Efficient implementation of essentially non-oscillatory shock-capturing schemes
Journal of Computational Physics
Bounds on multivariate polynomials and exponential error estimates for multiquadratic interpolation
Journal of Approximation Theory
Solving partial differential equations by collocation using radial basis functions
Applied Mathematics and Computation
An efficient numerical scheme for Burgers' equation
Applied Mathematics and Computation
Detection of Edges in Spectral Data II. Nonlinear Enhancement
SIAM Journal on Numerical Analysis
Overlapping domain decomposition method by radial basis functions
Applied Numerical Mathematics
Radial Basis Functions
A note on the Gibbs phenomenon with multiquadric radial basis functions
Applied Numerical Mathematics
Computers & Mathematics with Applications
The Runge phenomenon and spatially variable shape parameters in RBF interpolation
Computers & Mathematics with Applications
An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws
Journal of Computational Physics
Multiquadric collocation method with integralformulation for boundary layer problems
Computers & Mathematics with Applications
Recovery of High Order Accuracy in Radial Basis Function Approximations of Discontinuous Problems
Journal of Scientific Computing
Iterative adaptive RBF methods for detection of edges in two-dimensional functions
Applied Numerical Mathematics
Sparse Signal Reconstruction via Iterative Support Detection
SIAM Journal on Imaging Sciences
Brief paper: Set membership approximation of discontinuous nonlinear model predictive control laws
Automatica (Journal of IFAC)
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In [J.-H. Jung, Appl. Numer. Math. 57 (2007) 213-229], an adaptive multiquadric radial basis function method has been proposed for the reconstruction of discontinuous functions. Utilizing the vanishing shape parameters near the local jump discontinuity, the adaptive method considerably reduces the Gibbs oscillations and enhances convergence. In this paper, a new jump discontinuity detection method is developed based on the adaptive method. The global maximum of the expansion coefficients, @l"i, exists at the strongest jump discontinuity and its magnitude increases exponentially with the number of the center points, N. The adaptive method, however, dynamically reduces its magnitude to O(N) once applied. The global maximum of @l"i then exists at the next strongest jump discontinuity and its magnitude is exponentially large. In this way, the local jump discontinues are successively detected with the adaptive method applied iteratively. Numerical examples are provided using the piecewise analytic functions and the numerical solution of the shock interaction equations. Numerical results verify that the proposed method is efficient and accurate in finding local jump discontinuities.