曲率障碍下四阶变分不等式的交替方向乘子法

张霖森; 程兰; 张守贵

doi:10.21656/1000-0887.430243

曲率障碍下四阶变分不等式的交替方向乘子法

doi: 10.21656/1000-0887.430243

重庆师范大学数学科学学院，重庆 401331

基金项目:

国家自然科学基金项目 11971085

重庆市自然科学基金项目 cstc2020jcyj-msxmX0066

重庆市研究生教育教学改革研究项目 yjg213071

重庆市研究生科研创新项目 CYS22561

详细信息

作者简介:
张霖森(1997—)，男，硕士生(E-mail: 398780730@qq.com)

通讯作者:
张守贵(1973—)，男，教授，博士(通讯作者. E-mail: shgzhang@cqnu.edu.cn)

中图分类号: O241.82
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- 被引次数: 0
出版历程
- 收稿日期: 2022-07-23
- 修回日期: 2022-09-19
- 刊出日期: 2023-05-01

An Alternating Direction Multiplier Method for 4th-Order Variational Inequalities With Curvature Obstacle

School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P.R.China

摘要

摘要: 对于重调和算子和曲率障碍表示的变分不等式，提出了自适应交替方向乘子数值解法(SADMM). 对问题引入一个辅助变量表示曲率函数的增广Lagrange函数，导出一个约束极小值问题，并且该问题等价于一个鞍点问题. 然后采用交替方向乘子法(ADMM)求解这个鞍点问题. 通过采用平衡原理和迭代函数，得到了自动调整罚参数的自适应法则，从而提高了计算效率. 证明了该方法的收敛性，并给出了利用迭代函数近似罚参数的具体方法. 最后，用数值计算结果验证了该方法的有效性.
- 四阶变分不等式 /
- 曲率障碍 /
- 交替方向乘子法 /
- 自适应法则
Abstract: A self-adaptive alternating direction method of multipliers was proposed for the approximation solution of variational inequalities with biharmonic operators and curvature obstacle. An augmented Lagrange functional was introduced with an auxiliary variable to express the curvature function, and a constrained minimization problem equivalent to a saddle-point one was deduced. Then the alternating direction method of multipliers was applied to solve the saddle-point problem. By means of the balance principle and iterative functions, a self-adaptive rule was obtained to adjust the penalty parameter automatically, and improve the computation efficiency. The convergence of this method was proved and the penalty parameter approximation was given in detail with the iterative functions. The numerical results illustrate the effectiveness of the proposed method.
- 4th-order variational inequality /
- curvature obstacle /
- alternating direction multiplier method /
- self-adaptive rule

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图 1 u的数值解

Figure 1. Numerical solutions of u

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图 2 －Δu的数值解

Figure 2. Numerical solutions of －Δu

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图 3 精确解u

Figure 3. Analytical solutions of u

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图 4 逐点误差

Figure 4. Pointwise errors between numerical and analytical solutions

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表 1 算法随步长变化所需迭代次数的情况

Table 1. The numbers of iterations required for the algorithm to change with the step size

ρ	algorithm 1 (ADMM)				algorithm 2 (SADMM)
ρ	h=1/10	h=1/20	h=1/40	h=1/80	h=1/10	h=1/20	h=1/40	h=1/80
10^－2	*	*	*	*	25	28	24	28
10^－1	*	*	*	*	26	34	29	34
10⁰	47	59	93	101	30	39	33	39
10¹	*	*	*	*	32	41	35	41
10²	*	*	*	*	33	42	36	42
10³	*	*	*	*	34	43	37	43
10⁴	*	*	*	*	35	44	38	44

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表 2 算法随步长变化所需CPU时间情况

Table 2. CPU times required for the algorithm to change with the step size

ρ	algorithm 1 (ADMM)				algorithm 2 (SADMM)
ρ	h=1/10	h=1/20	h=1/40	h=1/80	h=1/10	h=1/20	h=1/40	h=1/80
10^－2	*	*	*	*	0.117 5	0.255 1	1.911 9	80.095 4
10^－1	*	*	*	*	0.020 0	0.174 4	2.126 9	95.011 4
10⁰	0.035 5	0.276 9	6.770 4	280.995 9	0.021 8	0.178 1	2.521 3	109.381 1
10¹	*	*	*	*	0.024 8	0.168 2	2.647 1	114.965 5
10²	*	*	*	*	0.024 2	0.181 6	2.620 6	117.895 8
10³	*	*	*	*	0.024 3	0.172 3	2.791 4	120.384 8
10⁴	*	*	*	*	0.025 6	0.183 3	2.875 5	123.236 0

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