Covariance regularity and H-matrix approximation for rough random fields

In an open, bounded domain D ⊂ R n with smooth boundary ∂ D or on a smooth, closed and compact, Riemannian n -manifold M ⊂ R n +1 , we consider the linear operator equation Au = f where A is a boundedly invertible, strongly elliptic pseudodifferential operator of order r ∈ R with analytic coefficients, covering all linear, second order elliptic PDEs as well as their boundary reductions. Here, f ∈ L 2 ( Ω ; H t ) is an H t -valued random field with finite second moments, with H t denoting the (isotropic) Sobolev space of (not necessarily integer) order t modelled on the domain D or manifold M , respectively. We prove that the random solution’s covariance kernel K u = ( A − 1 ⊗ A − 1 ) K f on D × D (resp. M×M ) is an asymptotically smooth function provided that the covariance function K f of the random data is a Schwartz distributional kernel of an elliptic pseudodifferential operator. As a consequence, numerical H -matrix calculus allows deterministic approximation of singular covariances K u of the random solution u = A − 1 f ∈ L 2 ( Ω ; H t − r ) in D × D with work versus accuracy essentially equal to that for the mean field approximation in D, overcoming the curse of dimensionality in this case

A fast sparse grid based space-time boundary element method for the nonstationary heat equation

This article presents a fast sparse grid based space-time boundary element method for the solution of the nonstationary heat equation. We make an indirect ansatz based on the thermal single layer potential which yields a first kind integral equation.This integral equation is discretized by Galerkin’s method with respect to the sparse tensor product of the spatial and temporal ansatz spaces. By employing the $\mathcal{H}$-matrix and Toeplitz structure of the resulting discretized operators, we arrive at an algorithm which computes the approximate solution in a complexity that essentially corresponds to that of the spatial discretization. Nevertheless, the convergence rate is nearly the same as in case of a traditional discretization in full tensor product spaces

Boundary integral equations in Kinetic Plasma Theory

In this thesis, we use boundary integral equations (BIE) as a powerful tool to gain new insights into the dynamics of plasmas. On the theoretical side, our work provides new results regarding the oscillation of bounded plasmas. With the analytical computation of the frequencies for a general ellipsoid we contribute a new benchmark for numerical methods. Our results are validated by an extensive numerical study of several three-dimensional problems, including a particle accelerator with complex geometry and mixed boundary conditions. The use of Boundary Element Methods (BEM) reduces the dimension of the problem from three to two, thus drastically reducing the number of unknowns. By employing hierarchical methods for the computation of the occurring nonlocal sums and integral operators, our method scales linearly with the number of particles and the number of surface triangles, where the error decays exponentially in the expansion parameter. Furthermore, our method allows the pointwise evaluation of the electric field without loss of convergence order. As we are able to compute the occurring boundary integrals analytically, we can precisely predict the electric field near the boundary. This property makes our method exceptionally well suited for the numerical simulation of plasma sheaths near irregular boundaries or of plasma-surface interaction such as etching of semiconductors.In der vorliegenden Arbeit nutzen wir Randintegralgleichungen als ein mächtiges Werkzeug, um neue Einsichten in die Dynamik von Plasmen zu gewinnen. Auf theoretischer Seite entwickelt diese Arbeit neue Resultate bezüglich der Oszillation beschränkter Plasmen. Durch die ana- lytische Berechnung der Frequenzen im Fall eines allgemeinen Ellipsoids stellen wir ein neues Testbeispiel für numerische Methoden bereit. Unsere Resultate werden durch umfangreiche numerische Untersuchen dreidimensionaler Beispiele validiert, etwa einen Partikelbeschleuniger mit komplexer Geometrie und gemischten Randwerten. Mithilfe der Randelementmethode reduziert sich die Dimension des Problems von drei auf zwei, womit sich die Anzahl der Un- bekannten drastisch reduziert. Dank der Nutzung hierarchischer Methoden zur Berechnung der auftauchenden nichtlokalen Summen und Integraloperatoren skaliert unsere Methode linear mit der Anzahl der Partikel und der Anzahl der Oberflächendreiecken, wobei der Fehler exponen- tiell im Entwicklungsparameter abfällt. Des Weiteren erlaubt unsere Methode die Berechnung des elektrischen Felds ohne Verringerung der Konvergenzordnung. Da wir die auftretenden Randintegrale analytisch berechnen können, können wir präzise Aussagen über das elektrische Feld nahe des Rands treffen. Dank dieser Eigenschaft ist unsere Methode außergewöhnlich gut geeignet, um Plasmaränder nahe irregulärer Ränder oder Plasma-Oberflächen-Interaktionen, etwa das Ätzen von Halbleitern, zu simulieren

