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An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations

Abstract: Based on the alternating direction method of multipliers (ADMM), we develop three numerical algorithms incrementally for solving the optimal control problems constrained by random Helmholtz equations. First, we apply the standard Monte Carlo technique and finite element method for the random and spatial discretization, respectively, and then ADMM is used to solve the resulting system. Next, combining the multi-modes expansion, Monte Carlo technique, finite element method, and ADMM, we propose the second algorithm. In the third algorithm, we preprocess certain quantities before the ADMM iteration, so that nearly no random variable is in the inner iteration. This algorithm is the most efficient one and is easy to implement. The error estimates of these three algorithms are established. The numerical experiments verify the efficiency of our algorithms.

简历:张凯,男,1976年9月生,吉林大学数学学院计算数学系教授。

张凯教授1999年本科毕业于吉林大学数学系,2006年获吉林大学与香港中文大学联合培养博士学位,博士论文被评为吉林省优秀博士论文。2008-2010年赴密歇根州立大学开展博士后研究。张凯教授先后赴伊利诺伊州立大学,奥本大学等开展合作研究,主要研究兴趣为偏微分方程的数值解法,主要从事SPDE控制优化问题的数值方法,期权定价,随机麦克斯韦方程和随机声波方程的研究。先后主持国家自然科学基金等项目8项,发表论文40余篇,其中SCI论文30余篇。