生物电磁成像中一类非线性椭圆型方程反问题的数值分析与计算

Electromagnetic imaging is an important field in mathematics and physics. Especially, a new medical imaging technique called magnetic resonance electrical impedance tomography (MREIT) is a research hotspot in recent years, with the physical backgrounds of reconstruction the interior electrical properties of a conducting object from some measurement data of electric currents given inside the medium. From mathematical point of view, this is a kind of coefficients inverse problems for elliptic equations, with the difficulties coming from the ill-posedness and nonlinearity. In this project, we will give numerical simulation and analysis for 3-dimensional electrical impedance tomography based on magnetic resonance imaging in the perspective of actual mathematical model. We mainly study the reconstruction algorithms of MREIT with anisotropic conductivity based on the Tikhonov regularization theories and methods, variational method, output least squares, iterative optimization and differential-integral equation method. Moreover, a modified symmetric finite volume element method is proposed solving direct problems at each iteration to lower the computational scale of the reconstruction algorithm. Some numerical experiments are presented to show that the algorithms are effective for the reconstruction. On the other hand, we also discuss reconstruction algorithms for anisotropic conductivity based on induced current magnetic resonance electrical impedance tomography. Our aim is to provide theoretical analysis and effective numerical methods for the MREIT problem. The study on these problems is very important in practice.

磁共振电阻抗成像(MREIT)是数学物理中的重要研究方向，它在生物医学领域有着非常重要的应用价值和应用前景。物理上是利用介质内的部分电流信息重构生物组织的电参数，从数学角度看，这是一类椭圆型偏微分方程系数反问题，难点在于问题的非线性性和不适定性。目前关于各向异性MREIT的理论分析和重构算法的研究还很少，本项目我们将从实际的数学模型出发，结合Tikhonov正则化和NOSER正则化两种方法混合的正则化方法，给出基于电流密度的迭代优化算法重构出各向异性电导率分布，同时给出一种修正的对称有限体积元方法对每一步迭代中的正问题进行求解，大大降低迭代算法的计算规模。通过数值模拟验证算法的可行性和有效性，并对算法的收敛性进行分析。本项目对于深入研究各向异性MREIT的稳定有效的重构算法具有重要的实际意义。