Anderson acceleration and application to the three-temperature energy equations
Anderson acceleration and application to the three-temperature energy equations
复制标题
安德森加速及其在三温能量方程中的应用
DOI：
10.1016/j.jcp.2017.06.031
复制
发表时间：
2017-10
影响因子：
4.1
通讯作者：
安恒斌
中科院分区：
文献类型：
--
作者：
安恒斌
安德森加速法是一种加速定点迭代收敛的算法，包括皮卡德法。安德森加速在1965年首次提出，并已成功地用于加速电子结构计算中自洽场迭代的收敛。与Newton-like方法相比，安德森加速方法的一个优点是不需要构造雅可比矩阵。因此，该方法易于实现。本文采用Anderson-accelerated Picard方法求解一类强非线性辐射扩散方程--三温能量方程。采用两种策略来提高安德森加速方法的鲁棒性。一种策略是在必要时调整迭代以满足物理约束。另一种策略是监控，并在必要时减少安德森加速实现中最小二乘问题的矩阵条件数，以便保证数值稳定性。数值结果表明，采用Anderson-Accelerated Picard方法可以有效地求解三温能量方程。与不加加速的Picard方法相比，安德森加速至少可以减少一半的迭代次数。本文对无Jacobian的Newton-Krylov方法、Picard方法和Anderson-accelerated Picard方法进行了比较。
The Anderson acceleration method is an algorithm for accelerating the convergence of fixed-point iterations, including the Picard method. Anderson acceleration was first proposed in 1965 and, for some years, has been used successfully to accelerate the convergence of self-consistent field iterations in electronic-structure computations. Recently, the method has attracted growing attention in other application areas and among numerical analysts.Compared with a Newton-like method, an advantage of Anderson acceleration is that there is no need to form the Jacobian matrix. Thus the method is easy to implement. In this paper, an Anderson-accelerated Picard method is employed to solve the three-temperature energy equations, which are a type of strong nonlinear radiation-diffusion equations. Two strategies are used to improve the robustness of the Anderson acceleration method. One strategy is to adjust the iterates when necessary to satisfy the physical constraint. Another strategy is to monitor and, if necessary, reduce the matrix condition number of the least-squares problem in the Anderson-acceleration implementation so that numerical stability can be guaranteed. Numerical results show that the Anderson-accelerated Picard method can solve the three-temperature energy equations efficiently. Compared with the Picard method without acceleration, Anderson acceleration can reduce the number of iterations by at least half. A comparison between a Jacobian-free Newton–Krylov method, the Picard method, and the Anderson-accelerated Picard method is conducted in this paper.
登录
查看更多内容
影响因子：
4.7
作者：
R. Haelterman;A. Bogaers;Klaudius Scheufele;B. Uekermann;M. Mehl
通讯作者：
R. Haelterman;A. Bogaers;Klaudius Scheufele;B. Uekermann;M. Mehl
DOI：
--
发表时间：
2012
期刊：
Chinese Journal of Computational Physics
影响因子：
--
作者：
An Hengbin;Mo Ze-yao
通讯作者：
An Hengbin;Mo Ze-yao
DOI：
10.1137/130919398
发表时间：
2015-03
期刊：
SIAM J. Numer. Anal.
影响因子：
--
作者：
A. Toth;C. Kelley
通讯作者：
A. Toth;C. Kelley
影响因子：
3.1
作者：
Lipnikov, K.;Svyatskiy, D.;Vassilevski, Y.
通讯作者：
Vassilevski, Y.
DOI：
10.1137/100817589
发表时间：
2012-05
期刊：
SIAM J. Math. Anal.
影响因子：
--
作者：
J. Karátson
通讯作者：
J. Karátson