Kernel-based Approximation Research Articles

An unsupervised [formula omitted]-means machine learning algorithm via overlapping to improve the nodes selection for solving elliptic problems

We propose an overlapping algorithm utilizing the K-means clustering technique to group scattered data nodes for discretizing elliptic partial differential equations. Unlike conventional kernel-based approximation methods, which select the closest points from the entire region for each center, our algorithm selects only the nearest points within each overlapping cluster. We present computational results to demonstrate the efficiency of our algorithm for both two-dimensional and three-dimensional problems. For evaluation and validation, these results are compared with results obtained using the RBF-FD+polynomial method with different kernels.

Non-differentiable kernel-based approximation of memory-dependent derivative for drug delivery applications

Although memory-dependent derivative (MDD) has recently been the subject of extensive study, only one numerical approximation has been reported in the literature. Hence, this study introduces a novel approximation for MDD. Moreover, a new form of the kernel function is presented. The convergence order of our approximation is Oh3−S,0<S<1 , where (1−S) is the exponent of the kernel function. The proposed approach is used to numerically solve the memory-dependent advection-diffusion problem, and the numerical scheme's stability and convergence are discussed. Also, memory-based models have been described to study drug delivery and its diffusion from multi-layer capsules/tablets. These models are based on the fractional derivative and the MDD to solve the paradox of the unphysical feature of the infinite propagation speed of the published Fickian-based models. The theoretical analysis is validated with numerical examples by investigating the convergence order that is 3−S in time. To illustrate the validity and efficiency of the proposed models, profiles of concentration and drug mass are compared with the observed in vivo data. It is observed to be highly accurate and compatible with the in vivo data when compared with Fick's model.

A simple and novel coupling method for CFD–DEM modeling with uniform kernel-based approximation

A simple and novel coupling method for CFD–DEM modeling with uniform kernel-based approximation

A stable meshless numerical scheme using hybrid kernels to solve linear Fredholm integral equations of the second kind and its applications

The main challenge in kernel-based approximation theory and its applications is the conflict between accuracy and stability. Hybrid kernels can be used as one of the simplest and most effective tools to manage this challenge. This article uses a meshless scheme based on hybrid radial kernels (HRKs) to solve second kind Fredholm integral equations (FIEs). The method estimates the solution via the discrete collocation procedure based on a hybrid kernel that combines appropriate kernels with suitable weight parameters. The optimal parameters in the hybrid kernels can be calculated using the particle swarm optimization (PSO) algorithm based on the root mean square (RMS) error. This technique converts the problem under investigation into a system of linear equations. Moreover, the convergence of the suggested hybrid approach is studied. Lastly, some numerical experiments are included to reveal the accuracy and stability of the hybrid kernels approach. The numerical results demonstrate that the presented hybrid procedure meaningfully reduces the ill-conditioning of the operational matrices, at the same time, it maintains the accuracy and stability for all values of shape parameters.

Full-rank orthonormal bases for conditionally positive definite kernel-based spaces

In kernel-based approximation, it is well-known that the direct approach to interpolation is prone to ill-conditioning of the interpolation matrix. One simple idea is to use other better-conditioned bases that span the same space of the translated kernels i.e. their associated native space. Pazouki and Schaback (2011) tracked this issue by investigating different factorization of the interpolation matrix in order to build stable and orthonormal bases for the corresponding native space of the positive definite kernels. In this paper, we work with the reproducing kernel K for the associated native Hilbert space NΦ(Ω) corresponding to a conditionally positive definite kernel Φ on the nonempty set Ω. We give a well-organized matrix formulation of the kernel matrix K by constructing the matrices corresponding to cardinal basis from monomials. Then, we present two possible ways to find full-rank data-dependent orthonormal bases that are discretely ℓ2 and NΦ-orthonormal. The first approach is given by the factorization of the kernel matrix K and the next one is based on the eigenpairs approximation of the linear operator associated with the reproducing kernel K given by Mercer’s theorem. In the sequel, we employ the truncated singular value decomposition technique to find an optimal low-rank basis with the coefficient matrix whose rank is less than that of the original matrix. Special attention is also given to error analysis, duality, and stability. Some numerical experiments are also provided.

High-dimensional approximation with kernel-based multilevel methods on sparse grids

Moderately high-dimensional approximation problems can successfully be solved by combining univariate approximation processes using an intelligent combination technique. While this has so far predominantly been done with either polynomials or splines, we suggest to employ a multilevel kernel-based approximation scheme. In contrast to those schemes built upon polynomials and splines, this new method is capable of combining arbitrary low-dimensional domains instead of just intervals and arbitrarily distributed points in these low-dimensional domains. We introduce the method and analyse its convergence in the so-called isotropic and anisotropic cases.

