We will discuss the solution of eigenvalue problems associated with partial differential equations (PDE)s that can be written in the generalised form Ax = λMx, where the matrices A and/or M may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilised formulations are used for the numerical approximation of PDEs. With the help of classical numerical examples we will show that the presence of one (or both) parameters can produce unexpected results.

Hessian-dependent functionals play a pivotal role in a wide latitude of problems in mathematics. Arising in the context of differential geometry and probability theory, this class of problems find applications in the mechanics of deformable media (mostly in elasticity theory) and the modelling of slow viscous fluids. We study such functionals from three distinct perspectives.

We present a theoretical analysis of the Weak Adversarial Networks (WAN) method, recently proposed in [1, 2], as a method for approximating the solution of partial differential equations in high dimensions and tested in the framework of inverse problems. In a very general abstract framework.

Analogies between financial markets and critical phenomena have long been observed empirically. So far, no convincing theory has emerged that can explain these empirical observations. Here, we take a step towards such a theory by modeling financial markets as a lattice gas.

Wave propagation and scattering problems are of huge importance in many applications in science and engineering - e.g., in seismic and medical imaging and more generally in acoustics and electromagnetics.

In this short course, we will introduce some elements in deriving the hp a posteriori error estimate for a high-order unfitted finite element method for elliptic interface problems. The key ingredient is an hp domain inverse estimate, which allows us to prove a sharp lower bound of the hp a posteriori error estimator.

Simulation tools capable of transient electromagnetic analysis are essential for designing and optimizing electromagnetic, photonic, and optoelectronic devices. In recent years, time-domain differential equation based solvers have found widespread use due to their advantages over integral equation counterparts in analyzing transient electromagnetic field/wave interactions and multiphysics problems. This dissertation develops a group of time-domain differential equation solvers for analyzing transient electromagnetic scattering from penetrable objects and multiphysics phenomena in optoelectronic devices. In addition to providing detailed formulations of these solvers, this dissertation presents numerical examples which demonstrate their accuracy, efficiency, and applicability to real-life problems.

The technological evolution has led to the current high-performing wireless communication systems that we use on a daily basis. However, coping with the increasing demand is becoming more and more challenging, especially since we are approaching the limits of what can be done with the available resources. One of these resources is bandwidth. This spectrum scarcity problem has motivated researchers to explore new frequencies for wireless communications. Due to this reason, the upper radio-frequency (RF) spectrum, from mmWave and THz to optical bands, is being pursued, which is termed as “Extreme Bandwidth Communication.” This conference brings world experts and the brightest minds from academia and industry to present the latest trends, challenges, results, and opportunities in the field of extreme bandwidth communication.

Time-domain methods are preferred over their frequency-domain counterparts for solving acoustic and electromagnetic scattering problems since they can produce wide-band data from a single simulation. Among the time-domain methods, time-domain surface integral equation solvers have recently found widespread use because they offer several benefits over differential equation solvers. This dissertation develops several second-kind surface integral equation solvers for analyzing transient acoustic scattering from rigid and penetrable objects and transient electromagnetic scattering from perfect electrically conducting and dielectric objects. For acoustically rigid, perfect electrically conducting, and dielectric scatterers, fully explicit marching-on-in-time schemes are developed for solving time domain Kirchhoff, magnetic field, and scalar potential integral equations, respectively.

In this thesis, efficient solutions are sought out to fundamental problems in Electromagnetic (EM) imaging that determines the shape, location, and material properties of an (unknown) object of interest in an investigation domain from the scattered field measured away from it. The solution of an EM inverse scattering problem inherently poses two main challenges: (i) non-linearity, since the scattered field is a non-linear function of the material properties and (ii) ill-posedness, since the integral operator has a smoothing effect and the number of measurements is finite in dimension and they are contaminated with noise. The non-linearity is tackled incorporating a multitude of techniques (ranging from Born approximation (linear), inexact Newton (linearized) to complete non-linear iterative Landweber schemes) that can account for weak to strong scattering problems. The ill-posedness of the EM inverse scattering problem is circumvented by formulating the above methods into a minimization problem with a sparsity constraint, which assumes that the dimension of the unknown object relative to the investigation domain is much smaller. Numerical experiments, which are carried out using synthetically generated measurements, show that the images recovered by these sparsity-regularized methods are sharper and more accurate than those produced by existing methods. The methods developed in this work have potential application areas ranging from oil/gas reservoir engineering to biological imaging where sparse domains naturally exist.

Hakan Bagci is an Associate Professor of Electrical Engineering (EE) and Principal Investigator of the Computational Electromagnetics Laboratory (CEML).  His scientific contribution are in advancing high-speed and long-distance communication, energy transfer, and medical imaging. Bagci’s research interests are in various aspects of applied and theoretical computational electromagnetics with emphasis on Time-domain integral-equations and their fast marching-on-in-time-based solutions and solvers to the characterization of wave interactions on complex integrated and electrically large system of photonics and optics. 

Semiconductor-based terahertz (THz) devices such as photoconductive antennas (PCAs) and photomixers (PMs) are widely studied as promising candidates of THz source generation and signal detection. Recent experimental research has shown that using nanostructures in the design of these devices dramatically enhances their optical-to-THz conversion efficiency, possibly allowing their use in widespread industrial applications. However, the nanostructures also increase the complexity of design and fabrication.