二维和三维区域中EIT最优控制问题的离散化与收敛性
Discretization and Convergence of the EIT Optimal Control Problem in 2D and 3D Domains
AI总结：
针对电阻抗断层成像逆问题，研究在最优控制框架下用有限差分法完全离散化问题，引入新Sobolev-Hilbert空间，证明二维三维区域中有限维问题序列在泛函和控制层面均收敛于原系数最优控制问题。
摘要：
我们研究电阻抗断层成像（Inverse Electrical Impedance Tomography，EIT）逆问题，该问题旨在基于给定电极电流下m个电极的边界电压测量值，恢复人体内部的电导率和电势。我们在最优控制框架下采用变分公式，将电导率和边界电压作为控制参数，代价泛函为边界电极电流与给定电流模式的范数偏差，以及边界电极电压与测量值的范数偏差。我们使用有限差分法对EIT最优控制问题进行完全离散化，引入了新的Sobolev-Hilbert空间，并证明了在二维和三维区域中，有限维最优控制问题序列在泛函和控制两个层面均收敛于EIT系数最优控制问题。
英文摘要：
We consider Inverse Electrical Impedance Tomography (EIT) problem on recovering electrical conductivity and potential in the body based on the measurement of the boundary voltages on the $m$ electrodes for a given electrode current. The variational formulation is pursued in the optimal control framework, where electrical conductivity and boundary voltages are control parameters, and the cost functional is the norm declinations of the boundary electrode current from the given current pattern and boundary electrode voltages from the measurements. EIT optimal control problem is fully discretized using the method of finite differences. New Sobolev-Hilbert space is introduced, and the convergence of the sequence of finite-dimensional optimal control problems to EIT coefficient optimal control problem is proved both with respect to functional and control in 2- and 3-dimensional domains.
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