Multiquadric Function Research Articles

Numerical study of stream-function formulation governing flows in multiply-connected domains by integrated RBFs and Cartesian grids

This paper describes a new numerical procedure, based on point collocation, integrated multiquadric functions and Cartesian grids, for the discretisation of the stream-function formulation for flows of a Newtonian fluid in multiply-connected domains. Three particular issues, namely (i) the derivation of the stream-function values on separate boundaries, (ii) the implementation of cross derivatives in irregular regions, and (iii) the treatment of double boundary conditions, are studied in the context of Cartesian grids and approximants based on integrated multiquadric functions in one dimension. Several test problems, i.e. steady flows between a rotating circular cylinder and a fixed square cylinder and also between eccentric cylinders maintained at different temperatures, are investigated. Results obtained are compared well with numerical data available in the literature.

Improved geologic surface approximation using a multiquadric method with additional constraints

Geologic surface approximation is profoundly affected by the presence, density and location of scattered geologic input data. Many studies have recognized the importance of utilizing varied sources of information when reconstructing a surface. This paper presents an improved geologic surface approximation method using a multiquadric function and borehole data. Additional information, i.e., inequality elevation and dip-strikes data extracted from outcrops or mining faces, is introduced in the form of physical constraints that control local changes in the estimated surface. Commonly accepted hypothesis states that geologic surfaces can be approximated to any desired degree of exactness by the summation of regular, mathematically defined, surfaces: in particular displaced quadric forms. The coefficients of the multiquadric functions are traditionally found by a least squares method. The addition of physical constraints in this work makes such an approach into a non-deterministic polynomial time problem. Hence we propose an objective function that represents the quality of the estimated surface and that includes the additional constraints by incorporation of a penalty function. Maximizing the smoothness of the estimated surface and its fitness to the additional constraints then allows the coefficients of the multiquadric function to be obtained by iterative methods. This method was implemented and demonstrated using data collected from the 81'st coal mining area of the Huaibei Coal Group.

Numerical solution of PDEs via integrated radial basis function networks with adaptive training algorithm

This paper develops a mesh-free numerical method for solving PDEs, based on integrated radial basis function networks (IRBFNs) with adaptive residual subsampling training scheme. The multiquadratic function is chosen as the transfer function of the neurons. The nonlinear algebraic equation systems for weights training are solved by Levenberg–Marquardt algorithm. The performance of the proposed method is demonstrated in numerical examples by approximating several functions and solving nonlinear PDEs. The result of numerical experiments shows that the IRBFNs with the adaptive procedure requires less neurons to attain the desired accuracy than conventional radial basis function networks.

Multiquadric Finite Difference (MQ-FD) Method and Its Application

The conventional finite difference (FD) schemes are based on the low or- der polynomial approximation in a local region. This paper shows that when the polynomial approximation is replaced by the multiquadric (MQ) function approx- imation in the same region, a new FD method, which is termed as MQ-FD method in this work, can be developed. The paper gives analytical formulas of the MQ-FD method and carries out a performance study for its derivative approximation and solution of Poisson equation and the incompressible Navier-Stokes equations. In addition, the effect of the shape parameter c in MQ on the formulas of the MQ- FD method is analyzed. Derivative approximation in one-dimensional space and Poisson equation in two-dimensional space are taken as model problems to study the accuracy of the MQ-FD method. Furthermore, a lid-driven flow problem in a square cavity is simulated by the MQ-FD method. The obtained results indicate that this method may solve the engineering problem very accurately with a proper choice of the shape parameter c.

A shape-preserving quasi-interpolation operator satisfying quadratic polynomial reproduction property to scattered data

In this paper, we construct a univariate quasi-interpolation operator to non-uniformly distributed data by cubic multiquadric functions. This operator is practical, as it does not require derivatives of the being approximated function at endpoints. Furthermore, it possesses univariate quadratic polynomial reproduction property, strict convexity-preserving and shape-preserving of order 3 properties, and a higher convergence rate. Finally, some numerical experiments are shown to compare the approximation capacity of our quasi-interpolation operator with that of Wu and Schaback’s quasi-interpolation scheme.

