Deep learning, the study of multi-layered artificial neural networks, has received tremendous attention over the course of the last few years. Neural networks are now able to outperform humans in a growing variety of tasks and increasingly have an impact on our day-to-day lives. There is a wide range of potential directions to advance deep learning, two of which we investigate in this thesis:(1) O

In this paper, we present an adaptive spectral projection based finite element method to numerically approximate the solution of the wave equation with memory. The adaptivity is not restricted to the mesh (hp-adaptivity), but it is also applied to the size of the computed spectrum (k-adaptivity). The meshes are refined using a residual based error estimator, while the size of the computed spectrum

BackgroundAmbulatory blood pressure (BP) monitoring has long been used to monitor BP in hypertension and lately emerged as a useful tool to detect hypotensive susceptibility in reflex syncope. However, hemodynamic characteristics in reflex syncope have not been sufficiently explored. The present study investigated the differences between ambulatory BP monitoring profiles associated with reflex syn

In this paper, we consider the numerical inverse Laplace transform for distributed order time-fractional equations, where a discontinuous Galerkin scheme is used to discretize the problem in space. The success of Talbot's approach for the computation of the inverse Laplace transform depends critically on the problem's spectral properties and we present a method to numerically enclose the spectrum

In this paper, we compare two approaches to numerically approximate the solution of second-order Gurtin–Pipkin type of integro-differential equations. Both methods are based on a high-order Discontinous Galerkin approximation in space and the numerical inverse Laplace transform. In the first approach, we use functional calculus and the inverse Laplace transform to represent the solution. The spect

Discontinuous Galerkin composite finite element methods (DGCFEM) are designed to tackle approximation problems on complicated domains. Partial differential equations posed on complicated domain are common when there are mesoscopic or local phenomena which need to be modelled at the same time as macroscopic phenomena. In this paper, an optical lattice will be used to illustrate the performance of t

In this paper, we study spectral properties and spectral enclosures for the Gurtin-Pipkin type of integro-differential equations in several dimensions. The analysis is based on an operator function and we consider the relation between the studied operator function and other formulations of the spectral problem. The theory is applied to wave equations with Boltzmann damping.

In this paper, we consider a sorting scheme for potentially spurious scattering resonant pairs in one- and two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can, therefore, be applied to structures with graded materials as well as to configurations including piece-wise

In this paper we consider scattering resonance computations in optics when the resonators consist of frequency dependent and lossy materials, such as metals at optical frequencies. The proposed computational approach combines a novel hp-FEM strategy, based on dispersion analysis for complex frequencies, with a fast implementation of the nonlinear eigenvalue solver NLEIGS. Numerical computations il

This article reviews both recent progress on the mathematics of dispersive and absorptive photonic crystals and well-established results on conservative photonic crystals. The focus is on properties of the photonic band structures and we also provide results that are of importance for the understanding of lossy metal-dielectric photonic crystals.

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of

We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains. We consider a variational formulation and show that the spectrum consists of isolated

In this paper we present equivalence results for several types of unbounded operator functions. A generalization of the concept equivalence after extension is introduced and used to prove equivalence and linearization for classes of unbounded operator functions. Further, we deduce methods of finding equivalences to operator matrix functions that utilizes equivalences of the entries. Finally, a met

In this paper, we discuss problems arising when computing resonances with a finite element method. In the pre-asymptotic regime, we detect for the one dimensional case, spurious solutions in finite element computations of resonances when the computational domain is truncated with a perfectly matched layer (PML) as well as with a Dirichlet-to-Neumann map (DtN). The new test is based on the Lippmann

In this paper we introduce an enclosure of the numerical range of a class of rational operator functions. In contrast to the numerical range the presented enclosure can be computed exactly in the infinite dimensional case as well as in the finite dimensional case. Moreover, the new enclosure is minimal given only the numerical ranges of the operator coefficients and many characteristics of the num

We establish new analytic results for a general class of rational spectral problems. They arise e.g. in modelling photonic crystals whose capability to control the flow of light depends on specific features of the eigenvalues. Our results comprise a complete spectral analysis including variational principles and two-sided bounds for all eigenvalues, as well as numerical implementations. They apply

We present a-posteriori analysis of higher order finite element approximations (hp-FEM) for quadratic Fredholm-valued operator functions. Residual estimates for approximations of the algebraic eigenspaces are derived and we reduce the analysis of the estimator to the analysis of an associated boundary value problem. For the reasons of robustness we also consider approximations of the associated in

We present an algorithm for approximating an eigensubspace of a spectral component of an analytic Fredholm valued function. Our approach is based on numerical contour integration and the analytic Fredholm theorem. The presented method can be seen as a variant of the FEAST algorithm for infinite dimensional nonlinear eigenvalue problems. Numerical experiments illustrate the performance of the algor

Metallic nano-antennas are devices used to concentrate the energy in light into regions that are much smaller than the wavelength. These structures are currently used to develop new measurement and printing techniques, such as optical microscopy with sub-wavelength resolution, and high-resolution lithography. Here, we analyze and design a nano-antenna in a two-dimensional setting with the source b

Galerkin spectral approximation theory for non-self-adjoint quadratic operator polynomials with periodic coefficients is considered. The main applications are complex band structure calculations in metallic photonic crystals, periodic waveguides, and metamaterials. We show that the spectrum of the considered operator polynomials consists of isolated eigenvalues of finite multiplicity with a nonzer