Kind Of Basis Functions Research Articles

A New Immersed Finite Element Method for Two-Phase Stokes Problems Having Discontinuous Pressure

Abstract In this paper, we develop a new immersed finite element method (IFEM) for two-phase incompressible Stokes flows. We allow the interface to cut the finite elements. On the noninterface element, the standard Crouzeix–Raviart element and the P 0 {P_{0}} element pair is used. On the interface element, the basis functions developed for scalar interface problems (Kwak et al., An analysis of a broken P 1 {P_{1}} -nonconforming finite element method for interface problems, SIAM J. Numer. Anal. (2010)) are modified in such a way that the coupling between the velocity and pressure variable is different. There are two kinds of basis functions. The first kind of basis satisfies the Laplace–Young condition under the assumption of the continuity of the pressure variable. In the second kind, the velocity is of bubble type and is coupled with the discontinuous pressure, still satisfying the Laplace–Young condition. We remark that in the second kind the pressure variable has two degrees of freedom on each interface element. Therefore, our methods can handle the discontinuous pressure case. Numerical results including the case of the discontinuous pressure variable are provided. We see optimal convergence orders for all examples.

A Fast Galerkin Approach for Solving the Fractional Rayleigh–Stokes Problem via Sixth-Kind Chebyshev Polynomials

Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. Some specific integer and fractional derivative formulas are used to introduce our proposed numerical algorithm. Moreover, the stability and convergence accuracy are derived in detail. As a final validation of our theoretical results, we present a few numerical examples.

Orthonormal piecewise Bernoulli functions: Application for optimal control problems generated using fractional integro-differential equations

In this study, the orthonormal piecewise Bernoulli functions are generated as a new kind of basis functions. An explicit matrix related to fractional integration of these functions is obtained. An efficient direct method is developed to solve a novel set of optimal control problems defined using a fractional integro-differential equation. The presented technique is based on the expressed basis functions and their fractional integral matrix together with the Gauss–Legendre integration method and the Lagrange multipliers algorithm. This approach converts the original problem into a mathematical programming one. Three examples are investigated numerically to verify the capability and reliability of the approach.

Dual Empirical Orthogonal Functions Restrained Tomographic Model for Ionosphere Imaging

Extra constraints are always brought into an ionosphere tomographic model to strengthen its robustness. However, existent models using the constraints either show a weak restriction, amplifying the model’s uncertainties, or demonstrate a strong restriction, decreasing the model’s flexibility. Empirical Orthogonal Function (EOF), as a kind of basis function and generally being extracted from a prior dataset, is an ideal constraint for the tomographic model. By using EOFs as the vertical and slant constraints of the tomographic model, a dual-EOFs restrained tomographic model (DERT) is proposed. In DERT, a smooth constraint, conducted by a moving window process, is firstly imposed, and then a vertical constraint and a slant constraint, which are carried out by fitting the estimated vertical and slant profiles with their dominant EOFs via a least square method, are appended. To reduce the impacts of the errors of the slant total electron content (STEC) measurements on the processes of moving window and least square fit, an algorithm that can reject abnormal STEC values is employed before the two processes. Validations Tests are conducted to validate the DERT and compare it with the Slepian method and the IRI-2016 model. Results show that DERT outperforms the reference methods in aspects of vertical profiles, F2 layer peak density (foF2), STEC measurement and vertical total electron content (VTEC). The root mean square errors (RMSE) improvements over the Slepian method are 14.17%, 0.47%, 16.24% and 40.46% (IGS map as the truth), respectively.

Numerical dispersion and dissipation of the triangle-based discontinuous Galerkin method for acoustic and elastic velocity-stress equations

In this work, we present the numerical dispersion and dissipation analyses of the triangle-based discontinuous Galerkin method (DGM) for acoustic and elastic velocity-stress equations. The analysis is based on the eigenvalue problem. The applicability of the flux-based DGM is affected by many factors, including numerical flux, basis function, mesh element, and time-stepping scheme. Based on the semi-discrete analysis, we compare different numerical fluxes—Centred flux, local Lax-Friedrichs (LLF) flux, and Godunov flux, different basis functions—Koornwinder-Dubiner (KD) polynomial, Legendre polynomial, and simple monomial function, and four types of triangular meshes. The fully discrete schemes include the third-order total variation diminishing Runge-Kutta (TVD RK) method and the third-order weighted Runge–Kutta (WRK) method. The results indicate that numerical dispersion and dissipation behave differently for different numerical fluxes and mesh types, which verifies the importance of considering numerical fluxes and mesh types. We find that using LLF and Godunov fluxes, and the mesh pattern of case 3 has attractive advantages for numerical simulation. The result of the dispersion-dissipation analysis with different basis functions demonstrates that for high-order integral accuracy, three kinds of basis functions have the same performance in suppressing numerical dispersion and dissipation in triangular elements. The fully discrete dispersion–dissipation analysis shows that the time-stepping scheme introduces numerical dispersion and dissipation and the fully discrete case with a small sampling ratio and Courant number has a better effect in suppressing numerical dispersion and dissipation. Finally, we provide several numerical experiments to validate our theoretical finding, and numerical results are consistent with theoretical dispersion-dissipation analyses.

