A cooperative radial basis function method for variable-fidelity surrogate modeling

By coupling the low-fidelity (LF) model with the high-fidelity (HF) samples, the variable-fidelity model (VFM) offers an efficient way to overcome the expensive computing challenge in multidisciplinary design optimization (MDO). In this paper, a cooperative radial basis function (Co-RBF) method for the VFM is proposed by modifying the basis function of RBF. The RBF method is constructed on the HF samples, while the Co-RBF method incorporates the entire information of the LF model with the HF samples. In Co-RBF, the LF model is regard as a basis function of Co-RBF and the HF samples are utilized to compute the Co-RBF model coefficients. Two numerical functions and three engineering problems are adopted to verify the proposed Co-RBF method. The predictive results of Co-RBF are compared with those of RBF and Co-Kriging, which show that the Co-RBF method improves the efficiency, accuracy and robustness of the existing VFMs.     

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.     

Global optimization based on active preference learning with radial basis functions

Machine Learning - This paper proposes a method for solving optimization problems in which the decision-maker cannot evaluate the objective function, but rather can only express a preference such...     

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.     

基于改进型径向基函数网络的功放非线性建模

针对功率放大器(PA)的非线性建模,提出了改进型径向基函数神经网络(RBFNN)模型。首先,在该模型的输入端加入延迟交叉项和输出反馈项,利用正交最小二乘法提取模型的权值以及隐含层的中心;然后,采用15MHz带宽的宽带码分多址（WCDMA）三载波信号对Doherty功放进行测试,其归一化均方误差(NMSE)可以达到-45dB;最后,通过逆F类功放对模型的普遍适用性进行验证。仿真结果表明,该模型能够更加真实地拟合功率放大器的特性。     

A two-level approach to implicit surface modeling with compactly supported radial basis functions

We describe a two-level method for computing a function whose zero-level set is the surface reconstructed from given points scattered over the surface and associated with surface normal vectors. The function is defined as a linear combination of compactly supported radial basis functions (CSRBFs). The method preserves the simplicity and efficiency of implicit surface interpolation with CSRBFs and the reconstructed implicit surface owns the attributes, which are previously only associated with globally supported or globally regularized radial basis functions, such as exhibiting less extra zero-level sets, suitable for inside and outside tests. First, in the coarse scale approximation, we choose basis function centers on a grid that covers the enlarged bounding box of the given point set and compute their signed distances to the underlying surface using local quadratic approximations of the nearest surface points. Then a fitting to the residual errors on the surface points and additional off-surface points is performed with fine scale basis functions. The final function is the sum of the two intermediate functions and is a good approximation of the signed distance field to the surface in the bounding box. Examples of surface reconstruction and set operations between shapes are provided.     

Parametric shape-from-shading by radial basis functions

We present a new method of shape from shading by using radial basis functions to parameterize the object depth. The radial basis functions are deformed by adjusting their centers, widths, and weights such that the intensity errors are minimized. The initial centers and widths are arranged hierarchically to speed up convergence and to stabilize the solution. Although the smoothness constraint is used, it can be eventually dropped out without causing instabilities in the solution. An important feature of our parametric shape-from-shading method is that it offers a unified framework for integration of multiple sensory information. We show that knowledge about surface depth and/or surface normals anywhere in the image can be easily incorporated into the shape from shading process. It is further demonstrated that even qualitative knowledge can be used in shape from shading to improve 3D reconstruction. Experimental comparisons of our method with several existing ones are made by using both synthetic and real images. Results show that our solution is more accurate than the others     

紧支径向基函数插值实现多维数据可视化

通过分析某城市空气质量数值预报数据的时空组织结构,构建出了多维空间数据的整体框架。论述了几种插值方法的优缺点,在比较的基础上,将新的紧支径向基函数局部径向点插值方法引入到多维数据处理中,在空间、时间维度上对数据进行局部插值,从而实现数据的重构。以新的基于封装回调函数的多线程方法实现了大规模空气质量预报数据的三维动态可视化。实验结果表明,以上方法应用于大规模数据可视化时,其质量和运算速度都能满足实际需要。     

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.     

A novel model modification method for support vector regression based on radial basis functions

Structural and Multidisciplinary Optimization - There are some inherent limitations to the performance of support vector regression (SVR), such as (i) the loss function, penalty parameter, and...     