Robust Design of Electromagnetic Systems Based on Interval Taylor Extension Applied to a Multiquadric Performance Function

A worst case robust design approach based on a suitable application of interval arithmetic (IA) is proposed. The use of IA leads to a definition of the interval extension of the Taylor series named interval Taylor extension (ITE). The width of the ITE (WITE) has been shown to be a valuable robustness index in the case of polynomial performance functions (PFs). In this paper, the method is extended to nonpolynomial PFs, as those ones obtained by multiquadric interpolation in computer assisted electromagnetic design (by FEM, BEM, etc.). In particular, it is shown that the ITE of a multiquadric function furnishes a rigorous overbounding of the PF and the WITE gives an overestimation of the PF range in presence of an assigned uncertainty of the design parameters, both in an analytical monodimensional case study and in the benchmark TEAM 22 optimization problem. Moreover, the most robust stationary solution can be achieved by looking at the minimum of WITE.

Fast Radial Basis Function Interpolation via Preconditioned Krylov Iteration

We consider a preconditioned Krylov subspace iterative algorithm presented by Faul, Goodsell, and Powell (IMA J. Numer. Anal. 25 (2005), pp. 1-24) for computing the coefficients of a radial basis function interpolant over $N$ data points. This preconditioned Krylov iteration has been demonstrated to be extremely robust to the distribution of the points and the iteration rapidly convergent. However, the iterative method has several steps whose computational and memory costs scale as $O(N^{2}),$ both in preliminary computations that compute the preconditioner and in the matrix-vector product involved in each step of the iteration. We effectively accelerate the iterative method to achieve an overall cost of $O(N\log N).$ The matrix vector product is accelerated via the use of the fast multipole method. The preconditioner requires the computation of a set of closest points to each point. We develop an $O(N\log N)$ algorithm for this step as well. Results are presented for multiquadric interpolation in $\mathbb{R}^{2}$ and biharmonic interpolation in $\mathbb{R}^{3}$. A novel FMM algorithm for the evaluation of sums involving multiquadric functions in $\mathbb{R}^{2}$ is presented as well.

A new meshless method based on MLPG for elastic dynamic problems

The basic concept and numerical implementation of a new local Petrov-Galerkin method for solving a dynamic problem are presented in this paper. It uses a radial basis function (RBF) coupled with a polynomial basis function as a trial function, and uses the Heaviside function as a test function of the weighted residual method. The shape function has the Kronecker Delta properties for the trial-function-interpolation, and so no additional treatment to impose essential boundary conditions. The method does not involve any domain and singular integrals to generate the global effective stiffness matrix except for the mass and damping matrice; it only involves a regular boundary integral. It possesses a great flexibility in dealing with the numerical model of the elastic dynamic problem under various boundary conditions with arbitrary shapes. The Newmark family of methods is adopted in computation. The numerical results also show that using a multiquadrics (MQ) function with the polynomial basis function as the interpolation function can give quite accurate numerical results. The a Q and a S are investigated which are parameters of the radii of the sub-domain and influence domain, respectively.

A comparative study of metamodeling methods for multiobjective crashworthiness optimization

The response surface methodology (RSM), which typically uses quadratic polynomials, is predominantly used for metamodeling in crashworthiness optimization because of the high computational cost of vehicle crash simulations. Research shows, however, that RSM may not be suitable for modeling highly nonlinear responses that can often be found in impact related problems, especially when using limited quantity of response samples. The radial basis functions (RBF) have been shown to be promising for highly nonlinear problems, but no application to crashworthiness problems has been found in the literature. In this study, metamodels by RSM and RBF are used for multiobjective optimization of a vehicle body in frontal collision, with validations by finite element simulations using the full-scale vehicle model. The results show that RSM is able to produce good approximation models for energy absorption, and the model appropriateness can be well predicted by ANOVA. However, in the case of peak acceleration, RBF is found to generate better models than RSM based on the same number of response samples, with the multiquadric function identified to be the most stable RBF. Although RBF models are computationally more expensive, the optimization results of RBF models are found to be more accurate.