Tau and Galerkin operational matrices of derivatives for treating singular and Emden–Fowler third-order-type equations

In this paper, our target is to implement and analyze numerical algorithms for the numerical solutions of initial and boundary third-order singular-type equations, and in particular the Emden–Fowler-type equations. Both linear and nonlinear equations are investigated. Two kinds of basis functions in terms of the shifted Chebyshev polynomials of the second kind are employed. Regarding the initial value problems, a new Galerkin operational matrix of derivatives is established and utilized, while in the case of boundary value problems (BVPs), the operational matrix of derivatives of the shifted Chebyshev polynomials of the second kind is employed. Convergence and error estimates of the two proposed expansions are given. The obtained solutions are spectral and they are consequences of the application of tau and collocation methods. These methods serve to convert the problems governed by their initial/boundary conditions into systems of linear or nonlinear algebraic equations which can be solved via suitable numerical solvers. Finally, we support our theoretical study by presenting several examples to ensure the accuracy and efficiency of the proposed algorithms.

Linear Fredholm Integro-Differential-Difference Equations and Their Effective Computation

In this work, a new computational approach is proposed to solve the linear Fredholm integro-differential-difference equations with variable coefficients under the mixed conditions. This method yields the solution of the governing equation with the form of a linear combination of a special kind of basis functions. Those unknown coefficients of the solution of the given problem are determined by solving a linear system of algebraic equations which has been derived with the help of the least square approximation method and the Lagrange-multiplier method. Moreover, an error estimation of the approximate solution is constructed by using the residual error function technique. The convergence of the approximate solution is proved. Several numerical examples are given to demonstrate the accuracy and efficiency. Comparisons are made between the proposed method and other existing approaches.

The use of the edge basis function for non-conformal solution of electric current volume integral equation

A novel kind of basis function which is defined only in a single tetrahedron element and is along the edge of a tetrahedron element is derived in this paper. Then, it is used for the discretization of the electric current volume integral equation. Compared with the traditional used Schaubert–Wilton–Glisson (SWG) basis function, the proposed one permits the use of non-conformal meshes. Details for the calculation of impedance matrix elements are developed. The high order singularity of the Green’s function caused by the gradient-divergence operator has been decreased to the order of 1/R. Therefore, it is much easier to be implemented than other non-conformal solution schemes presented in literatures. To validate the proposed scheme, numerical results for electromagnetic scattering from several inhomogeneous dielectric objects are presented. It is shown that the proposed scheme gives accurate results for electromagnetic scattering analysis from dielectric bodies. Particularly, the use of the non-conformal meshes greatly improves the solution efficiency of volume integral equation for high-contrast permittivity or multiscale cases.

Adaptive neural dynamic global PID sliding mode control for MEMS gyroscope

In this paper, a dynamic global proportional integral derivative (PID) sliding mode controller based on an adaptive radial basis function (RBF) neural estimator is developed to guarantee the stability and robustness in the presence of a lumped uncertainty for a micro electromechanical systems (MEMS) gyroscope. This approach gives a new dynamic global PID sliding mode manifold, which not only enables system trajectory to run on the global sliding mode surface at the start point more quickly and eliminates the reaching phase of the conventional sliding mode control, but also restrains the steady-state error and reduces the chattering via a dynamic PID sliding surface. A RBF neural network (NN) system is employed to estimate the lumped uncertainty and eliminate the chattering phenomenon at the same time. Additionally, adaptive laws and dynamic global PID sliding control gains that ensure the asymptotic stability of the close-loop system are proposed, together with the techniques for deciding which kind of basis function should be selected. Finally, simulation results demonstrate the effectiveness of RBFNN dynamic global PID sliding mode control method, meanwhile some comparisons are made to verify the good properties of the suggested control approach.

Constructing diabatic representations using adiabatic and approximate diabatic data--Coping with diabolical singularities.

