On variational eigenvalue approximation of semidefinite operators

Eigenvalue problems for semidefinite operators with infinite dimensional kernels appear for instance in electromagnetics. Variational discretizations with edge elements have long been analyzed in terms of a discrete compactness property. As an alternative, we show here how the abstract theory can be developed in terms of a geometric property called the vanishing gap condition. This condition is shown to be equivalent to eigenvalue convergence and intermediate between two different discrete variants of Friedrichs estimates. Next we turn to a more practical means of checking these properties. We introduce a notion of compatible operator and show how the previous conditions are equivalent to the existence of such operators with various convergence properties. In particular the vanishing gap condition is shown to be equivalent to the existence of compatible operators satisfying an Aubin-Nitsche estimate. Finally we give examples demonstrating that the implications not shown to be equivalences, indeed are not.Comment: 26 page

The Final Remnant of Binary Black Hole Mergers: Multipolar Analysis

Methods are presented to define and compute source multipoles of dynamical horizons in numerical relativity codes, extending previous work from the isolated and dynamical horizon formalisms in a manner that allows for the consideration of horizons that are not axisymmetric. These methods are then applied to a binary black hole merger simulation, providing evidence that the final remnant is a Kerr black hole, both through the (spatially) gauge-invariant recovery of the geometry of the apparent horizon, and through a detailed extraction of quasinormal ringing modes directly from the strong-field region.Comment: 12 pages, 13 figures. Published version. Some references have been added and reordered, and the figures cleaned up

Parallel eigensolvers in plane-wave Density Functional Theory

We consider the problem of parallelizing electronic structure computations in plane-wave Density Functional Theory. Because of the limited scalability of Fourier transforms, parallelism has to be found at the eigensolver level. We show how a recently proposed algorithm based on Chebyshev polynomials can scale into the tens of thousands of processors, outperforming block conjugate gradient algorithms for large computations

Fast randomized iteration: diffusion Monte Carlo through the lens of numerical linear algebra

We review the basic outline of the highly successful diffusion Monte Carlo technique commonly used in contexts ranging from electronic structure calculations to rare event simulation and data assimilation, and propose a new class of randomized iterative algorithms based on similar principles to address a variety of common tasks in numerical linear algebra. From the point of view of numerical linear algebra, the main novelty of the Fast Randomized Iteration schemes described in this article is that they work in either linear or constant cost per iteration (and in total, under appropriate conditions) and are rather versatile: we will show how they apply to solution of linear systems, eigenvalue problems, and matrix exponentiation, in dimensions far beyond the present limits of numerical linear algebra. While traditional iterative methods in numerical linear algebra were created in part to deal with instances where a matrix (of size $\mathcal{O}(n^2)$) is too big to store, the algorithms that we propose are effective even in instances where the solution vector itself (of size $\mathcal{O}(n)$) may be too big to store or manipulate. In fact, our work is motivated by recent DMC based quantum Monte Carlo schemes that have been applied to matrices as large as $10^{108} \times 10^{108}$. We provide basic convergence results, discuss the dependence of these results on the dimension of the system, and demonstrate dramatic cost savings on a range of test problems.Comment: 44 pages, 7 figure

A survey of topics related to Functional Analysis and Applied Sciences

This survey is the result of investigations suggested by recent publications on functional analysis and applied sciences. It contains short accounts of the above theories not usually combined in a single document and completes the work of D. Huet 2017. The main topics which are dealt with involve spectrum and pseudospectra of partial differential equations, Steklov eigenproblems, harmonic Bergman spaces, rotation number and homeomorphisms of the circle, spectral flow, homogenization. Applications to different types of natural sciences such as echosystems, biology, elasticity, electromagnetisme, quantum mechanics, are also presented. It aims to be a useful tool for advanced students in mathematics and applied sciences

Zolotarev Quadrature Rules and Load Balancing for the FEAST Eigensolver

The FEAST method for solving large sparse eigenproblems is equivalent to subspace iteration with an approximate spectral projector and implicit orthogonalization. This relation allows to characterize the convergence of this method in terms of the error of a certain rational approximant to an indicator function. We propose improved rational approximants leading to FEAST variants with faster convergence, in particular, when using rational approximants based on the work of Zolotarev. Numerical experiments demonstrate the possible computational savings especially for pencils whose eigenvalues are not well separated and when the dimension of the search space is only slightly larger than the number of wanted eigenvalues. The new approach improves both convergence robustness and load balancing when FEAST runs on multiple search intervals in parallel.Comment: 22 pages, 8 figure

Riesz-projection-based methods for the numerical simulation of resonance phenomena in nanophotonics

