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In this research, a radial basis functions weighted essentially non-oscillatory (RBF-WENO) scheme in the framework of finite difference is developed for solving non-linear degenerate parabolic (NDP) equations which may contain discontinuous solutions. The traditional WENO schemes for solving NDP equations are based on the polynomial interpolation. In this paper, a non-polynomial WENO finite difference scheme in order to enhance the local accuracy is proposed. For a non-polynomial interpolation basis, the infinitely smooth RBFs such as the multi-quadratic are adopted. By optimizing the free parameter in RBFs, the local error generated by the RBFs interpolation can be easily controlled. The formulation of scheme will be described in detail and numerical tests have been prepared to demonstrate the efficiency of the scheme in one and two-dimension. The numerical results demonstrate that the developed non-polynomial WENO schemes enhance the local accuracy and give sharper solution profile than the original WENO based on the polynomial interpolation for solving NDP equations. [ABSTRACT FROM AUTHOR]

In this article, we proposed an enhancement to the convergence rate of Landweber's method by incorporating the concept of momentum acceleration. Landweber's method is commonly used to solve least squares problems of the form min x ‖ A x − b ‖. Our approach is based on Landweber's method, which is acknowledged as a particular case of the methodologies outlined in Ding and Chen (2006). Through optimizing the momentum parameter, we were able to demonstrate the superior performance of the momentum-accelerated Landweber method. Specifically, we established that when A is a nonsquare m × n matrix with Rank (A) = n and σ min (A) ≠ σ max (A) , the momentum-accelerated Landweber method with the optimal parameter consistently outperforms the standard Landweber method. Our numerical experiments have confirmed the theoretical findings, demonstrating a notable improvement in the convergence rate of the Landweber method. [ABSTRACT FROM AUTHOR]