The optimisation of induction heating system based on multiquadric function approximation

PurposeThe investigation of the efficiency of optimisation technique based on approximation of objective function by multiquadric (MQ) function, used for induction heating devices was the aim of the paper.Design/methodology/approachThe optimisation package based on Matlab language and using Flux2D commercial program for calculation of electromagnetic and thermal fields was built. It allows the use of different optimisation techniques for induction heating devices, e.g. based on MQ function approximation. In the paper two algorithms of approximated points generating have been tested.FindingsThe efficiency of MQ optimisation method strongly depends on the applied algorithm of approximated point generating. To ensure high efficiency of MQ optimisation method, the stochastic element of the algorithm of approximated point generating should have a significant role.Research limitations/implicationsThe efficiency of elaborated algorithms of MQ function approximated point generating should be proved in other applications.Practical implicationsThe efficient optimisation technique of induction heating devices has been proposed.Originality/valueThe two new algorithms for generation of MQ function approximated points have been proposed. The paper could be useful for designers of induction heating devices.

A Krylov subspace algorithm for multiquadric interpolation in many dimensions

We consider the version of multiquadric interpolation where the interpolation conditions are the equations s(x i ) = f i , i = 1, 2,..., n, and where the interpolant has the form s(x) = Σ n j=1 λ j (∥x - x j ∥ 2 + c 2 ) 1/2 + α, x ∈ R d , subject to the constraint Σ n j=1 λ j = 0. The points x i E R d , the right-hand sides f i , i = 1, 2,..., n, and the constant c are data. The equations and the constraint define the parameters λ j , j = 1, 2,..., n, and a. The resultant approximation s f is useful in many applications, but the calculation of the parameters by direct methods requires O(n 3 ) operations, and n may be large. Therefore iterative procedures for this calculation have been studied at Cambridge since 1993, the main task of each iteration being the computation of s(x i ), i = 1, 2,..., n, for trial values of the required parameters. These procedures are based on approximations to Lagrange functions, and often they perform very well. For example, ten iterations usually provide enough accuracy in the case d = 2 and c = 0, for general positions of the data points, but the efficiency deteriorates if d and c are increased. Convergence can be guaranteed by the inclusion of a Krylov subspace technique that employs the native semi-norm of multiquadric functions. An algorithm of this kind is specified, its convergence is proved, and careful attention is given to the choice of the operator that defines the Krylov subspace, which is analogous to pre-conditioning in the conjugate gradient method. Finally, some numerical results are presented and discussed, for values of d and n from the intervals [2, 40] and [200, 100001, respectively.

Investigation of the use of radial basis functions in local collocation method for solving diffusion problems

Multidimensional diffusion problems can be solved by a local collocation method using radial basis functions. This method is computationally efficient because it is meshless and yields a sparse system of algebraic equations. Three types of radial basis functions—the generalized multiquadric function, the Gaussian function, and the generalized thin-plate spline function—are considered as interpolation functions in this method. Analysis of truncation error indicates that the use of the generalized multiquadric function or the Gaussian function produces satisfactory results, but the generalized thin-plate spline function is not a good interpolation function.

A Novel Approach to the Design of the Class of Triplet Halfband Filterbanks

A new approach is presented for designing the recently introduced class of triplet halfband filterbank which are defined by three kernels. The Parametric Bernstein Polynomial is used to construct the kernels. The filterbanks have the advantage of structural perfect reconstruction and structural regularity. The design of the free parameters of the Bernstein Polynomial is achieved through a least squares method. A novel iterative procedure is employed to optimize the objective function which is a multiquadratic function of the free parameters. The design technique is flexible in that it allows filters with different characteristics to be designed with ease. Filter regularity can be traded for increased sharpness in the frequency response and regular scaling function and wavelets can be readily obtained.

