Multiscale Radial Basis Functions

Wendland, Holger

doi:10.1007/978-3-319-55550-8_12

Cited by 4 publications

(3 citation statements)

References 40 publications

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“…And the applications of this algorithm in solving linear partial differential equations on spheres were proposed in [19] and on bounded domains were proposed in [20][21][22]. A detailed discussion on the stationary multilevel algorithm is reviewed in the latest papers [23,24]. Algorithm 2 is the specific form for nonlinear situation.…”

Section: Stationary Multilevel Methodmentioning

confidence: 99%

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

2020

Mathematical Problems in Engineering

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This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

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Section: Stationary Multilevel Methodmentioning

confidence: 99%

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

2020

Mathematical Problems in Engineering

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“…Theorem 3: ''Let , be two matrices such that and for all . Then if is an M-matrix, so is the matrix '' [16].…”

Section: Theorem 1: ''A Linear Iteration With the Iteration Matrix [ ...mentioning

confidence: 99%

“…Theorem 7: ''Let be a regular splitting of a matrix . Then if and only if is nonsingular and is nonnegative'' [16].…”

Section: (Ii)mentioning

confidence: 99%

A New Improved Classical Iterative Algorithm for Solving System of Linear Equations

Baloch¹,

Kalhoro²,

Khalil³

et al. 2022

PPASA

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The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

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Advanced Algorithm for Interpolation with Wendland Functions

Björnsson

Hafstein

2020

Informatics in Control, Automation and Robotics

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Multiscale Radial Basis Functions

Cited by 4 publications

References 40 publications

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

A New Improved Classical Iterative Algorithm for Solving System of Linear Equations

Advanced Algorithm for Interpolation with Wendland Functions

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