The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dim...The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dimension of the finite element space and N is the number of time steps.展开更多
自适应全变分(adaptive total variation,ATV)模型可以利用差分曲率自适应地选择基于Lp范数的正则项,并且能自适应调节正则项与保真项的权重,能够有效地去除噪声和保持图像边缘.使用半隐式梯度下降法求解ATV模型时,误差的高频分量会快...自适应全变分(adaptive total variation,ATV)模型可以利用差分曲率自适应地选择基于Lp范数的正则项,并且能自适应调节正则项与保真项的权重,能够有效地去除噪声和保持图像边缘.使用半隐式梯度下降法求解ATV模型时,误差的高频分量会快速衰减而低频分量却衰减缓慢,从而导致收敛速度缓慢.为了加快低频误差衰减的速度,利用半隐式梯度下降法设计了光滑化方法,构造了求解ATV模型的非线性多重网格法,并通过与不动点迭代法、半隐式梯度下降法的对比实验,验证了新方法的去噪效果更好且计算速度更快.展开更多
A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either...A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either a direct step on the current level or a coarse subspace correction step. In the coarse subspace correction step, we augment the traditional coarse grid space by a two-dimensional subspace spanned by the coordinate direction and the gradient direction at the current point. Global convergence is proved and convergence rate is studied under some mild conditions on the discretized functions. Preliminary numerical experiments on a few variational problems show that our two-level subspace method is promising.展开更多
In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlin...In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.展开更多
文摘The multigrid algorithm in [13] is developed for solving nonlinear parabolic equations arising from the finite element discretization. The computational cost of the algorithm is approximate O(NkN) where N-k is the dimension of the finite element space and N is the number of time steps.
文摘自适应全变分(adaptive total variation,ATV)模型可以利用差分曲率自适应地选择基于Lp范数的正则项,并且能自适应调节正则项与保真项的权重,能够有效地去除噪声和保持图像边缘.使用半隐式梯度下降法求解ATV模型时,误差的高频分量会快速衰减而低频分量却衰减缓慢,从而导致收敛速度缓慢.为了加快低频误差衰减的速度,利用半隐式梯度下降法设计了光滑化方法,构造了求解ATV模型的非线性多重网格法,并通过与不动点迭代法、半隐式梯度下降法的对比实验,验证了新方法的去噪效果更好且计算速度更快.
文摘A new two-level subspace method is proposed for solving the general unconstrained minimization formulations discretized from infinite-dimensional optimization problems. At each iteration, the algorithm executes either a direct step on the current level or a coarse subspace correction step. In the coarse subspace correction step, we augment the traditional coarse grid space by a two-dimensional subspace spanned by the coordinate direction and the gradient direction at the current point. Global convergence is proved and convergence rate is studied under some mild conditions on the discretized functions. Preliminary numerical experiments on a few variational problems show that our two-level subspace method is promising.
基金The research of the first author is supported by the Hong Kong Baptist University. The research of the second author is partially supported by a USA-AR0 grant 43751-MA and USA- NFS grants DMS0201094 and DMS-0412654. The third author is partially supported by CERG Grants of Hong Kong Research Grant Council, FRG grants of Hong Kong Baptist University, and an NSAF Grant （#10476032） of National Science Foundation of Chian.
文摘In this work, we propose an efficient numerical method for computing the electrostatic interaction between two like-charged spherical particles which is governed by the nonlinear Poisson-Boltzmann equation. The nonlinear problem is solved by a monotone iterative method which leads to a sequence of linearized equations. A modified central finite difference scheme is developed to solve the linearized equations on an exterior irregular domain using a uniform Cartesian grid. With uniform grids, the method is simple, and as a consequence, multigrid solvers can be employed to speed up the convergence. Numerical experiments on cases with two isolated spheres and two spheres confined in a charged cylindrical pore are carried out using the proposed method. Our numerical schemes are found efficient and the numerical results are found in good agreement with the previous published results.
