Multiquadric and its shape parameter -- a numerical investigation of error estimate, condition number, and round-off error by arbitrary precision computation

DOI10.1016/j.enganabound.2011.07.008zbMath1245.65162OpenAlexW2028050352MaRDI QIDQ444824

Alexander H.-D. Cheng

Publication date: 24 August 2012

Published in: Engineering Analysis with Boundary Elements (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1016/j.enganabound.2011.07.008

zbMATH Keywords

collocation methodcondition numbererror estimateradial basis functionshape parametermultiquadricarbitrary precision computation

Mathematics Subject Classification ID

Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Interpolation in approximation theory (41A05) Approximation by other special function classes (41A30)

Related Items

Numerical simulation for a time-fractional coupled nonlinear Schrödinger equations, A meshless method based on the fundamental solution and radial basis function for solving an inverse heat conduction problem, Enhancing finite differences with radial basis functions: experiments on the Navier-Stokes equations, The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods, Effective condition number for the selection of the RBF shape parameter with the fictitious point method, Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions, A global particular solution meshless approach for the four-sided lid-driven cavity flow problem in the presence of magnetic fields, A numerical study of Asian option with radial basis functions based finite differences method, Numerical solution of stochastic elliptic partial differential equations using the meshless method of radial basis functions, An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein-Gordon equations, An algorithm with \(m\)-step residual history for solving linear equations: data interpolation by a multi-shape-factors RBF, The use of radial basis functions (RBFs) collocation and RBF-QR methods for solving the coupled nonlinear sine-Gordon equations, Meshless simulation of stochastic advection-diffusion equations based on radial basis functions, Numerical investigation of the meshless radial basis integral equation method for solving 2D anisotropic potential problems, Meshless analysis of soil-structure interaction using an MFS-MLPG coupled approach, A meshless technique based on the local radial basis functions collocation method for solving parabolic-parabolic Patlak-Keller-Segel chemotaxis model, The local Kansa's method for solving Berger equation, The use of element free Galerkin method based on moving Kriging and radial point interpolation techniques for solving some types of Turing models, PDE centres enhancement in the localized regular dual reciprocity method, A new approximate method for an inverse time-dependent heat source problem using fundamental solutions and RBFs, Buckling analysis of functionally graded thin plate with in-plane material inhomogeneity, Radial basis function collocation method for an elliptic problem with nonlocal multipoint boundary condition, A multiple-scale MQ-RBF for solving the inverse Cauchy problems in arbitrary plane domain, Numerical solution of a non-classical two-phase Stefan problem via radial basis function (RBF) collocation methods, Numerical simulation of metal removal in laser drilling using radial point interpolation method, On the ill-conditioned nature of \(C^\infty\) RBF strong collocation, An improved radial basis-pseudospectral method with hybrid Gaussian-cubic kernels, Finite integration method with radial basis function for solving stiff problems, Doubly stochastic radial basis function methods, An alternative local collocation strategy for high-convergence meshless PDE solutions, using radial basis functions, Improved Kansa RBF method for the solution of nonlinear boundary value problems, A novel dual reciprocity boundary element formulation for two-dimensional transient convection-diffusion-reaction problems with variable velocity, Unnamed Item, Extending meshless method of approximate particular solutions (MAPS) to two-dimensional convection heat transfer problems., New approximations for one-dimensional 3-point and two-dimensional 5-point compact integrated RBF stencils, A trustable shape parameter in the kernel-based collocation method with application to pricing financial options, The use of proper orthogonal decomposition (POD) meshless RBF-FD technique to simulate the shallow water equations, Pharmacokinetic parameter identification using a meshless method approach for transdermal drug delivery, \(h\)-adaptive radial basis function finite difference method for linear elasticity problems, Kansa-RBF algorithms for elliptic BVPs in annular domains with mixed boundary conditions, A Kansa-radial basis function method for elliptic boundary value problems in annular domains, Automatic update of Gaussian and multiquadric shape parameter for sequential metamodels based optimization, A shape parameter insensitive CRBFs‐collocation for solving nonlinear optimal control problems, Transient