New Frontiers in Computational Electromagnetics: Towards High-Accuracy Solutions and Reliable Microwave Imaging

快速无线通信、可靠的电力传输和生物医学成像是决定数百万加拿大人生活质量的重要技术。这些技术进步的关键是依靠计算电磁学（CEM）的方法和工具的新设备和系统的设计。这些系统包括手机、数字电路板、GPS装置、雷达和断层扫描系统——仅举几例。尽管近年来取得了各种根本性的进步，但目前可用的CEM算法仍然存在各种根本性的缺陷。这些缺点阻碍了新技术的发展，由于设计错误和上市时间延长，这些行业损失了数百万美元。其中最具挑战性的是相关算法的计算复杂性和应用于实际模型的电磁分析时精度不足。提出的研究计划侧重于开发新的算法和高精度求解方法，克服计算复杂性障碍，实现复杂大型结构的鲁棒电磁分析。该计划的主要贡献在于开发了新的、更经济的、更健壮的CEM积分方程公式，以及新的高阶数值解格式。在程序中提出的新的积分方程将允许减少一半的计算模型的大小源于现实模型的离散化。与行业中传统的低阶方法相比，所提出的高阶解方法具有指数级的效率，从而进一步降低了计算复杂度。

Fast wireless communications, reliable power transmission, and biomedical imaging are the important technologies which define the quality of life for millions of Canadians. Critical to the advancements of these technologies is the design of novel devices and systems which rely on the methods and tools of computational electromagnetics (CEM). Among such systems are the cell phones, digital circuit boards, GPS units, radars, and tomography systems - just to name a few. Despite various fundamental advances made in the recent past, currently available algorithms of CEM suffer from various fundamental shortcomings. These shortcomings handicap development of the new technologies costing the industries millions of dollars in losses from faulty designs and prolonged time to market. The most challenging of these are large computational complexity of the pertinent algorithms and their inadequate accuracy when applied to electromagnetic analysis of realistic models. The proposed research program focuses on development of novel algorithms and high-accuracy solution methods, which overcome the computational complexity barrier and enable robust electromagnetic analysis of complex large-scale structures. The key contributions of the program are in development of novel, more economical, and more robust integral equation formulations for CEM as well as the novel higher-order numerical schemes for their solution. The novel integral equations proposed in the program will allow to reduce in half the size of the computational models stemming from the discretization of the realistic models. The proposed higher-order solution methods will further reduce the computational complexity by taking advantage of their exponentially higher efficiency compared to the low-order methods traditionally used in the industry. The other thrust area of the proposed research program is in development of new methods of microwave imaging. The latter suffers from non-uniqueness of the images which can be reconstructed from the information contained in the field scattered by the object of interest. This non-uniqueness stems from the inherent incorrectness (ill-posedness) of the mathematical formulations of the imaging problem. It presents a grand challenge which for decades has been preventing microwaves from being used for reliable non-invasive imaging may it be in biomedical screening of patients for cancer, for underground oil and gas exploration or non-destructive testing of an aircraft structural health. The research program will build on prior work of the applicant which has shown that this non-uniqueness in the image reconstruction using microwaves can be overcome. This novel approach to microwave tomography promises to place it as a viable imaging modality alongside with MRI and X-rays as well as into the fields of underground detection and material testing.