization of the ONERA M6 wing at transonic speed using the interpolation algorithm. The results are used for comparison with another technique of mesh deformation. The quality of the mesh obtained by the new algorithm, and the interpolation error, are analyzed with respect to the parameters of the interpolation scheme: the type of RBF, the RBF’s shape parameter, and the sets of control points.     

3D garment positioning using Hermite radial basis functions

Virtual Reality - Fitting an elegant 3D garment model onto a target 3D human model is crucial for garment design industry and virtual try-on systems. The alignment of the garment onto virtual...     

一种改进的RBF全局优化方法

径向基函数（Radial Basis Functions）由于具有良好的近似效果和运算简单的特点,被应用于全局优化中,成为解决黑箱函数全局优化问题的有效方法。然而现有的基于RBF的全局优化算法存在迭代过程中RBF模型重构效率低下,以及采样方法不合理导致函数估值次数过多等问题。在此提出几个改进思路：采用基于矩阵分块的增量RBF方法以减少模型重构时间提高效率;采用增量LHD采样方法以确保具有更好的空间填充性;采用算法重启策略以降低估值次数。通过实验验证改进方法的优势。     

Reconstructing surfaces by volumetric regularization using radial basis functions

We present a new method of surface reconstruction that generates smooth and seamless models from sparse, noisy, nonuniform, and low resolution range data. Data acquisition techniques from computer vision, such as stereo range images and space carving, produce 3D point sets that are imprecise and nonuniform when compared to laser or optical range scanners. Traditional reconstruction algorithms designed for dense and precise data do not produce smooth reconstructions when applied to vision-based data sets. Our method constructs a 3D implicit surface, formulated as a sum of weighted radial basis functions. We achieve three primary advantages over existing algorithms: (1) the implicit functions we construct estimate the surface well in regions where there is little data, (2) the reconstructed surface is insensitive to noise in data acquisition because we can allow the surface to approximate, rather than exactly interpolate, the data, and (3) the reconstructed surface is locally detailed, yet globally smooth, because we use radial basis functions that achieve multiple orders of smoothness.     

Robustness of radial basis functions

Neural networks are intended to be used in future nanoelectronic technology since these architectures seem to be robust to malfunctioning elements and noise in its inputs and parameters. In this work, the robustness of radial basis function networks is analyzed in order to operate in noisy and unreliable environment. Furthermore, upper bounds on the mean square error under noise contaminated parameters and inputs are determined if the network parameters are constrained. To achieve robuster neural network architectures fundamental methods are introduced to identify sensitive parameters and neurons.     

Sequential approximate multi-objective optimization using radial basis function network

In industrial design optimization, objectives and constraints are generally given as implicit form of the design variables, and are evaluated through computationally intensive numerical simulation. Under this situation, response surface methodology is one of helpful approaches to design optimization. One of these approaches, known as sequential approximate optimization (SAO), has gained its popularity in recent years. In SAO, the sampling strategy for obtaining a highly accurate global minimum remains a critical issue. In this paper, we propose a new sampling strategy using sequential approximate multi-objective optimization (SAMOO) in radial basis function (RBF) network. To identify a part of the pareto-optimal solutions with a small number of function evaluations, our proposed sampling strategy consists of three phases: (1) a pareto-optimal solution of the response surfaces is taken as a new sampling point; (2) new points are added in and around the unexplored region; and (3) other parts of the pareto-optimal solutions are identified using a new function called the pareto-fitness function. The optimal solution of this pareto-fitness function is then taken as a new sampling point. The upshot of this approach is that phases (2) and (3) add sampling points without solving the multi-objective optimization problem. The detailed procedure to construct the pareto-fitness function with the RBF network is described. Through numerical examples, the validity of the proposed sampling strategy is discussed.     

Nonlinear systems parameters estimation using radial basis function network

In this paper, a new on-line scheme for the state and parameter estimation of a large class of nonlinear systems is presented. This scheme uses a radial basis function neuronal predictor with the on-line learning of weights. The algorithms developed are potentially useful for adjusting the controller parameters of variable speed drives. The other interesting feature of the proposed method is its application to failure and fault detection. The parameter identification scheme is an algebraic method combined with state estimation. The asymptotic convergence of the estimates to their nominal values is achieved using the Lyapunov's arguments. The simulation results and the real-time estimation of both rotor resistance and speed of an induction motor based on this approach, show rapidly converging estimates in spite of the measurements noise, discretization effects, parameters uncertainties (e.g. inaccuracies on motor inductance values) and modeling inaccuracies. The other applications of the proposed method include the on-line estimation of the parameters of a synchronous generator.     

Robust adaptive fault accommodation for a robot system using a radial basis function neural network

A discrete-time radial basis function (RBF) neural network is designed for the fault accommodation of robotic systems. A robust learning algorithm using the adaptive dead-zone technique is presented to train the network parameters (weights and centres). This scheme assures the convergence of the estimate errors of both the neural network and the fault-monitoring system in the presence of system uncertainties. Simulations have been done on applying the RBF-network-based fault accommodation scheme to a two-link robotic manipulator. The main advantage of the adaptive algorithm is that the upper bound of system uncertainties is not known in advance, which makes the system more practical for the fault accommodation scheme as demonstrated.     

A forecasting system for car fuel consumption using a radial basis function neural network

A predictive system for car fuel consumption using a radial basis function (RBF) neural network is proposed in this paper. The proposed work consists of three parts: information acquisition, fuel consumption forecasting algorithm and performance evaluation. Although there are many factors affecting the fuel consumption of a car in a practical drive procedure, in the present system the relevant factors for fuel consumption are simply decided as make of car, engine style, weight of car, vehicle type and transmission system type which are used as input information for the neural network training and fuel consumption forecasting procedure. In fuel consumption forecasting, to verify the effect of the proposed RBF neural network predictive system, an artificial neural network with a back-propagation (BP) neural network is compared with an RBF neural network for car fuel consumption prediction. The prediction results demonstrated the proposed system using the neural network is effective and the performance is satisfactory in terms of fuel consumption prediction.     

Sequential improvement for robust optimization using an uncertainty measure for radial basis functions

The performance of the sequential metamodel based optimization procedure depends strongly on the chosen building blocks for the algorithm, such as the used metamodeling method and sequential improvement criterion. In this study, the effect of these choices on the efficiency of the robust optimization procedure is investigated. A novel sequential improvement criterion for robust optimization is proposed, as well as an improved implementation of radial basis function interpolation suitable for sequential optimization. The leave-one-out cross-validation measure is used to estimate the uncertainty of the radial basis function metamodel. The metamodeling methods and sequential improvement criteria are compared, based on a test with Gaussian random fields as well as on the optimization of a strip bending process with five design variables and two noise variables. For this process, better results are obtained in the runs with the novel sequential improvement criterion as well as with the novel radial basis function implementation, compared to the runs with conventional sequential improvement criteria and kriging interpolation.     

An unified approach for DTC design using interactive tools

This paper presents an interactive tool for dead-time compensator design. The tool is based on an unified dead-time compensator and considers models commonly used in industrial process control. The main contribution of this work is that the proposed tool and controller are simple to analyze and tune as they are based on an unique modified structure of the Smith predictor valid for every type of dead-time process (including integrating and unstable systems). Simple frequency analysis and block diagram transformations are used together with simulations to illustrate the main problems associated to the control of dead-time processes. Several examples of typical processes are presented to illustrate the fundamental concepts associated to the control of these systems. The interactive tool is not only useful for designing and analyzing but also for training and educational purposes.