半无界退化抛物系统的参数识别问题的最优控制与数值模拟

项目名称： 半无界退化抛物系统的参数识别问题的最优控制与数值模拟

项目编号： No.11461039

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 邓醉茶

作者单位： 兰州交通大学

项目金额： 36万元

中文摘要： 本项目研究一类半无界退化抛物型方程的参数识别问题。此类问题在人口预测与控制，多孔介质流体力学，气候演化，以及金融衍生产品定价等许多应用科学领域有重要意义。相对于其他参数反演问题，本项目有两个主要特点：（1）方程的主项系数在左边界处退化为零；（2）定解域是半无界的。系数的退化性会导致边界条件的缺失，而定解域的无界性则会给反问题的数值求解带来本质性困难，通常的计算方法，如有限元方法和有限差分方法都不能直接应用于数值求解这类问题。我们的研究主要分为理论和数值计算两个方面。理论方面：研究反问题的解的唯一性，以及基于Tikhonov正则化理论框架下的最优控制问题解的存在性，唯一性，稳定性和收敛性；数值计算方面：首先利用人工边界方法及有限差分方法求正问题的数值解，进而设计迭代算法求反问题的数值解，进行数值试验，并作误差分析。

中文关键词： 无界域；退化抛物型方程；参数识别；最优控制；数值模拟

英文摘要： This project studies the parameter-identification problems for degenerate parabolic equations in semi-infinite domain. These kinds of problems have important applications in many fields of applied science, such as population prediction and control, porous media fluid mechanics, climate evolution and financial derivative pricing,etc. Compared with other parameter inversion problems, there are two main features for the project: (1) the principle coefficient of the equation degenerates into zero on the left boundary; (2) the domain is semi-infinite. The degeneracy of the coefficient may cause the boundary condition missing, while the unboundedness of the domain may bring essential difficulties for the numerical computation of the inverse problem. Common computation methods, such as the finite element method and the finite difference method, can not be directly applied to obtain the numerical solutions of these kinds of problems. Our work can be divided into two parts. In the theoretical part, we will study the uniqueness of the solution for the inverse problem. Correspondingly, on the basis of Tikhonov regularization framework, the existence, uniqueness, stability and convergence for the solution of the optimal control problem will be considered. In the numerical part, the numerical solution for the forward problem will be obtained by the artificial boundary method and the finite difference method. Then, we will design iteration algorithms for the inverse problem, perform numerical experiments and do error analysis.

英文关键词： unbounded domain;degenerate parabolic equation;parameter identification;optimal control;numerical simulation