Convection-Diffusion-Reaction Problems with Variable Velocity Field by Means of DRBEM with Different Radial Basis Functions, Formulation of Kansa's method in the frequency domain for the analysis of transient heat conduction, Local radial basis function collocation method for linear thermoelasticity in two dimensions, Development of the Kansa method for solving seepage problems using a new algorithm for the shape parameter optimization, An energy regularization of the MQ-RBF method for solving the Cauchy problems of diffusion-convection-reaction equations, On the selection of a better radial basis function and its shape parameter in interpolation problems, An overview of the method of fundamental solutions -- solvability, uniqueness, convergence, and stability, A comparison of the meshless RBF collocation method with finite element and boundary element methods in neutron diffusion calculations, The numerical solution of Fokker-Planck equation with radial basis functions (RBFs) based on the meshless technique of Kansa's approach and Galerkin method, Numerical solution to time-dependent 4D inviscid Burgers' equations, Stability estimate on meshless unsymmetric collocation method for solving boundary value problems, Optimization, conditioning and accuracy of radial basis function method for partial differential equations with nonlocal boundary conditions -- a case of two-dimensional Poisson equation, Optimal shape parameter in the MQ-RBF by minimizing an energy gap functional, Meshless modeling of coupled transdermal pharmacokinetics with analytical validation, Particular solutions of products of Helmholtz-type equations using the Matern function, Heat transfer for the generalized Newtonian fluid flow through a fibrous porous media, A stabilized collocation method based on the efficient gradient reproducing kernel approximations for the boundary value problems, An interval for the shape parameter in radial basis function approximation, Local RBF Algorithms for Elliptic Boundary Value Problems in Annular Domains, Selection of an interval for variable shape parameter in approximation by radial basis functions, Numerical solution of two-dimensional elliptic PDEs with nonlocal boundary conditions, The sample solution approach for determination of the optimal shape parameter in the multiquadric function of the Kansa method, Fully meshless solution of the one-dimensional multigroup neutron transport equation with the radial basis function collocation method, The method of approximate particular solutions for the time-fractional diffusion equation with a non-local boundary condition, The method of variably scaled radial kernels for solving two-dimensional magnetohydrodynamic (MHD) equations using two discretizations: the Crank-Nicolson scheme and the method of lines (MOL), Two numerical meshless techniques based on radial basis functions (RBFs) and the method of generalized moving least squares (GMLS) for simulation of coupled Klein-Gordon-Schrödinger (KGS) equations, Remediation of contaminated groundwater by meshless local weak forms, On the search of the shape parameter in radial basis functions using univariate global optimization methods, An accurate and stable RBF method for solving partial differential equations, Kansa-RBF Algorithms for Elliptic Problems in Axisymmetric Domains, Kansa-RBF algorithms for elliptic problems in regular polygonal domains, Scattered data interpolation: strictly positive definite radial basis/cardinal functions, Numerical solution of the system of second-order boundary value problems using the local radial basis functions based differential quadrature collocation method, Application of radial basis functions to the problem of elasto-plastic torsion of prismatic bars, On the new variable shape parameter strategies for radial basis functions, On elastoplastic analysis of some plane stress problems with meshless methods and successive approximations method, Method of particular solutions using polynomial basis functions for the simulation of plate bending vibration problems, On a generalized Gaussian radial basis function: analysis and applications, Gaussian RBF-FD weights and its corresponding local truncation errors, The shape parameter in the shifted surface spline. III., Acoustic analysis of heterogeneous domains coupling the BEM with Kansa's method, The quasi-optimal radial basis function collocation method: a technical note, A bounded randomly variable shape multi-quadric interpolation method in dual reciprocity boundary element method, Variable shape parameter Kansa RBF method for the solution of nonlinear boundary value problems, A meshless generalized finite difference method for 2D elasticity problems, A class of RBFs-based DQ methods for the space-fractional diffusion equations on 3D irregular domains, Residual-error cross-validation method for selecting a suitable shape parameter for RBF interpolation, Educating local radial basis functions using the highest gradient of interest in three dimensional geometries, Computing high-order derivatives in compact integrated-RBF stencils

