With Newton＇s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the ex...With Newton＇s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton＇s polynomial inter- polation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.展开更多
为了解决现有压缩感知图像重构算法中对大规模数据处理复杂度高且计算量大和存储量较大的问题,分别介绍了梯度追踪算法、拟牛顿法和限域拟牛顿法的核心思想并对以上算法的优缺点进行了分析。在分块压缩感知理论的基础上,对梯度追踪(Grad...为了解决现有压缩感知图像重构算法中对大规模数据处理复杂度高且计算量大和存储量较大的问题,分别介绍了梯度追踪算法、拟牛顿法和限域拟牛顿法的核心思想并对以上算法的优缺点进行了分析。在分块压缩感知理论的基础上,对梯度追踪(Gradient Pursuit,GP)算法进行改进,通过L-BFGS算法寻找梯度追踪算法中的更新方向并不断修正,将其运用到分块压缩感知的图像重构中,形成了基于L-BFGS方法的GP算法(L-BFGS Method based on GP algorithm,LMGP)。通过对分块后的图像进行单独处理,既避免了牛顿算法中需要进行Hesse矩阵的计算,降低了计算量和复杂度,节省了重构时间,也大大提高了重构效果。该文还对提出的LMGP算法的收敛性进行了分析,并通过LMGP算法对标准图像和一般图像分别进行了重构。仿真实验表明,提出的LMGP算法在重构时间、均方误差及峰值信噪比三个方面均优于其他传统的贪婪算法,说明LMGP算法的重构性能更具有优势。展开更多
A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-bal...A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.展开更多
基金Supported by the Key Project Foundation of the Department of Education of Anhui Province(No.KJ2008A027)the Project Foundation of the Department of Education of Anhui Province(No.KJ2010B182)
文摘With Newton＇s interpolating formula, we construct a kind of block based Newton-like blending osculatory interpolation.The interpolation provides us many flexible interpolation schemes for choices which include the expansive Newton＇s polynomial inter- polation as its special case. A bivariate analogy is also discussed and numerical examples are given to show the effectiveness of the interpolation.
文摘为了解决现有压缩感知图像重构算法中对大规模数据处理复杂度高且计算量大和存储量较大的问题,分别介绍了梯度追踪算法、拟牛顿法和限域拟牛顿法的核心思想并对以上算法的优缺点进行了分析。在分块压缩感知理论的基础上,对梯度追踪(Gradient Pursuit,GP)算法进行改进,通过L-BFGS算法寻找梯度追踪算法中的更新方向并不断修正,将其运用到分块压缩感知的图像重构中,形成了基于L-BFGS方法的GP算法(L-BFGS Method based on GP algorithm,LMGP)。通过对分块后的图像进行单独处理,既避免了牛顿算法中需要进行Hesse矩阵的计算,降低了计算量和复杂度,节省了重构时间,也大大提高了重构效果。该文还对提出的LMGP算法的收敛性进行了分析,并通过LMGP算法对标准图像和一般图像分别进行了重构。仿真实验表明,提出的LMGP算法在重构时间、均方误差及峰值信噪比三个方面均优于其他传统的贪婪算法,说明LMGP算法的重构性能更具有优势。
基金Project supported by the National Natural Science Foundation of China(Nos.91330205and 11421101)the National Key Research and Development Program of China(No.2016YFB0200603)
文摘A Newton multigrid method is developed for one-dimensional （1D） and two- dimensional （2D） steady-state shallow water equations （SWEs） with topography and dry areas. The nonlinear system arising from the well-balanced finite volume discretization of the steady-state SWEs is solved by the Newton method as the outer iteration and a geometric multigrid method with the block symmetric Gauss-Seidel smoother as the inner iteration. The proposed Newton multigrid method makes use of the local residual to regularize the Jacobian matrix of the Newton iteration, and can handle the steady- state problem with wet/dry transition. Several numerical experiments are conducted to demonstrate the efficiency, robustness, and well-balanced property of the proposed method. The relation between the convergence behavior of the Newton multigrid method and the distribution of the eigenvalues of the iteration matrix is detailedly discussed.
