Reproducing Kernel Hilbert Space Method Research Articles

Group preserving scheme and reproducing kernel method for the Poisson–Boltzmann equation for semiconductor devices

This paper introduces that the nonlinear Poisson–Boltzmann equation for semiconductor devices describing potential distribution in a double-gate metal oxide semiconductor field effect transistor (DG-MOSFET) is exactly solvable. The DG-MOSFET shows one of the most advanced device structures in semiconductor technology and is a primary focus of modeling efforts in the semiconductor industry. Lie symmetry properties of this model is investigated in order to extract some exact solutions. The reproducing kernel Hilbert space method and group preserving scheme also have been applied to the nonlinear equation. Numerical results show that the present methods are very effective.

On solutions of fractional Riccati differential equations

We apply an iterative reproducing kernel Hilbert space method to get the solutions of fractional Riccati differential equations. The analysis implemented in this work forms a crucial step in the process of development of fractional calculus. The fractional derivative is described in the Caputo sense. Outcomes are demonstrated graphically and in tabulated forms to see the power of the method. Numerical experiments are illustrated to prove the ability of the method. Numerical results are compared with some existing methods.

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm.

Numerical Computation of Fractional Second-Order Sturm-Liouville Problems

In this article, we investigate the eigenvalues of nonsingular fractional second-order Sturm-Liouville problem. The fractional derivative in this paper is in the conformable fractional derivative sense. We implement the reproducing kernel Hilbert space method to approximate the eigenvalues. Convergence of the proposed method is discussed. The main properties of the Sturm-Liouville problem are investigated. Numerical results demonstrate the accuracy of the present algorithm. Comparisons with other methods are presented.

Kernel Mean Embedding of Distributions: A Review and Beyond

A Hilbert space embedding of a distribution---in short, a kernel mean embedding---has recently emerged as a powerful tool for machine learning and inference. The basic idea behind this framework is to map distributions into a reproducing kernel Hilbert space (RKHS) in which the whole arsenal of kernel methods can be extended to probability measures. It can be viewed as a generalization of the original "feature map" common to support vector machines (SVMs) and other kernel methods. While initially closely associated with the latter, it has meanwhile found application in fields ranging from kernel machines and probabilistic modeling to statistical inference, causal discovery, and deep learning. The goal of this survey is to give a comprehensive review of existing work and recent advances in this research area, and to discuss the most challenging issues and open problems that could lead to new research directions. The survey begins with a brief introduction to the RKHS and positive definite kernels which forms the backbone of this survey, followed by a thorough discussion of the Hilbert space embedding of marginal distributions, theoretical guarantees, and a review of its applications. The embedding of distributions enables us to apply RKHS methods to probability measures which prompts a wide range of applications such as kernel two-sample testing, independent testing, and learning on distributional data. Next, we discuss the Hilbert space embedding for conditional distributions, give theoretical insights, and review some applications. The conditional mean embedding enables us to perform sum, product, and Bayes' rules---which are ubiquitous in graphical model, probabilistic inference, and reinforcement learning---in a non-parametric way. We then discuss relationships between this framework and other related areas. Lastly, we give some suggestions on future research directions.

Fitted reproducing kernel Hilbert space method for the solutions of some certain classes of time-fractional partial differential equations subject to initial and Neumann boundary conditions

Latterly, many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, to a series of time-fractional partial differential equations. Unlike the normal case derivative, the differential order in such equations is with a fractional order, which will lead to new challenges for numerical simulation. The purpose of this analysis is to introduce the reproducing kernel Hilbert space method for treating classes of time-fractional partial differential equations subject to Neumann boundary conditions with parameters derivative arising in fluid-mechanics, chemical reactions, elasticity, anomalous diffusion, and population growth models. The method provides appropriate representation of the solutions in convergent series formula with accurately computable components. Numerical experiments with different order derivatives degree are performed to support the theoretical analyses which are acquired by interrupting the n-term of the exact solutions. Finally, the obtained outcomes showed that the proposed method is competitive in terms of the quality of the solutions found and is very valid for solving such time-fractional Neumann problems.

Reproducing kernel method for numerical simulation of downhole temperature distribution

This paper research downhole temperature distribution in oil production and water injection using reproducing kernel Hilbert space method (RKHSM) for the first time. The aim of this paper is that using the highly accurate RKHSM can solve downhole temperature problems effectively. According to 2-D mathematical models of downhole temperature distribution, the analytical solution was given in a series expansion form and the approximate solution was expressed by n-term summation of reproducing kernel functions which initial and boundary conditions were selected properly. Numerical results of downhole temperature distribution with multiple pay zones, in which different radial distance and different injection–production conditions (such as injection rate, injection temperature, injection time, oil layer thickness), were carried out by mathematical 7.0, and numerical results correspond to general knowledge and show that use RKHSM to research downhole temperature distribution is feasible and effective.