Uses Software

• Algorithm 719

Cites Work

• On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs
• Effective condition number for simplified hybrid Trefftz methods
• Study on effective condition number for collocation methods
• A localized collocation meshless method (LCMM) for incompressible flows CFD modeling with applications to transient hemodynamics
• Interpolation of scattered data: distance matrices and conditionally positive definite functions
• Theory and applications of the multiquadric-biharmonic method. 20 years of discovery 1968-1988
• Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I: Surface approximations and partial derivative estimates
• Numerical algorithm for triphasic model of charged and hydrated soft tissues
• Multistage trivariate surfaces
• The role of the multiquadric shape parameters in solving elliptic partial differential equations
• Stability analysis of method of fundamental solutions for mixed boundary value problems of Laplace's equation
• On convergent numerical algorithms for unsymmetric collocation
• Effective condition number for the finite element method using local mesh refinements
• Multiquadric method for the numerical solution of a biphasic mixture model
• Eigenvalues of euclidean distance matrices
• Norm estimates for the inverses of a general class of scattered-data radial-function interpolation matrices
• Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation
• On the sensitivity of radial basis interpolation to minimal data separation distance
• The asymptotic cardinal function of the multiquadratic \(\varphi{}(r)=(r^ 2+c^ 2)^{1/2}\) as \(c{\rightarrow{}}\infty\)
• Improved accuracy of multiquadric interpolation using variable shape parameters
• Miscellaneous error bounds for multiquadric and related interpolators
• The method of fundamental solutions for elliptic boundary value problems
• The uniform convergence of thin plate spline interpolation in two dimensions
• Lower bounds for norms of inverses of interpolation matrices for radial basis functions
• On condition numbers associated with radial-function interpolation
• The effective condition number applied to error analysis of certain boundary collocation methods
• Some observations regarding interpolants in the limit of flat radial basis functions
• Circumventing the ill-conditioning problem with multiquadric radial basis functions: Applications to elliptic partial differential equations
• Adaptive greedy techniques for approximate solution of large RBF systems
• On the optimal shape parameters of radial basis functions used for 2-D meshless methods
• Interpolation in the limit of increasingly flat radial basis functions
• Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary
• A quasi-radial basis functions method for American options pricing.
• The mystery of the shape parameter. IV
• Stable computation of multiquadric interpolants for all values of the shape parameter
• A least-squares preconditioner for radial basis functions collocation methods
• Multiquadrics -- a scattered data approximation scheme with applications to computational fluid-dynamics. II: Solutions to parabolic, hyperbolic and elliptic partial differential equations
• An adaptive greedy algorithm for solving large RBF collocation problems
• Error estimates and condition numbers for radial basis function interpolation
• Support-vector networks
• The conditioning of some numerical methods for first kind boundary integral equations
• Radial basis function approximation methods with extended precision floating point arithmetic
• Effective condition number and superconvergence of the Trefftz method coupled with high-order FEM for singularity problems
• Error estimate, optimal shape factor, and high precision computation of multiquadric collocation method
• An algorithm for selecting a good value for the parameter \(c\) in radial basis function interpolation
• Fast fitting of radial basis functions: Methods based on preconditioned GMRES iteration
• A new formulation and computation of the triphasic model for mechano-electrochemical mixtures
• The parameter \(R^ 2\) in multiquadric interpolation
• Multivariate interpolation by polynomials and radial basis functions
• Effective condition number for finite difference method
• The Runge phenomenon and spatially variable shape parameters in RBF interpolation
• Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions
• Preconditioning for radial basis functions with domain decomposition methods
• Multivariate cardinal interpolation with radial-basis functions
• Norms of inverses and condition numbers for matrices associated with scattered data
• Solution of Incompressible Turbulent Flow by a Mesh-Free Method
• Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation
• Meshless local radial basis function collocation method for convective‐diffusive solid‐liquid phase change problems
• Numerical experiments on optimal shape parameters for radial basis functions
• The crucial constants in the exponential-type error estimates for Gaussian interpolation
• A Stable Algorithm for Flat Radial Basis Functions on a Sphere
• Trefftz method: Fitting boundary conditions
• Numerical Procedures for Surface Fitting of Scattered Data by Radial Functions
• Effectively Well-Conditioned Linear Systems
• Error Analysis of Direct Methods of Matrix Inversion
• Scattered Data Interpolation: Tests of Some Method
• Local error estimates for radial basis function interpolation of scattered data
• A numerical method for heat transfer problems using collocation and radial basis functions
• The Lack of Influence of the Right-Hand Side on the Accuracy of Linear System Solution
• Element‐free Galerkin methods
• Algorithm 719: Multiprecision translation and execution of FORTRAN programs
• Radial Basis Functions
• Exponential convergence andH-c multiquadric collocation method for partial differential equations
• Multivariate Interpolation and Conditionally Positive Definite Functions. II
• Direct solution of ill-posed boundary value problems by radial basis function collocation method
• Polynomials and Potential Theory for Gaussian Radial Basis Function Interpolation
• Scattered Data Approximation
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item