Forward and Inverse Problems of Electromagnetics: Novel Algorithms and Their Implementations

计算电磁学（CEM）是一门结合了电气工程、数学和计算机科学的基础学科。它能够为微电子电路、新材料、散射和辐射系统、室内和细胞级通信系统以及其他应用中的无线链路的电磁分析构建方法和计算工具。这种分析对于创建提供更高级别连接性、安全性和医疗保健的新产品和服务至关重要。提出的方案解决当前的挑战，开发新的算法，并探索在CEM的正问题和反问题的方法之间的重要关系。构建新的快速算法，打破CEM中所需数据随问题规模的指数增长。“维度的诅咒”)是这个节目的主题之一。这种算法将使解决比以前更复杂和更大规模的电磁分析问题成为可能。开发新的电磁学积分方程及其解决方案的数值方案，可以解决诸如低频不稳定性，伪共振，过采样击穿等基本挑战，这些问题一直困扰着基于经典电磁学积分方程的计算软件。所提出的方法将大大增强现有的学术和商业计算软件包，并导致新一代的电磁求解器具有显著提高的鲁棒性和容量。消除微波层析成像（MWT）逆问题中的病态性（解的非唯一性）是本研究计划的另一个主题。几十年来，这种不健康的姿势一直阻碍着微波与x射线、超声波和核磁共振成像一起被用作主流成像方式。我们将利用最近出现的生物启发天线系统的原型，该系统已被证明具有前所未有的指向性。这些新的成像系统有望消除现有系统中的病态特征，从而使它们无法达到理想的分辨率。提出的新的MWT方法有望在放射学、机械系统的无损材料检测、安全筛查、遥感和其他领域找到广泛的应用。

Computational electromagnetics (CEM) is foundational discipline that combines electrical engineering, mathematics, and computer science. It enables construction of methods and computational tools for electromagnetic analysis of microelectronic circuits, novel materials, scattering and radiation systems, wireless links in both indoors and cell level communication systems and other applications. Such analysis is critical for creation of new products and services providing higher levels of connectivity, security, and healthcare. Proposed program addresses current challenges, develops new algorithms, and explores important relationships between the methods in the forward and inverse problems of CEM. Construction of novel fast algorithms that break exponential growth in required data with the size of problem in CEM (a.k.a. 'curse of dimensionality') is one theme of the program. Such algorithms will make possible to solve more sophisticated and larger-scale problems of electromagnetic analysis than was possible previously. Development of new integral equations of electromagnetics and numerical schemes for their solution that can address fundamental challenges such as low-frequency instability, the presence of spurious resonances, oversampling breakdown and others, which have been plaguing computational software based on the classical integral equations of electromagnetics. The proposed methods will significantly enhance existing academic and commercial computational software packages and lead to new generation of electromagnetic solvers with drastically increased robustness and capacity. Elimination of ill-posedness (non-uniqueness of solution) in the inverse problems of microwave tomography (MWT) is another theme of the research program. Such ill-posedness for decades has been preventing microwaves from being used as main stream imaging modality alongside X-rays, ultrasounds, and MRI. We will exploit recently emerging prototypes of biology inspired antenna systems which have been demonstrated to possess unprecedented directivity. These new imaging system promises to eliminate ill-posedness featured in existing systems that prevents them from achieving desirable resolution. Proposed new MWT approach promises to find broad applications in radiology, non-destructive material testing of mechanical systems, security screening, remote sensing, and other areas.