Nonlinear spectral problems in electromagnetics: asymptotics and inversion.

尽管在计算速度方面取得了非凡的进步，但在从生物医学成像到地球物理勘探和材料缺陷的无损检测等广泛的应用中，电磁学中的3D成像问题的解决仍然构成了重大挑战。例如，通过从钻孔中的探头发送电磁源并读取产生的电位，原则上可以推断出石油矿床的位置，因为石油矿床的电磁特性与周围地面不同。然而，对于大规模的非线性问题，蛮力计算方法往往很麻烦，甚至难以处理。在提出的研究中，数学方法将被开发，我们将找到明确的公式，用于数据对电磁参数的依赖，从而导致快速算法微调到特定的几何形状，并具有明确的物理解释。几名学生将在项目过程中接受培训，其中包括一名全职博士生。该项目还将为本科生提供为期六个月的机会，深入研究一个研究问题，并可能增加他们在数学或相关领域攻读更高学位的可能性。本提案的目的是通过建立基于材料性质摄动的新的渐近公式来解决电磁学中的一些非线性谱问题和相关的逆问题。该项目将包括五个主要部分：(i)线性和非线性小体积电磁散射体存在时共振值的渐近公式。渐近性将基于解易于计算的简化极限问题的摄动。（ii）在薄的高对比材料存在下的电场和共振的尺寸简化公式。这个公式将提供一个高对比度的玻恩近似，我们可以反转它。（iii）对不同材料性质的透射特征值的扰动进行显式计算。这将包括材料中的小的不均匀性、空洞或孔洞，以及周期性的微观结构。这些公式将揭示从读取远场传输特征值可以恢复的材料特性。(4)超声调制光学层析成像的快速非线性反演技术和唯一性和稳定性理论的发展，这是一种新的混合成像方式，为光学参数的恢复提供非线性内部数据。(v)非线性弦的逆谱方法，德雷克塞尔大学本科生课题。材料摄动渐近在非线性谱问题中的应用是所有这些项目的一个统一主题。

Despite extraordinary advances in computing speed, the resolution of 3D imaging problems in electromagnetics continues to pose significant challenges in a wide range of applications, from biomedical imaging to geophysical exploration and non-destructive detection of material defects. By sending electromagnetic sources from probes in boreholes and reading back resulting potentials, for instance, one can in principle infer the location of oil deposits, which have different electromagnetic properties than the surrounding ground. Brute-force computational approaches, however, are often cumbersome or even intractable for large scale non-linear problems. In the proposed research, mathematical methods will be developed where we find explicit formulas for the dependence of the data on electromagnetic parameters, leading to fast algorithms fine-tuned to specific geometries and having clear physical interpretations. Several students will be trained in the course of the project, including a full time PhD student. The project will also provide undergraduate students the opportunity to spend six months delving deeply into a research problem, and may increase the likelihood they will pursue a higher degree in mathematics or a related field.The objective of this proposal is to solve several nonlinear spectral problems and related inverse problems in electromagnetics by developing novel asymptotic formulas based on material property perturbations. The project will involve five main components: (i) Asymptotic formulas for resonance values in the presence of linear and nonlinear small volume electromagnetic scatters. The asymptotics will be based on perturbations of simplified limiting problems whose solutions are easily computed. (ii) A dimensionally reduced formulation for the electric field and resonances in the presence of thin high contrast materials. This formulation will provide a high contrast Born approximation which we can invert. (iii) Explicit calculations of the perturbations of transmission eigenvalues for varying material properties. This will include small inhomogeneities, voids, or holes in the materials, and periodic microstructures. The formulas will reveal what material properties can be recovered from reading transmission eigenvalues in the far field. (iv) The development of a fast nonlinear inversion technique and a theory of uniqueness and stability in ultrasound modulated optical tomography, a novel hybrid imaging modality which provides nonlinear internal data for the recovery of optical parameters. (v) An inverse spectral method for nonlinear strings, a project for Drexel undergraduate students. The application of material perturbation asymptotics to nonlinear spectral problems is a unifying theme in all of these projects.