A Coupled Hybridizable Discontinuous Galerkin and Boundary Integral Method for Analyzing Electromagnetic Scattering

A coupled hybridizable discontinuous Galerkin (HDG) and boundary integral (BI) method is proposed to efficiently analyze electromagnetic scattering from inhomogeneous/composite objects. The coupling between the HDG and the BI equations is realized using the numerical flux operating on the equivalent current and the global unknown of the HDG. This approach yields sparse coupling matrices upon discretization. Inclusion of the BI equation ensures that the only error in enforcing the radiation conditions is the discretization. However, the discretization of this equation yields a dense matrix, which prohibits the use of a direct matrix solver on the overall coupled system as often done with traditional HDG schemes. To overcome this bottleneck, a "hybrid" method is developed. This method uses an iterative scheme to solve the overall coupled system but within the matrix-vector multiplication subroutine of the iterations, the inverse of the HDG matrix is efficiently accounted for using a sparse direct matrix solver. The same subroutine also uses the multilevel fast multipole algorithm to accelerate the multiplication of the guess vector with the dense BI matrix. The numerical results demonstrate the accuracy, the efficiency, and the applicability of the proposed HDG-BI solver

Broadband Multilevel Fast Multipole Methods

Numerical simulations of electromagnetic fields are very important for a plethora of modern applications like antenna design, wireless communication systems, optical systems, high-frequency circuits and so on. As a consequence, there is much interest in finding algorithms that make these simulations as computationally efficient as possible. One of the leading classes of algorithms consists of the so-called Fast Multipole Methods. These methods use a subdivision of the geometry into boxes on multiple levels, in combination with a decomposition of the Green function. For high frequency simulations, where the wavelength is smaller then the smallest features of the geometry, a propagating plane wave decomposition leads to a very efficient algorithm. Unfortunately, this decomposition fails when the geometry contains features smaller than the wavelength, which is the case for broadband simulations. Broadband simulations are becoming increasingly important, for example in the simulation of high frequency printed circuit boards and microwave circuits, metamaterials or the scattering of radar waves off complex shapes. Because of the failure of the propagating plane wave decomposition, performing broadband simulations requires the construction of a hybrid algorithm which uses the propagating plane wave decomposition when the boxes are large enough and some low frequency decomposition when they are not. However, the known low frequency decompositions are usually suboptimal compared to the theoretical performance of the propagating plane wave decomposition. In this work, the focus will be on these low frequency decompositions. First, an improvement over a known low frequency decomposition (the spectral decomposition) is presented. Among other techniques, the well-known Beltrami decomposition of electromagnetic fields is shown to significantly reduce the computational burden in this scheme. Secondly, entirely novel ways of decomposing the Green function are developed in both two and three dimensions. These decompositions use evanescent plane waves, so they can handle small boxes. Nevertheless, they have the same convergence characteristics as the propagating plane wave decomposition. Therefore, these decompositions are also very efficient. Finally, the novel techniques are applied in the full-wave homogenization of various metamaterials

Fast solution to electromagnetic scattering by large-scale complicated structures using adaptive integral method

Ph.DDOCTOR OF PHILOSOPH

Network Approaches to the Study of Genomic Variation in Cancer

Advances in genomic sequencing technologies opened the door for a wider study of cancer etiology. By analyzing datasets with thousands of exomes (or genomes), researchers gained a better understanding of the genomic alterations that confer a selective advantage towards cancerous growth. A predominant narrative in the field has been based on a dichotomy of alterations that confer a strong selective advantage, called cancer drivers, and the bulk of other alterations assumed to have a neutral effect, called passengers. Yet, a series of studies questioned this narrative and assigned potential roles to passengers, be it in terms of facilitating tumorigenesis or countering the effect of drivers. Consequently, the passenger mutational landscape received a higher level of attention in attempt to prioritize the possible effects of its alterations and to identify new therapeutic targets. In this dissertation, we introduce interpretable network approaches to the study of genomic variation in cancer. We rely on two types of networks, namely functional biological networks and artificial neural nets. In the first chapter, we describe a propagation method that prioritizes 230 infrequently mutated genes with respect to their potential contribution to cancer development. In the second chapter, we further transcend the driver-passenger dichotomy and demonstrate a gradient of cancer relevance across human genes. In the last two chapters, we present methods that simplify neural network models to render them more interpretable with a focus on functional genomic applications in cancer and beyond