Showing papers by "Guo-Wei Wei published in 2000"

Wavelets generated by using discrete singular convolution kernels

Abstract: This paper explores the connection between wavelet methods and an efficient computational algorithm - the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cases of the present DSC kernel-generated wavelets. A family of wavelet generators proposed in this paper are found to form an infinite-dimensional Lie group which has an invariant subgroup of translation and dilation. If DSC kernels form an orthogonal system, they are found to span a wavelet subspace in a multiresolution analysis.

Solving quantum eigenvalue problems by discrete singular convolution

Abstract: This paper explores the utility of a discrete singular convolution (DSC) algorithm for solving the Schrodinger equation. DSC kernels of Shannon, Dirichlet, modified Dirichlet and de la Vallee Poussin are selected to illustrate the present algorithm for obtaining eigenfunctions and eigenvalues. Four benchmark physical problems are employed to test numerical accuracy and speed of convergence of the present approach. Numerical results indicate that the present approach is efficient and reliable for solving the Schrodinger equation.

A unified approach for the solution of the Fokker-Planck equation

Abstract: This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test the accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.

A Note on Regularized Shannon's Sampling Formulae

Abstract: Error estimation is given for a regularized Shannon's sampling formulae, which was found to be accurate and robust for numerically solving partial differential equations.

Shock capturing by anisotropic diffusion oscillation reduction

Abstract: This paper introduces the method of anisotropic diffusion oscillation reduction (ADOR) for shock wave computations. The connection is made between digital image processing,in particular, image edge detection, and numerical shock capturing. Indeed, numerical shock capturing can be formulated on the lines of iterative digital edge detection. Various anisotropic diffusion and super diffusion operators originated from image edge detection are proposed for the treatment of hyperbolic conservation laws and near-hyperbolic hydrodynamic equations of change. The similarity between anisotropic diffusion and artificial viscosity is discussed. Physical origins and mathematical properties of the artificial viscosity is analyzed from the kinetic theory point of view. A form of pressure tensor is derived from the first principles of the quantum mechanics. Quantum kinetic theory is utilized to arrive at macroscopic transport equations from the microscopic theory. Macroscopic symmetry is used to simplify pressure tensor expressions. The latter provides a basis for the design of artificial viscosity. The ADOR approach is validated by using (inviscid) Burgers' equation in one and two spatial dimensions, the incompressible Navier-Stokes equation and the Euler equation. A discrete singular convolution (DSC) algorithm is utilized for the spatial discretization.

A unified approach for the solution of the Fokker-Planck equation

Abstract: This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is demonstrated that different implementations of the present algorithm, such as global, local, Galerkin, collocation, and finite difference, can be deduced from a single starting point. Three benchmark stochastic systems, the repulsive Wong process, the Black-Scholes equation and a genuine nonlinear model, are employed to illustrate the robustness and to test accuracy of the present approach for the solution of the Fokker-Planck equation via a time-dependent method. An additional example, the incompressible Euler equation, is used to further validate the present approach for more difficult problems. Numerical results indicate that the present unified approach is robust and accurate for solving the Fokker-Planck equation.