Inverse Problems: Discrete Meets Continuum

反问题：离散遇见连续体

• 批准号: 1411577
• 负责人: Fernando Guevara Vasquez
• 金额: $ 18.16万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1411577&HistoricalAwards=false
• 关键词: Inverse; Problems; Discrete; Meets; Continuum

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.

该项目的主要目的是设计有效的方法，用于从间接测量中成像某些物理量。我们研究的问题是在应用中从皮肤上的电测量中找到人体中的电导率，从皮肤受控振动中找到组织的弹性特性，甚至使用电气测量来监测管道的弹性。我们设计的方法是有效的，因为它们基于模仿问题基础物理的网络。例如，为了成像电导率，我们使用电阻网络。网络上的反问题包括使用边界数据（类似于连续逆问题的Neumann Map类似于Neumann Map）来查找某些（标量或矢量值）节点数量和（标量或矩阵值值或矩阵值）在已知图上定义的边缘数量。我们使用复杂几何光学的想法来猜想，如果可以解决此类问题的线性化，那么非线性问题也可以解决。这将提供一种相对简单的方法来检查此类离散反向问题的可溶性。然后，通过将网络问题解释为管理部分微分方程的离散化，将网络上的可溶性结果和网络上的重建算法用于解决连续性逆问题。