Techniques for second-order convergent weakly compressible smoothed particle hydrodynamics schemes without boundaries

Despite the many advances in the use of weakly compressible smoothed particle hydrodynamics (SPH) for the simulation of incompressible fluid flow, it is still challenging to obtain second-order convergence even for simple periodic domains. In this paper, we perform a systematic numerical study of convergence and accuracy of kernel-based approximation, discretization operators, and weakly compressible SPH (WCSPH) schemes. We explore the origins of the errors and issues preventing second-order convergence despite having a periodic domain. Based on the study, we propose several new variations of the basic WCSPH scheme that are all second-order accurate. Additionally, we investigate the linear and angular momentum conservation property of the WCSPH schemes. Our results show that one may construct accurate WCSPH schemes that demonstrate second-order convergence through a judicious choice of kernel, smoothing length, and discretization operators in the discretization of the governing equations.

A stochastic extended Rippa’s algorithm for LpOCV

In kernel-based approximation, the tuning of the so-called shape parameter is a fundamental step for achieving an accurate reconstruction. Recently, the popular Rippa’s algorithm Rippa (1999) [1] has been extended to a more general cross validation setting. In this work, we propose a modification of such extension with the aim of further reducing the computational costs. The resulting Stochastic Extended Rippa’s Algorithm (SERA) is first detailed and then tested by means of various numerical experiments, which show its efficacy and effectiveness in different approximation settings.

Kernel-based approximation of variable-order diffusion models

In this paper, a numerical scheme is constructed that is based on radial basis functions (RBF) and the Coimbra variable time fractional derivative of order 0 < α(t, x) ≤ 1. The derivative due to Coimbra can efficiently model a dynamical system whose fractional order behaviour varies with time as well as space. The stability and convergence of the RBF-based numerical scheme are discussed and the developed numerical scheme is validated for various 1D and 2D anomalous diffusion models with different fractional variable orders either a function of t or x . The accuracy and efficiency of the numerical scheme are achieved by comparing the results of available results in the literature.

Kernel-based approximation of variable-order diffusion models

Kernel-based approximation of variable-order diffusion models

Fast approximation by periodic kernel-based lattice-point interpolation with application in uncertainty quantification

This paper deals with the kernel-based approximation of a multivariate periodic function by interpolation at the points of an integration lattice—a setting that, as pointed out by Zeng et al. (Monte Carlo and Quasi-Monte Carlo Methods 2004, Springer, New York, 2006) and Zeng et al. (Constr. Approx. 30: 529–555, 2009), allows fast evaluation by fast Fourier transform, so avoiding the need for a linear solver. The main contribution of the paper is the application to the approximation problem for uncertainty quantification of elliptic partial differential equations, with the diffusion coefficient given by a random field that is periodic in the stochastic variables, in the model proposed recently by Kaarnioja et al. (SIAM J Numer Anal 58(2): 1068–1091, 2020). The paper gives a full error analysis, and full details of the construction of lattices needed to ensure a good (but inevitably not optimal) rate of convergence and an error bound independent of dimension. Numerical experiments support the theory.

Hyperspectral classification using an adaptive spectral-spatial kernel-based low-rank approximation

ABSTRACTThis paper presents a novel adaptive spectral-spatial kernel-based low-rank approximation method for spectral-spatial hyperspectral image (HSI) classification. In the first of three steps of the proposed method, superpixel and image patch are used together to calculate the weights in the homogeneous region. Second, an adaptive spectral-spatial kernel is defined to capture the spectral and spatial feature of HSIs. In the final step, an adaptive spectral-spatial kernel and low-rank approximation are integrated into a decision model to perform HSI classification. Extensive experimental results on Indian Pines and Pavia University demonstrate the superiority of the proposed classifier when compared with other competing classifiers.

Kernel-based probability measures for generalized interpolations: A deterministic or stochastic problem?

In this article, we propose a kernel-based algorithm for generalized interpolations by kernel-based probability measures on Banach spaces. Different from the classical kernel-based approximation methods, we construct and analyze the deterministic kernel-based estimators based on the generalized data by the stochastic approaches including formulas and convergence. Specially, we do not need the regularization parameter estimations and the noise distribution estimations in the computational processes of the kernel-based estimators based on the noisy data.

The Role of Hilbert–Schmidt SVD basis in Hermite–Birkhoff interpolation in fractional sense

The Hermite–Birkhoff (HB) interpolation is an extension of polynomial interpolation that appears when observation gives operational information. Additional capabilities in HB interpolation by considering fractional operators are created, as it occur in many applied systems. Due to the nice properties in kernel-based approximation, we intend to apply it to solve the HB interpolation in the fractional sense. We show the standard basis in kernel-based approximation is often insufficient for computing a stable solution in fractional HB interpolation. Because of the inherent ill-condition of kernel-based methods, we investigate the fractional HB interpolation using alternate Hilbert–Schmidt SVD (HS–SVD) bases, since it provides a linear transformation which can be applied analytically, and therefore, is able to remove a significant portion of the ill-conditioning. Also, the convergence and stability of the fractional HB interpolation using HS–SVD method are discussed. Numerical results show that in solving fractional HB interpolation, although the standard basis for many positive definite kernels is ill-conditioned in the flat limit, the HS–SVD basis solves the existing problem.