Genome-wide prediction using Bayesian additive regression trees

BackgroundThe goal of genome-wide prediction (GWP) is to predict phenotypes based on marker genotypes, often obtained through single nucleotide polymorphism (SNP) chips. The major problem with GWP is high-dimensional data from many thousands of SNPs scored on several thousands of individuals. A large number of methods have been developed for GWP, which are mostly parametric methods that assume statistical linearity and only additive genetic effects. The Bayesian additive regression trees (BART) method was recently proposed and is based on the sum of nonparametric regression trees with the priors being used to regularize the parameters. Each regression tree is based on a recursive binary partitioning of the predictor space that approximates an unknown function, which will automatically model nonlinearities within SNPs (dominance) and interactions between SNPs (epistasis). In this study, we introduced BART and compared its predictive performance with that of the LASSO, Bayesian LASSO (BLASSO), genomic best linear unbiased prediction (GBLUP), reproducing kernel Hilbert space (RKHS) regression and random forest (RF) methods.ResultsTests on the QTLMAS2010 simulated data, which are mainly based on additive genetic effects, show that cross-validated optimization of BART provides a smaller prediction error than the RF, BLASSO, GBLUP and RKHS methods, and is almost as accurate as the LASSO method. If dominance and epistasis effects are added to the QTLMAS2010 data, the accuracy of BART relative to the other methods was increased. We also showed that BART can produce importance measures on the SNPs through variable inclusion proportions. In evaluations using real data on pigs, the prediction error was smaller with BART than with the other methods.ConclusionsBART was shown to be an accurate method for GWP, in which the regression trees guarantee a very sparse representation of additive and complex non-additive genetic effects. Moreover, the Markov chain Monte Carlo algorithm with Bayesian back-fitting provides a computationally efficient procedure that is suitable for high-dimensional genomic data.Electronic supplementary materialThe online version of this article (doi:10.1186/s12711-016-0219-8) contains supplementary material, which is available to authorized users.

Analytical Investigation of Jeffery–Hamel Flow by Modified Adomian Decomposition Method

Abstract The objective of this paper is to present a reliable approach to compute an approximate analytical solution of magneto-hydrodynamic (MHD) Jeffery–Hamel flow by using a new modification of Adomian decomposition method. The approximate solution of this problem is calculated here in the form of a rapidly convergent series at one and both grid points. Results for velocity profiles in divergent and convergent channels are presented for different values of Hartmann and Reynolds numbers. The validity and applicability of this new technique are illustrated through a comparison between reproducing Kernel Hilbert Space Method (RKHSM) (Inc et al., 2013), Modified Homotopy Perturbation Method (MHPM) (Singh, 2014) and numerical solutions.

Error analysis of reproducing kernel Hilbert space method for solving functional integral equations

This paper is devoted to error analysis of reproducing kernel Hilbert space method for solving linear and nonlinear Volterra integral equations in some reproducing kernel Hilbert spaces. Finally, some examples are given to illustrate and confirm error analysis of the method.

Solutions to Uncertain Volterra Integral Equations by Fitted Reproducing Kernel Hilbert Space Method

We present an efficient modern strategy for solving some well-known classes of uncertain integral equations arising in engineering and physics fields. The solution methodology is based on generating an orthogonal basis upon the obtained kernel function in the Hilbert spaceW21a,bin order to formulate the analytical solutions in a rapidly convergent series form in terms of theirα-cut representation. The approximation solution is expressed byn-term summation of reproducing kernel functions and it is convergent to the analytical solution. Our investigations indicate that there is excellent agreement between the numerical results and the RKHS method, which is applied to some computational experiments to demonstrate the validity, performance, and superiority of the method. The present work shows the potential of the RKHS technique in solving such uncertain integral equations.

Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integrodifferential equations

In this article, we propose the reproducing kernel Hilbert space method to obtain the exact and the numerical solutions of fuzzy Fredholm–Volterra integrodifferential equations. The solution methodology is based on generating the orthogonal basis from the obtained kernel functions in which the constraint initial condition is satisfied, while the orthonormal basis is constructing in order to formulate and utilize the solutions with series form in terms of their r-cut representation form in the Hilbert space \( W_{2}^{2} \left( \varOmega \right) \oplus W_{2}^{2} \left( \varOmega \right) \). Several computational experiments are given to show the good performance and potentiality of the proposed procedure. Finally, the utilized results show that the present method and simulated annealing provide a good scheduling methodology to solve such fuzzy equations.

PICARD-REPRODUCING KERNEL HILBERT SPACE METHOD FOR SOLVING GENERALIZED SINGULAR NONLINEAR LANE-EMDEN TYPE EQUATIONS

An iterative method is discussed with respect to its effectiveness and capability of solving singular nonlinear Lane-Emden type equations using reproducing kernel Hilbert space method combined with the Picard iteration. Some new error estimates for application of the method are established. We prove the convergence of the combined method. The numerical examples demonstrates a good agreement between numerical results and analytical predictions.

Some error estimates for the reproducing kernel Hilbert spaces method

In this paper we derive some effective error estimates for the reproducing kernel Hilbert space method applied to a general class of linear initial or boundary value problems. The first error estimate is computable and yields a worst case bound in the form of a percentage of the norm of the true solution which has not yet been discussed according to the knowledge of the authors. The second error estimate is a residual based error estimate, which is expressed in terms of the fill distance, so that convergence is studied for the fill distance tends to zero. This is a generalization and improvement of the existing error estimates. Some numerical results are presented to demonstrate the applicability of the estimates.