Moving-boundary problems solved by adaptive radial basis functions

The objective of this paper is to present an alternative approach to the conventional level set methods for solving two-dimensional moving-boundary problems known as the passive transport. Moving boundaries are associated with time-dependent problems and the position of the boundaries need to be determined as a function of time and space. The level set method has become an attractive design tool for tracking, modeling and simulating the motion of free boundaries in fluid mechanics, combustion, computer animation and image processing. Recent research on the numerical method has focused on the idea of using a meshless methodology for the numerical solution of partial differential equations. In the present approach, the moving interface is captured by the level set method at all time with the zero contour of a smooth function known as the level set function. A new approach is used to solve a convective transport equation for advancing the level set function in time. This new approach is based on the asymmetric meshless collocation method and the adaptive greedy algorithm for trial subspaces selection. Numerical simulations are performed to verify the accuracy and stability of the new numerical scheme which is then applied to simulate a bubble that is moving, stretching and circulating in an ambient flow to demonstrate the performance of the new meshless approach.     

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.     

全局正定径向基函数在图像分割中的应用

将全局正定径向基函数和图像分割中基于偏微分方程水平集方法的发展方程相结合,提出了一种基于全局正定径向基函数的图像分割算法。用全局正定径向基函数插值发展方程中的水平集函数,得到的插值函数具有较高的精度和光滑性,克服了传统水平集方法中复杂费时的重新初始化过程和水平集对初始轮廓位置敏感等缺点,非线性发展方程最终被转化成常微分方程组并用Euler法求解。实验结果表明该算法不需要重新初始化过程,并且在没有初始轮廓时也能够快速正确地分割图像。     

Multiscale approximation with hierarchical radial basis functions networks

An approximating neural model, called hierarchical radial basis function (HRBF) network, is presented here. This is a self-organizing (by growing) multiscale version of a radial basis function (RBF) network. It is constituted of hierarchical layers, each containing a Gaussian grid at a decreasing scale. The grids are not completely filled, but units are inserted only where the local error is over threshold. This guarantees a uniform residual error and the allocation of more units with smaller scales where the data contain higher frequencies. Only local operations, which do not require any iteration on the data, are required; this allows to construct the network in quasi-real time. Through harmonic analysis, it is demonstrated that, although a HRBF cannot be reduced to a traditional wavelet-based multiresolution analysis (MRA), it does employ Riesz bases and enjoys asymptotic approximation properties for a very large class of functions. HRBF networks have been extensively applied to the reconstruction of three-dimensional (3-13) models from noisy range data. The results illustrate their power in denoising the original data, obtaining an effective multiscale reconstruction of better quality than that obtained by MRA.     

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

A new class of oscillatory radial basis functions

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).     

Development and validation of a dynamic metamodel based on stochastic radial basis functions and uncertainty quantification

A dynamic radial basis function (DRBF) metamodel is derived and validated, based on stochastic RBF and uncertainty quantification (UQ). A metric for assessing metamodel efficiency is developed and used. The validation includes comparisons with a dynamic implementation of Kriging (DKG) and static metamodels for both deterministic test functions (with dimensionality ranging from two to six) and industrial UQ problems with analytical and numerical benchmarks, respectively. DRBF extends standard RBF using stochastic kernel functions defined by an uncertain tuning parameter whose distribution is arbitrary and whose effects on the prediction are determined using UQ methods. Auto-tuning based on curvature, adaptive sampling based on prediction uncertainty, parallel infill, and multiple response criteria are used. Industrial problems are two UQ applications in ship hydrodynamics using high-fidelity computational fluid dynamics for the high-speed Delft catamaran with stochastic operating and environmental conditions: (1) calm water resistance, sinkage and trim with variable Froude number; and (2) mean value and root mean square of resistance and heave and pitch motions with variable regular head wave. The number of high-fidelity evaluations required to achieve prescribed error levels is considered as the efficiency metric, focusing on fitting accuracy and UQ variables. DKG is found more efficient for fitting low-dimensional test functions and one-dimensional UQ, whereas DRBF has a greater efficiency for fitting higher-dimensional test functions and two-dimensional UQ.     

Gaussian process as complement to test functions for surrogate modeling

It is common for papers on surrogate fitting to select test functions for testing algorithms. This raises the issue of how well the algorithms generalize t