A semi-resolved CFD–DEM approach for particulate flows with kernel based approximation and Hilbert curve based searching strategy

Particulate flow has a wide range of industrial applications and is frequently modeled with coupled CFD–DEM approaches. Herein, we first identified a simulation gap between the resolved CFD–DEM and unresolved CFD–DEM through a size effect study. Then we analyzed the error sources of the conventional unresolved CFD–DEM when modeling particulate flows with comparable mesh size and particle diameter. We finally developed a semi-resolved CFD–DEM model, which combines the advantages of both resolved and unresolved CFD–DEM models. The semi-resolved CFD–DEM uses a drag force model to characterize particle–fluid interaction, while the relative velocity in the drag force model is corrected through kernel-based approximations on the neighboring fluid cells rather than simply taking values in the local cell containing the concerned particle, and the void fraction in the force model is corrected as well. In order to improve the computational efficiency, a Hilbert curve based searching strategy is used to identify the fluid cells covered by the influencing area of the kernel function. A number of numerical simulations have been conducted and numerical results from different CFD–DEM approaches are compared together with experimental data. It is shown that the presented semi-resolved CFD–DEM bridges the simulation gap between the resolved CFD–DEM and unresolved CFD–DEM while it is as efficient as the conventional unresolved CFD–DEM and as accurate as the resolved CFD–DEM.

Localized kernel-based approximation for pricing financial options under regime switching jump diffusion model

In this paper, we consider European and American option pricing problems under regime switching jump diffusion models which are formulated as a system of partial integro-differential equations (PIDEs) with fixed and free boundaries. For free boundary problem arising in pricing American option, we use operator splitting method to deal with early exercise feature of American option. For developing a numerical technique we employ localized radial basis function generated finite difference (RBF-FD) approximation to overcome the ill-conditioning and high density issues of discretized matrices. The proposed method leads to linear systems with tridiagonal and diagonal dominant matrices. Also, in this paper the convergence and consistency of the proposed method are discussed. Numerical examples presented in the last section illustrate the robustness and practical performance of the proposed algorithm for pricing European and American options.

Kernel-Based Discretization for Solving Matrix-Valued PDEs

In this paper, we discuss the numerical solution of certain matrix-valued partial differential equations. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyse a new meshfree discretisation scheme using kernel-based approximation spaces. However, since these pproximation spaces have now to be matrix-valued, the kernels we need to use are fourth order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems

Fully Symmetric Kernel Quadrature

Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs to be solved. In this article we show that the weights of a kernel quadrature rule can be computed efficiently and exactly for up to tens of millions of nodes if the kernel, integration domain, and measure are fully symmetric and the node set is a union of fully symmetric sets. This is based on the observations that in such a setting there are only as many distinct weights as there are fully symmetric sets and that these weights can be solved from a linear system of equations constructed out of row sums of certain submatrices of the full kernel matrix. We present several numerical examples that show feasibility, both for a large number of nodes and in high dimensions, of the developed fully symmetric kernel quadrature rules. Most prominent of the fully symmetric kernel quadrature rules we propose are those that use sparse grids.

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

We provide a framework for the sparse approximation of multilinear problems and show that several problems in uncertainty quantification fit within this framework. In these problems, the value of a multilinear map has to be approximated using approximations of different accuracy and computational work of the arguments of this map. We propose and analyze a generalized version of Smolyak’s algorithm, which provides sparse approximation formulas with convergence rates that mitigate the curse of dimension that appears in multilinear approximation problems with a large number of arguments. We apply the general framework to response surface approximation and optimization under uncertainty for parametric partial differential equations using kernel-based approximation. The theoretical results are supplemented by numerical experiments.

Fundamental kernel-based method for backward space–time fractional diffusion problem

Based on kernel-based approximation technique, we devise in this paper an efficient and accurate numerical scheme for solving a backward space–time fractional diffusion problem (BSTFDP). The kernels used in the approximation are the fundamental solutions of the space–time fractional diffusion equation expressed in terms of inverse Fourier transform of Mittag-Leffler functions. The use of Inverse fast Fourier transform (IFFT) technique enables an accurate and efficient evaluation of the fundamental solutions and gives a robust numerical algorithm for the solution of the BSTFDP. Since the BSTFDP is intrinsic ill-posed, we apply the standard Tikhonov regularization technique to obtain a stable solution to the highly ill-conditioned resultant system of linear equations. For choosing optimal regularization parameter, we combine the regularization technique with the generalized cross validation (GCV) method for an optimal placement of the source points in the use of fundamental solutions. Meanwhile, the proposed algorithm also speeds up the previous method given in Dou and Hon (2014). Several numerical examples are constructed to verify the accuracy and efficiency of the proposed method.