Indirect RBFN method with thin plate splines for numerical solution of differential equations

This paper reports a mesh-free Indirect Radial Basis Function Network method (IRBFN) using Thin Plate Splines (TPSs) for numerical solution of Differential Equations (DEs) in rectangular and curvilinear coordinates. The adjustable parameters required by the method are the number of centres, their positions and possibly the order of the TPS. The first and second order TPSs which are widely applied in numerical schemes for numerical solution of DEs are employed in this study. The advantage of the TPS over the multiquadric basis function is that the former, with a given order, does not contain the adjustable shape parameter (i.e. the RBF's width) and hence TPS-based RBFN methods require less parametric study. The direct TPS-RBFN method is also considered in some cases for the purpose of comparison with the indirect TPS-RBFN method. The TPS-IRBFN method is verified successfully with a series of problems including linear elliptic PDEs, nonlinear elliptic PDEs, parabolic PDEs and Navier-Stokes equations in rectangular and curvilinear coordinates. Numerical results obtained show that the method achieves the norm of the relative error of the solution of O(10 -6) for the case of 1D second order DEs using a density of 51, of O(10 -7) for the case of 2D elliptic PDEs using a density of 20 × 20 and a Reynolds number Re = 200 for the case of Jeffery-Hamel flow with a density of 43 × 12.

Prediction of log kw of disubstituted benzene derivatives in reversed-phase high-performance liquid chromatography using multiple linear regression and radial basis function neural network

A study of the relationships between the extrapolated capacity factor (log k w) of a group of 54 disubstituted benzene derivatives and a set of eight molecular descriptors was made. By using multiple linear regression (MLR), we obtained an empirical function, which included five descriptors. The performance of a radial basis function neural network (RBFNN) was evaluated. The network used thin plate spline and multi-quadratic functions, which showed better than MLR. Semi-empirical quantum chemical method PM3 implemented in HyperChem 4.0 was employed to calculate the molecular descriptors of the compounds. The results gave a relative minor root mean squared (rms) error (0.070 and 0.084) and indicated that the quantitative structure–retention relationships (QSRR) models proposed were very satisfactory.

Approximation by radial bases and neural networks

In this paper, we study approximation by radial basis functions including Gaussian, multiquadric, and thin plate spline functions, and derive order of approximation under certain conditions. Moreover, neural networks are also constructed by wavelet recovery formula and wavelet frames.

Approximation of the objective function: multiquadrics versus neural networks

Global optimization in electrical engineering using stochastic methods requires usually a large amount of CPU time to locate the optimum, if the objective function is calculated either with the finite element method (FEM) or the boundary element method (BEM). One approach to reduce the number of FEM or BEM calls using neural networks and another one using multiquadric functions have been introduced recently. This paper compares the efficiency of both methods, which are applied to a couple of test problems and the results are discussed.

Complexity reduction in radial basis function (RBF) networks by using radial B-spline functions

In this paper, new basis consisting of radial cubic and quadratic B-spline functions are introduced together with the CORDIC algorithm, within the context of RBF networks as a means of reducing computational complexity in real-time signal-processing applications. The new basis are compared with two other existing and popularly used basis families, namely the Gaussian functions and the inverse multiquadratic functions (IVMQ) in terms of approximation performance and computational requirements. The new basis are shown to achieve approximation performance very similar to the Gaussian basis functions and are better than the IVMQ functions with less computational load and without any need for approximation methods such as table-lookup.

Multiquadric Prewavelets on Nonequally Spaced Knots in One Dimension

In this paper, we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers (or "knots"). Although the multiquadric function and its relations are our prime examples, the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well.

Multiquadric prewavelets on nonequally spaced knots in one dimension

In this paper, we identify univariate prewavelets on spaces spanned by translates of multiquadric functions and other radial basis functions with nonequally spaced centers (or "knots"). Although the multiquadric function and its relations are our prime examples, the theory is sufficiently broad to admit prewavelets from other radial basis function spaces as well.