Resonance effects are ubiquitous in physics and essential for understanding wave propagation and interference. In the field of nanophotonics, devices are often based on the strong confinement of light by resonances. The numerical simulation of resonances plays a crucial role for the design and optimization of the devices. The resonances are electromagnetic field solutions to the time-harmonic source-free Maxwell's equations with loss mechanisms. The corresponding eigenproblems are non-Hermitian due to the losses leading to complex-valued eigenvalues. The material dispersion, which is typically significant in nanophotonics, results in nonlinear eigenproblems. In this thesis, we develop an approach based on Riesz projections for the expansion of electromagnetic fields caused by light sources into resonances. The Riesz projection expansion is computed by contour integration in the complex frequency plane. The numerical realization essentially relies on solving Maxwell's equations with a source term, meaning solving linear systems of equations. For this, Maxwell's equations are directly evaluated at the given frequencies on the integration contours, which implies that linearization of the corresponding nonlinear eigenproblems is not required. This makes Riesz-projection-based approaches a natural choice for dealing with eigenproblems from the field of nanophotonics. We further extend the Riesz projection expansion approach to optical far-field quantities, which is not straightforward due to the spatial divergence of the resonances with increasing distance from the underlying resonators. Based on the ideas of the Riesz projection expansion, we introduce approaches for the calculation of physically relevant eigenvalues and for computing eigenvalue sensitivities. Physically relevant means that the eigenvalues are significant with respect to the resonance expansion of the physical observable of interest. By using physical solutions to Maxwell's equations for the contour integration, the developed numerical methods have a strong relation to physics. The methods can be applied to any material system and to any measurable physical quantity that can be derived from the electric field. We apply the numerical methods to several recent nanophotonic applications, for example, single-photon sources from the field of quantum technology, plasmonic nanostructures characterized by nonlocal material properties, and nanoantennas based on bound states in the continuum. The approaches introduced in this thesis are developed for nanophotonic systems, but can be applied to any resonance problem.Resonanzeffekte treten in allen physikalischen Systemen auf, die durch Wellen beschrieben werden, und sie sind für die Beschreibung von Wellenausbreitung und Interferenz unerlässlich. Auf dem Gebiet der Nanophotonik basieren viele Geräte auf den durch Lichtquellen angeregten Resonanzen mit ihren stark erhöhten elektromagnetischen Feldern. Die numerische Simulation von Resonanzen ist ein wichtiges Hilfsmittel für die Entwicklung und Optimierung der Geräte. Die Resonanzen sind die Lösungen der zeitharmonischen quellenfreien Maxwell-Gleichungen mit Verlustmechanismen. Die entsprechenden Eigenwertprobleme sind aufgrund der Verluste nicht-Hermitesch, was zu komplexwertigen Eigenwerten führt. Die Materialdispersion, die in der Nanophotonik typischerweise signifikant ist, führt zu nichtlinearen Eigenwertproblemen. In dieser Dissertation entwickeln wir einen auf der Riesz-Projektion basierenden Ansatz für die Expansion von elektromagnetischen Feldern, die von Lichtquellen erzeugt werden, in Resonanzen. Wir berechnen die Riesz-Projektionen durch Konturintegration in der komplexen Frequenzebene. Die numerische Realisierung basiert im Wesentlichen auf der Lösung der Maxwell-Gleichungen mit einem Quellterm, das heißt der Lösung von linearen Gleichungssystemen. Dabei werden die Maxwell-Gleichungen direkt bei den gegebenen Frequenzen auf den Integrationskonturen ausgewertet, sodass eine Linearisierung der entsprechenden nichtlinearen Eigenwertprobleme nicht erforderlich ist. Das macht die auf der Riesz-Projektion basierenden Methoden zu einer natürlichen Wahl für die Behandlung von Eigenwertproblemen aus dem Bereich der Nanophotonik. Wir erweitern den Ansatz der Riesz-Projektions-Expansion auf optische Größen im Fernfeld, was aufgrund der räumlichen Divergenz der Resonanzen mit zunehmender Entfernung von den zugrunde liegenden Resonatoren problematisch ist. Basierend auf den Ideen der Riesz-Projektions-Expansion entwickeln wir außerdem Methoden zur Berechnung physikalisch relevanter Eigenwerte und zur Berechnung von Sensitivitäten von Eigenwerten. Physikalisch relevant bedeutet, dass die Eigenwerte in Bezug auf die Resonanzexpansion der interessierenden physikalischen Größe signifikant sind. Durch die Verwendung physikalischer Lösungen der Maxwell-Gleichungen für die Konturintegration haben die entwickelten numerischen Methoden einen starken Bezug zur zugrunde liegenden Physik. Die Methoden können auf jedes Materialsystem und auf jede messbare physikalische Größe angewendet werden, die sich aus dem elektrischen Feld herleiten lässt. Wir wenden die numerischen Methoden auf mehrere aktuelle nanophotonische Strukturen an, wie zum Beispiel Einzelphotonenquellen aus dem Bereich der Quantentechnologie, plasmonische Nanostrukturen, die sich durch nichtlokale Materialeigenschaften auszeichnen, und Nanoantennen, die auf gebundenen Zuständen im Kontinuum basieren. Die in dieser Dissertation vorgestellten Ansätze werden für nanophotonische Systeme entwickelt, lassen sich aber auf jedes Resonanzproblem anwenden