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，以便从生物医学、经济学、地球物理学和材料科学中发现的各种物理场的外部测量中恢复源、障碍物和介质特征。特别是，研究结果将显著提高一种廉价、快速、安全的诊断方法——电阻抗断层扫描的质量。具体应用的例子包括推导用于发现金融市场波动性和监测经济的算法，对离子通道中的蛋白质进行非侵入性评估，以及勘探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。目前，电阻抗层析成像对分辨率的要求很高。本项目通过将完整麦克斯韦系统的高频数据简化为矢量薛定谔方程，并使用该系统的复和实几何光学解，寻求对该方法的改进。为了取得进展，首席研究员主要对上三角系统使用各向异性约简，并在标量Carleman估计中使用第二大参数。目前，描述各向异性系统的唯一性和稳定性的Carleman估计的分析结果很少，包括横向各向同性弹性的具有挑战性和重要的情况。复杂漂移扩散系统的反问题，如半导体和离子通道，充满了数学挑战，包括唯一性的分析条件和可靠的数值算法。主要研究者通过使用各种渐近简化、对偶问题和类似于用于标量椭圆和抛物方程的逆障碍问题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.