We have recently introduced a diabatization scheme, which simultaneously fits and diabatizes adiabatic ab initio electronic wave functions, Zhu and Yarkony J. Chem. Phys. 140, 024112 (2014). The algorithm uses derivative couplings in the defining equations for the diabatic Hamiltonian, H(d), and fits all its matrix elements simultaneously to adiabatic state data. This procedure ultimately provides an accurate, quantifiably diabatic, representation of the adiabatic electronic structure data. However, optimizing the large number of nonlinear parameters in the basis functions and adjusting the number and kind of basis functions from which the fit is built, which provide the essential flexibility, has proved challenging. In this work, we introduce a procedure that combines adiabatic state and diabatic state data to efficiently optimize the nonlinear parameters and basis function expansion. Further, we consider using direct properties based diabatizations to initialize the fitting procedure. To address this issue, we introduce a systematic method for eliminating the debilitating (diabolical) singularities in the defining equations of properties based diabatizations. We exploit the observation that if approximate diabatic data are available, the commonly used approach of fitting each matrix element of H(d) individually provides a starting point (seed) from which convergence of the full H(d) construction algorithm is rapid. The optimization of nonlinear parameters and basis functions and the elimination of debilitating singularities are, respectively, illustrated using the 1,2,3,4(1)A states of phenol and the 1,2(1)A states of NH3, states which are coupled by conical intersections.

The use of the constant vector basis functions for the magnetic field integral equation

The magnetic field integral equation (MFIE) is widely used in the analysis of electromagnetic scattering problems for conducting objects. Usually, the MFIE is solved by the method of moments (MoM) using the Rao–Wilton–Glisson (RWG) basis functions. In this paper, a new kind of basis function which is named the piece-wise constant vector basis function is proposed and used to solve the MFIE by MoM. Definition of this kind of basis function is given. The calculation of the impedance matrix entries is presented in detail. This kind of basis function is then used for the solution of the MFIE for electromagnetic scattering problems. The radar cross section (RCS) results and the iterative property of both kinds of basis functions are presented. It is shown that the piece-wise constant vector basis functions give similar RCS results as those of the RWG basis functions. Particularly, when iterative solver is used to solve the resultant linear system, the solution scheme using the piece-wise constant vector basis functions iterates much faster than that using the RWG basis functions.

Change of image quality by kinds of basis functions in PET dynamic image of four-dimensional reconstruction

Change of image quality by kinds of basis functions in PET dynamic image of four-dimensional reconstruction

Compressive sensing image-fusion algorithm in wireless sensor networks based on blended basis functions

Compressive sensing (CS) has given us a new idea at data acquisition and signal processing. It has proposed some novel solutions in many practical applications. Focusing on the pixel-level multi-source image-fusion problem in wireless sensor networks, the paper proposes an algorithm of CS image fusion based on multi-resolution analysis. We present the method to decompose the images by nonsubsampled contourlet transform (NSCT) basis function and wavelet basis function successively and fuse the images in compressive domain. It means that the images can be sparsely represented by more than one basis function. We named this process as blended basis functions representation. Since the NSCT and wavelet basis functions have complementary advantages in multi-resolution image analysis, and the signals are sparser after being decomposed by two kinds of basis functions, the proposed algorithm has perceived advantages in comparison with CS image fusion in wavelet domain which is widely reported by literatures. The simulations show that our method provides promising results.

Approximate dynamic programming for stochastic N-stage optimization with application to optimal consumption under uncertainty

Stochastic optimization problems with an objective function that is additive over a finite number of stages are addressed. Although Dynamic Programming allows one to formally solve such problems, closed-form solutions can be derived only in particular cases. The search for suboptimal solutions via two approaches is addressed: approximation of the value functions and approximation of the optimal decision policies. The approximations take on the form of linear combinations of basis functions containing adjustable parameters to be optimized together with the coefficients of the combinations. Two kinds of basis functions are considered: Gaussians with varying centers and widths and sigmoids with varying weights and biases. The accuracies of such suboptimal solutions are investigated via estimates of the error propagation through the stages. Upper bounds are derived on the differences between the optimal value of the objective functional and its suboptimal values corresponding to the use at each stage of approximate value functions and approximate policies. Conditions under which the number of basis functions required for a desired approximation accuracy does not grow “too fast” with respect to the dimensions of the state and random vectors are provided. As an example of application, a multidimensional problem of optimal consumption under uncertainty is investigated, where consumers aim at maximizing a social utility function. Numerical simulations are provided, emphasizing computational pros and cons of the two approaches (i.e., value-function approximation and optimal-policy approximation) using the above-mentioned two kinds of basis functions. To investigate the dependencies of the performances on dimensionality, the numerical analysis is performed for various numbers of consumers. In the simulations, discretization techniques exploiting low-discrepancy sequences are used. Both theoretical and numerical results give insights into the possibility of coping with the curse of dimensionality in stochastic optimization problems whose decision strategies depend on large numbers of variables.

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein–Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.