A new efficient method for cases of the singular integral equation of the first kind

Various cases of Cauchy type singular integral equation of the first kind occur rather frequently in mathematical physics and possess very unusual properties. These equations are usually difficult to solve analytically, and it is required to obtain approximate solutions. This paper investigates the numerical solution of various cases of Cauchy type singular integral equations using reproducing kernel Hilbert space (RKHS) method. The solution u(x) is represented in the form of a series in the reproducing kernel space, afterwards the n-term approximate solution un(x) is obtained and it is proved to converge to the exact solution u(x). The major advantage of the method is that it can produce good globally smooth approximate solutions. Moreover, in this paper, an efficient error estimation of the RKHS method is introduced. Finally, numerical experiments show that our reproducing kernel method is efficient.

New approach for the Fornberg–Whitham type equations

This paper applies the reproducing kernel Hilbert space method to the solutions of three types of Fornberg–Whitham equations: original, modified and time fractional. Comparison with Adomian decomposition method, homotopy analysis method and the variational iteration method shows the validity and applicability of the technique.

Solving a class of linear nonlocal boundary value problems using the reproducing kernel

Recently, the reproducing kernel Hilbert space methods(RKHSM) (see Wang et al (2011) [2]; Lin and Lin (2010) [3]; Wu and Li (2011) [4]; Zhou et al. (2009) [6]; Jiang and Chen (2014) [7]; Wang et al. (2010) [8]; Du and Cui (2008) [9]; Akram et al. (2013) [10]; Lü and Cui (2010) [11]; Wang et al. (2008) [12]; Yao and Lin (2009) [13]; Geng et al. (2014) [14] ; Arqub et al. (2013) [15]) emerged one after the other. But, a lot of difficult work should be done to deal with multi-point boundary value problems(BVPs). Our work is aimed at giving a new reproducing kernel method for multi-point BVPs. We do not put the homogenization conditions into the reproducing kernel space which can avoid to compute the reproducing kernel satisfying boundary conditions and the orthogonal system. Three numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by our method indicate that new algorithm has the following advantages: small computational work, fast convergence speed and high precision.

Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method

Modeling of uncertainty differential equations is very important issue in applied sciences and engineering, while the natural way to model such dynamical systems is to use fuzzy differential equations. In this paper, we present a new method for solving fuzzy differential equations based on the reproducing kernel theory under strongly generalized differentiability. The analytic and approximate solutions are given with series form in terms of their parametric form in the space $$W_2^2 [a,b]\oplus W_2^2 [a,b].$$W22[a,b]źW22[a,b]. The method used in this paper has several advantages; first, it is of global nature in terms of the solutions obtained as well as its ability to solve other mathematical, physical, and engineering problems; second, it is accurate, needs less effort to achieve the results, and is developed especially for the nonlinear cases; third, in the proposed method, it is possible to pick any point in the interval of integration and as well the approximate solutions and their derivatives will be applicable; fourth, the method does not require discretization of the variables, and it is not effected by computation round off errors and one is not faced with necessity of large computer memory and time. Results presented in this paper show potentiality, generality, and superiority of our method as compared with other well-known methods.

A fast hybrid method for constructing multidimensional potential energy surfaces from ab initio calculations: A new global analytic PES of NH2 system

A hybrid fitting scheme is proposed for the construction of global analytic ab initio potential energy surfaces (PESs) by means of applying reproducing kernel Hilbert space (RKHS) interpolation and cubic spline interpolation onto different dimensions of the molecular configuration space. In addition to inheriting most advantages of the pure RKHS method, this scheme offers the following extra benefits: short initiation time and enhanced stability and accuracy. We also propose a fast algorithm for the scheme allowing the PES computation time to be independent of the number of ab initio points. We have constructed an adiabatic PES of N(2D)+H2→NH+H reactive system from more than twenty thousand ab initio points using this scheme. The accurate quantum dynamics results calculated on the constructed PES demonstrate high accuracy and efficiency of this new scheme.

Spectral method for solving high order nonlinear boundary value problems via operational matrices

A spectral method based on operational matrices of Bernstein polynomials using collocation method is elaborated and employed for solving nonlinear ordinary and partial differential equations with multi-point boundary conditions. First, properties of Bernstein polynomial, operational matrices of integration, differentiation and product are introduced and then utilized to reduce the given differential equation to the solution of a system of algebraic equations. This new approach provides a significant computational advantage by converting the given original problem to an equivalent integro-differential equation which implies all boundary condition. Approximate solution is achieved by expanding the desired function in terms of a Bernstein basis and employing operational matrices. Unknown coefficients are determined by collocation. The method is compared with modified Adomian decomposition method, Birkhoff-type interpolation method, reproducing kernel Hilbert space method, fixed point method, finite-difference Keller-box method, multilevel augmentation method and shooting method. Illustrative examples are included to demonstrate the high precision, validity and good performance of the new scheme even for solving nonlinear singular differential equations.