一类抛物型最优控制问题Crank-Nicolson格式的预处理方法  

Preconditioned Iterative Method of Crank-Nicolson Scheme for a Parabolic Optimal Control Problem

作　　者：李直幸 LI Zhixing(School of Mathematical Science,Chongqing Normal University,Chongqing 401331,China)

机构地区：[1]重庆师范大学数学科学学院,重庆401331

出　　处：《乐山师范学院学报》2024年第8期1-11,共11页Journal of Leshan Normal University

摘　　要：针对一类抛物型偏微分方程约束的分布式最优控制问题,通过此问题的一阶最优性条件得到了一组耦合的初值和终值问题,对这组耦合的问题运用Crank-Nicolson格式进行离散,采取块三角预处理策略来求解2*2的线性系统,然后通过匹配策略和基于Kronecker积的分裂技术建立了基于Kronecker积的Schur补的近似,并由此提出了一种基于Crank-Nicolson离散格式的预处理迭代方法,最后用数值实验证明了该方法的精确性和计算效率。结果表明,由于Schur补的近似具有Kronecker积的结构形式,该方法可以有效地实现预处理条件;该方法能极大地减小迭代次数,节省运算时间,降低运算成本,具有较好的结果。研究结果在解决工程技术和社会科学领域的相关最优控制问题具有应用价值。For the distributed optimal control problem constrained by parabolic partial differential equations(PDE),a set of coupled initial and final value problems are obtained by the first-order optimality conditions of the problem.The coupled problems are discretized by the Crank-Nicolson scheme,and the block triangle preconditioned strategy is adopted to solve the 2*2 linear sys-tem.Then,the approximation of Schur complement based on Kronecker products is established by the matching strategy and split-ting technique based on Kronecker products.Based on this,a preconditioned iterative method based on Crank-Nicolson scheme is proposed.Finally,numerical experiments demonstrate the accuracy and eficiency of the proposed method.The results show that the approximation of Schur complement with the structural form of Kronecker products can effectively implement the precondi-tioning condition.The method can greatly help to reduce the number of iterations,save computing time,and reduce computing costs,with good results.The research results have application value in solving related optimal control problems in engineering technology and social sciences.

关 键 词：预处理 CRANK-NICOLSON格式 初值和终值问题 偏微分方程约束优化 

分 类 号：O241.8[理学—计算数学]