Some inverse problems: increasing stability and drift-diffusion models

该项目的目标是提高数值算法的分辨率，用于从生物医学，经济学，生物物理学和材料科学中发现的各种物理领域的外部测量中恢复源，障碍物和介质特性。特别是，这些结果将大大提高一种廉价，快速和安全的诊断方法的质量，称为电阻抗断层扫描。具体应用的例子包括推导用于发现金融市场波动和监测经济的算法，执行离子通道中的蛋白质的非侵入性评估，以及探测弹性结构的残余应力，这可以确定材料的老化或缺陷以及地球的弹性参数。该项目通过将完整的麦克斯韦系统的高频数据简化为矢量薛定谔方程并使用该系统的复杂和真实的几何光学解来寻求对该方法的改进。为了取得进展，主要研究人员使用各向异性减少主要是上三角系统，并使用标量Carleman估计中的第二个大参数。目前，很少有分析结果Carleman估计，描述各向异性系统的唯一性和稳定性，包括具有挑战性和重要的情况下，横观各向同性弹性。复杂的漂移扩散系统，如半导体和离子通道的反问题，丰富的数学挑战，包括分析条件的唯一性和可靠的数值算法。主要研究者通过使用各种渐近简化，对偶问题和类似于标量椭圆和抛物方程逆障碍问题中使用的方法来解决这些问题。

The goal of this project is to improve the resolution of numerical algorithms for the recovery of sources, obstacles, and medium characteristics from exterior measurements of various physical fields found in biomedicine, economics, geophysics, and material science. In particular, the results will dramatically enhance the quality of a cheap, fast, and safe diagnostic method called electrical impedance tomography. Examples of specific applications include deriving algorithms for finding volatility of financial markets and monitoring the economy, performing the non-invasive evaluation of protein in ion channels, and prospecting elastic structures for residual stress, which can determine the aging or defects of materials and the elastic parameters of the Earth.Currently, there is a great need for better resolution in electrical impedance tomography. This project seeks improvements for this method using higher frequency data for the complete Maxwell system by its reduction to a vectorial Schroedinger equation and using complex and real geometrical optics solutions of this system. To make progress, the principal investigator uses anisotropic reductions to mainly upper-triangular systems and uses the second large parameter in scalar Carleman estimates. At present, there are few analytical results on Carleman estimates, describing the uniqueness and stability of anisotropic systems, including the challenging and important case of transversely isotropic elasticity. Inverse problems for complicated drift-diffusion systems, such as semiconductors and ion channels, abound with mathematical challenges including analytic conditions for uniqueness and reliable numerical algorithms. The principal investigator addresses these by using various asymptotic simplifications, dual problems, and methods similar to those used in inverse obstacle problems for scalar elliptic and parabolic equations.