论文标题
有限的视图中电导率方程的解决方案的雅各布
Jacobian of solutions to the conductivity equation in limited view
论文作者
论文摘要
杂种反问题（例如声音层析成像或电流密度成像）的目的是重建一个只能从其外部访问的域中的电导率。在反转程序中，电导率方程的解决方案起着核心作用。特别是，重要的是，解决方案的雅各布式是不变的。在本文中，我们介绍了二维有限视图设置，其中只能通过非零的dirichlet条件来控制域的边界的一部分，而在其余边界上则存在零dirichlet条件。对于这种设置，我们在边界函数上提出了足够的条件，以使相应解决方案的雅各布式无逐渐呈现。在这方面，我们允许不连续的边界函数，这需要在加权Sobolev空间中使用解决方案。我们实施了从功率密度数据重建电导率的过程，并研究了这种有限的视图设置如何影响Jacobian和重建质量。
The aim of hybrid inverse problems such as Acousto-Electric Tomography or Current Density Imaging is the reconstruction of the electrical conductivity in a domain that can only be accessed from its exterior. In the inversion procedure, the solutions to the conductivity equation play a central role. In particular, it is important that the Jacobian of the solutions is non-vanishing. In the present paper we address a two-dimensional limited view setting, where only a part of the boundary of the domain can be controlled by a non-zero Dirichlet condition, while on the remaining boundary there is a zero Dirichlet condition. For this setting, we propose sufficient conditions on the boundary functions so that the Jacobian of the corresponding solutions is non-vanishing. In that regard we allow for discontinuous boundary functions, which requires the use of solutions in weighted Sobolev spaces. We implement the procedure of reconstructing a conductivity from power density data numerically and investigate how this limited view setting affects the Jacobian and the quality of the reconstructions.
