抛物方程最优控制问题高效有限元算法及其在分子扩散过程的应用

The aim of this project is to study the high efficient finite element method for optimal control problems governed by parabolic equations and its application in molecular diffusion. The detection of molecular diffusion in the ultra microstructure of living organisms is the frontier of life science and an important direction for the treatment of encephalopathy. It is difficult to solve the characteristic parameters of three dimensional parabolic equation in molecular diffusion process. We study the time and space variables posteriori error estimate of the finite element method and time and spatial parallel the finite element algorithm of the two optimal control problems governed by parabolic equations. Based on the above methods, we develop an efficient adaptive parallel finite element program for solving parabolic equations. Based on the above algorithm, we compile the program software of parameters optimal control and distributed optimal control of parabolic equation, and establish the integrated system of program and nuclear magnetic equipment. The reliability and validity of the above algorithms and programs are verified by the design of medical experiments. These efficient adaptive parallel finite element programs are used to solve the parameters of the diffusion of drug molecules in vivo. It can guide clinical medication and diagnose disease by analyzing parameters.

本项目拟研究扩散方程最优控制问题的高效有限元方法及其在分子扩散过程中应用。生物体超微结构空间内分子扩散的活体检测是生命科学的前沿领域，是治疗脑病突破的重要方向，其中快速数值求解三维抛物方程特征参数存在本质困难。我们研究两种抛物方程最优控制问题的时间和空间变量的有限元后验误差估计算法和时间空间并行算法，在此基础上建立高效并行自适应有限元算法。基于以上算法我们编制抛物方程参数最优控制与分布最优控制问题的程序软件，建立程序与核磁设备的集成系统。 设计医学实验验证以上算法和程序的可靠性和有效性，并应用高效自适应算法软件求解药物分子在活体扩散过程中的方程参数,指导临床用药、诊断疾病。

研究了抛物方程最优控制问题的后验误差估计与并行计算方法，给出最优控制问题高效算法的收敛性理论分析结果。研究了输运方程及其控制问题的后验误差估计。集成了高效并行扩散程序模块，应用到抛物方程最优控制的程序编制。设计动物实验并计算脑细胞外间隙参数，验证程序的可靠性和实用性。研制程序与核磁设备的集成系统，建立三维可视化的药物分子扩散与显示分析平台，利用数值模拟程序指导经脑间质途径给药，初步实现了临床应用。