Inverse Problems: Discrete Meets Continuum

该项目的主要目的是设计有效的方法，从间接测量中成像某些物理量。我们研究的问题出现在应用中，例如从皮肤上的电测量中找到人体的电导率，从皮肤的受控振动中找到组织的弹性特性，甚至使用电测量来监测管道。我们设计的方法是有效的，因为它们是基于模拟问题底层物理的网络。网络上的反问题是使用边界数据（类似于连续统反问题中的Dirichlet到Neumann映射）来找到定义在已知图上的某些（标量或向量值）节点量和（标量或矩阵值）边量。我们使用的想法，从复杂的几何光学猜想，如果线性化这样的问题是可解的，那么非线性问题也是可解的。这将提供一个相对简单的方法来检查这种离散逆问题的可解性。可解性的结果和重建算法的网络，然后用来解决一个连续的反问题，解释网络问题作为一个离散化的偏微分方程。

The main purpose of this project is to design efficient methods for imaging certain physical quantities from indirect measurements. The problems that we study arise in applications such as finding the electrical conductivity in the human body from electrical measurements on the skin, finding the elastic properties of tissue from controlled vibrations of the skin or even monitoring a pipeline using electrical measurements. The methods we design are efficient because they are based on networks that mimic the underlying physics of the problem. For example, to image the electrical conductivity we use a network of resistors.The inverse problems on networks consist on using boundary data (analogous to the Dirichlet to Neumann map in continuum inverse problems) to find certain (scalar or vector valued) node quantities and (scalar or matrix valued) edge quantities defined on a known graph. We use ideas from Complex Geometric Optics to conjecture that if the linearization of such a problem is solvable, then the non-linear problem is also solvable. This would give a relatively simple way of checking solvability of such discrete inverse problems. Solvability results and reconstruction algorithms on networks are then used to solve a continuum inverse problem by interpreting the network problem as a discretization of the governing partial differential equation.