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Characterisation and Classification of Hidden Conducting Security Threats using Magnetic Polarizability Tensors

The early detection of terrorist threat objects, such as guns and knives, through improved metal detection, has the potential to reduce the number of attacks and improve public safety and security. Walk through metal detectors (WTMDs) are commonly deployed for security screening purposes in applications where these attacks are of particular con-cern such as in airports, transport hubs, government buildings and at concerts. However, there is scope to improve the identification of an object’s shape and its material proper-ties. Using current techniques there is often the requirement for any metallic objects to be inspected or scanned separately before a patron may be determined to pose no threat, making the process slow. This can often lead to build ups of large queues of unscreened people waiting to be screened which becomes another security threat in itself. To improve the current method, there is considerable potential to use the fields applied and measured by a metal detector since, hidden within the field perturbation, is object characterisation information. The magnetic polarizability tensor (MPT) offers an economical characteri-sation of metallic objects and its spectral signature provides additional object character-isation information. The MPT spectral signature can be determined from measurements of the induced voltage over a range of frequencies for a hidden object. With classification in mind, it can also be computed in advance for different threat and non-threat objects, producing a dataset of these objects from which a machine learning (ML) classifier can be trained. There is also potential to generate this dataset synthetically, via the application of a method based on finite elements (FE). This concept of training an ML classifier trained on a synthetic dataset of MPT based characterisations is at the heart of this work.In this thesis, details for the production and use of a first of its kind synthetic dataset of realistic object characterisations are presented. To achieve this, first a review of re-cent developments of MPT object characterisations is provided, motivating the use of MPT spectral signatures. A problem specific, H(curl) based, hp-finite element discreti-sation is presented, which allows for the development of a reduced order model (ROM), using a projection based proper orthogonal decomposition (PODP), that benefits from a-posteriori error estimates. This allows for the rapid production of MPT spectral signatures the accuracy of which is guaranteed. This methodology is then implemented in Python, using the NGSolve finite element package, where other problem specific efficiencies are also included along with a series of additional outputs of interest, this software is then packaged and released as the open source MPT-Calculator. This methodology and software are then extensively tested by application to a series of illustrative examples. Using this software, MPT spectral signatures are then produced for a series of realistic threat and non-threat objects, creating the first of its kind synthetic dataset, which is also released as the open source MPT-Library dataset. Lastly, a series of ML classifiers are documented and applied to several supervised classification problems using this new syn-thetic dataset. A series of challenging numerical examples are included to demonstrate the success of the proposed methodology

Brain and Human Body Modeling 2020

​This open access book describes modern applications of computational human modeling in an effort to advance neurology, cancer treatment, and radio-frequency studies including regulatory, safety, and wireless communication fields. Readers working on any application that may expose human subjects to electromagnetic radiation will benefit from this book’s coverage of the latest models and techniques available to assess a given technology’s safety and efficacy in a timely and efficient manner. Describes computational human body phantom construction and application; Explains new practices in computational human body modeling for electromagnetic safety and exposure evaluations; Includes a survey of modern applications for which computational human phantoms are critical

1-D broadside-radiating leaky-wave antenna based on a numerically synthesized impedance surface

A newly-developed deterministic numerical technique for the automated design of metasurface antennas is applied here for the first time to the design of a 1-D printed Leaky-Wave Antenna (LWA) for broadside radiation. The surface impedance synthesis process does not require any a priori knowledge on the impedance pattern, and starts from a mask constraint on the desired far-field and practical bounds on the unit cell impedance values. The designed reactance surface for broadside radiation exhibits a non conventional patterning; this highlights the merit of using an automated design process for a design well known to be challenging for analytical methods. The antenna is physically implemented with an array of metal strips with varying gap widths and simulation results show very good agreement with the predicted performance