Spectral-domain analysis of multilayer cylindrical–rectangular microstrip antennas

In this paper, the resonance and radiation characteristics of multilayered cylindrical–rectangular microstrip antennas are analysed. The problem is formulated, in the spectral-domain, using an electric field integral equation and the spectral-domain Green's function. An efficient compact method is developed to obtain the Green's function of the electric field due to the current distribution of a patch located in an arbitrary layer of a cylindrical stratified medium. To accomplish this, efficient matrix representations are used to describe the electric and magnetic fields in each layer of the geometry in terms of ABCD matrices characteristic of its material properties. The boundary integral equation for the unknown patch current is solved numerically by applying the boundary element method using three kinds of basis functions. Taking into account the symmetry properties of the elements of the moment method matrix, the CPU time is reduced significantly. The convergence of the method is proven by performing the resonant frequencies for a single layer cylindrical–rectangular microstrip patch vs. the substrate thickness. The computed data are found to be in very good agreement with those found in the literature, using only two basis functions. Once the validity of the method is checked, further results for various antennas structures are presented proving the generality of the method. Finally, the effect of an air gap between the substrate layer and the ground conducting cylinder on far zone radiation field is investigated using the steepest-descent method.

Optimized Approximation Algorithm in Neural Networks Without Overfitting

In this paper, an optimized approximation algorithm (OAA) is proposed to address the overfitting problem in function approximation using neural networks (NNs). The optimized approximation algorithm avoids overfitting by means of a novel and effective stopping criterion based on the estimation of the signal-to-noise-ratio figure (SNRF). Using SNRF, which checks the goodness-of-fit in the approximation, overfitting can be automatically detected from the training error only without use of a separate validation set. The algorithm has been applied to problems of optimizing the number of hidden neurons in a multilayer perceptron (MLP) and optimizing the number of learning epochs in MLP's backpropagation training using both synthetic and benchmark data sets. The OAA algorithm can also be utilized in the optimization of other parameters of NNs. In addition, it can be applied to the problem of function approximation using any kind of basis functions, or to the problem of learning model selection when overfitting needs to be considered.

Full-wave analysis of single cylindrical striplines and microstriplines with multilayer dielectrics

In this paper, the spectral-domain method is used to calculate the propagation characteristics of cylindrical microstrip transmission lines. The problem is formulated using an electric field integral equation and the spectral-domain Green's function. The solutions of the field components are obtained in matrix forms, which facilitate the calculations of the Green's function and the power flowing over the lines. The Green's functions are obtained in terms of transition matrices over the dielectric layers. The obtained integral equation is solved by moment method using four kinds of basis functions. The convergence of the method is proven. Based on the power–current definition, a stationary expression for the characteristic impedance has been derived analytically. Numerical results of the effective dielectric constant and the characteristic impedance for various line parameters are calculated and analysed. The computed data are found to be in good agreement with results obtained using other methods. The formulation is then applied to covered microstripline, microstripline and stripline with air gaps, for which data are not found in the literature to date. The presented method is used to guide design of microstrip coil for magnetic resonance imaging. This method is also suitable for investigation of multiconductor strip lines. Copyright © 2006 John Wiley & Sons, Ltd.

A simple and efficient method of analyzing mechanical behaviors of multi-layered orthotropic plates in rectangular shape

We apply the Rayleigh-Ritz method to analyze multi-layered plates with residual stresses for membrane and bending deformation. The method is very simple, straightforward, and easily programmable, but it should be applied to structures only with simple shapes. We derive coupled variational equations using the principle of virtual displacement, and investigate what kind of basis functions are desirable for the analysis of rectangular plates with various boundary conditions. We demonstrate the effectiveness and usefulness of the present method through several examples. The analysis results obtained with the present method are in good agreement with those available in the literature. A multi-layered microelectromechanical plate example shows that the coupling effect between membrane and bending deformation should not be ignored and that residual stresses do significantly influence the stiffness of the structure.

A Multiscale Moment Method for Solving Fredholm Integral Equation of the First Kind - Summary

The aim of this paper is to deal with the problem of numerical solution of Fredholm integral equation of the first kind. First, a new kind of basis functions has been proposed based on multiscale method and wavelet-like basis. Second, by using this basis, the multiscale moment method for solving Fredholm integral equation of the first kind in one dimension has been proposed. Furthermore, the adaptive algorithm of the multiscale moment method has been presented according to the characteristics of the integral equation. Many numerical simulations are carried out to test the feasibility of the multiscale method and the implemented adaptive algorithm. It is found that the two multiscaling algorithms can provide good results. The adaptive algorithm can reduce the orders of linear equations constructed by multiscale moment method for Fredholm integral equation. It is a very effective algorithm to solve integral equations.