发展型偏微分方程组中基于部分观测数据的系数辨识问题

发展型偏微分方程组的系数辨识问题是受工程领域应用驱动的重要问题之一。在实际问题中，因为观测条件的局限性，仅利用解的部分分量的局部观测数据来获取整个系统中相互作用和影响的全部分量信息，即基于部分观测数据的系数辨识问题，具有重要研究价值。由于测量数据不完整，上述系数辨识问题在条件稳定性和数值计算方面均面临巨大挑战。..本项目拟针对Schrödinger方程组、Maxwell方程和弹性方程组等三类发展型偏微分方程组，研究局部观测解的部分分量的系数辨识问题。特别地，利用单个方程Carleman估计和拟微分算子等研究工具，通过建立满足观测条件的Carleman估计，得到相应系数辨识问题阶数最优的条件稳定性；并对具体问题提出稳定、收敛的数值求解方法，进行数值模拟。最终逐步形成基于部分观测数据的发展型偏微分方程组系数辨识问题的系统研究方法。

The coefficients identification problems (CIPs) of evolution partial differential equations (PDEs) are important issues driven by applications in the engineering. In practical problems, due to the limitation of observable conditions, using only the local observation data of a part of components of the solutions to obtain information of all the components of mutual effects and influence in the whole system, that is, CIPs with partial observational data of a part of components of the solution, is valuable to study. Because of the incompleteness of measurements, the conditional stability and numerical approximation of the above-mentioned coefficients identification problems are facing great challenges, and the related research is still in its infancy...The project devotes to studying the CIPs under the condition of local observational data of a part of components of the solutions in three types of evolution PDEs, i.e., Schrödinger equations, Maxwell equations, and elastic equations. Firstly, establishing Carleman estimates of the corresponding systems under suitable observation conditions. Then, by using Carleman estimates of single equations and some tools, such as pseudo-differential operators and harmonic analysis, to obtain conditional stabilities of optimal order for the corresponding CIPs; and propose stable and convergent numerical methods for the specified problems and simulate the approximation simultaneously. Finally, develop a systematic method for the CIPs of general evolution PDEs with partial observational data.