Systematic Data Extraction in High-Frequency Electromagnetic Fields

The focus of this work is on the investigation of billiards with its statistical eigenvalue properties. Specifically, superconducting microwave resonators with chaotic characteristics are simulated and the eigenfrequencies that are needed for the statistical analysis are computed. The eigenfrequency analysis requires many (in order of thousands) eigenfrequencies to be calculated and the accurate determination of the eigenfrequencies has a crucial significance. Consequently, the research interests cover all aspects from accurate numerical calculation of many eigenvalues and eigenvectors up to application development in order to get good performance out of the programs for distributed-memory and shared-memory multiprocessors. Furthermore, this thesis provides an overview and detailed evaluation of the used numerical approaches for large-scale eigenvalue calculations with respect to the accuracy, the computational time, and the memory consumption. The first approach for an accurate eigenfrequency extraction takes into consideration the evaluated electric field computations in Time Domain (TD) of a superconducting resonant structure. Upon excitation of the cavity, the electric field intensity is recorded at different detection probes inside the cavity. Thereafter, Fourier analysis of the recorded signals is performed and by means of signal-processing and fitting techniques, the requested eigenfrequencies are extracted by finding the optimal model parameters in the least squares sense. The second numerical approach is based on a numerical computation of electromagnetic fields in Frequency Domain (FD) and further employs the Lanczos method for the eigenvalue determination. Namely, when utilizing the Finite Integration Technique (FIT) to solve an electromagnetic problem for a superconducting cavity, which enclosures excited electromagnetic fields, the numerical solution of a standard large-scale eigenvalue problem is considered. Accordingly, if the numerical solution of the same problem is treated by the Finite Element Method (FEM) based on curvilinear tetrahedrons, it yields to the generalized large-scale eigenvalue problem. Afterward, the desired eigenvalues are calculated with the direct solution of the large (generalized) eigenvalue formulations. For this purpose, the implemented Lanczos solvers combine two major ingredients: the Lanczos algorithm with polynomial filtering on the one hand and its parallelization on the other

Dynamic Density-Matrix Renormalization for the Symmetric Single Impurity Anderson Model

In this thesis we investigated, implemented, gauged, optimized, and applied a numerical approach given by the dynamic density-matrix renormalization (D-DMRG). The dynamic density-matrix renormalization is the extension of standard DMRG to the calculation of dynamic quantities. D-DMRG provides valuable numerical information on dynamic correlations by computing convolutions of the corresponding spectral densities. The dynamics at zero temperature is determined by computing the expectation values in the local propagator. This can be realized by targeting not only at the ground state and the excited state, but also at the resolvent applied to the excited state. This additional targeted state is called the correction vector. The calculation of the frequency-dependent correction vector is numerically extremely demanding due to the inversion of an almost singular non-hermitian matrix. We compared the performance of several iterative solvers for linear equation systems and managed to stabilize the inversion problem of the D-DMRG by using optimized algorithms. The main limitation of the correction vector D-DMRG is that one cannot obtain data for purely real frequencies but only for frequencies with a certain imaginary part. Hence the extraction of the behavior at purely real frequencies is one of the main problems to be solved in using the D-DMRG. We illustrated how and to which extent such data can be deconvolved to retrieve the wanted spectral densities. We discussed and compared various algorithms to achieve this extraction and presented a non-linear approach from the family of maximum entropy methods. This approach provides a continuous, positive ansatz for the wanted spectral density with the least bias (LB). Even relative abrupt changes of the spectral density can be reproduced satisfactorily. In the vicinity of singularities spurious oscillations occur. The least-bias ansatz can be made more robust towards small numerical inaccuracies and finite-size effects by including besides the entropy functional a xi-functional in the functional to be minimized. We investigated the dynamic propagator of the symmetric single impurity Anderson model by D-DMRG. The Abrikosov-Suhl resonance (ASR) was calculated to gauge the D-DMRG and to demonstrate that features at low energies can be resolved. The low energy scale of the SIAM was reproduced over two orders of magnitude. The Hubbard satellites were investigated. From the broadened D-DMRG raw data we extracted the width and the shifts from the atomic positions. By means of the LB extraction we were able to address the line shape of the satellites. It was found that the Hubbard satellites are strongly asymmetric, the peak is very pronounced and the maximum value is very high. The Hubbard model on the Bethe lattice with infinite coordination number was investigated by means of the dynamic mean-field theory (DMFT) using the D-DMRG as impurity solver. A high-resolution investigation of the electron spectra close to the metal-to-insulator transition in dynamic mean-field theory was presented. The all-numerical, consistent confirmation of a smooth transition at zero temperature is provided

Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials

We study the seismic inverse problem for the recovery of subsurface properties in acoustic media. In order to reduce the ill-posedness of the problem, the heterogeneous wave speed parameter to be recovered is represented using a limited number of coefficients associated with a basis of eigenvectors of a diffusion equation, following the regularization by discretization approach. We compare several choices for the diffusion coefficient in the partial differential equations, which are extracted from the field of image processing. We first investigate their efficiency for image decomposition (accuracy of the representation with respect to the number of variables and denoising). Next, we implement the method in the quantitative reconstruction procedure for seismic imaging, following the Full Waveform Inversion method, where the difficulty resides in that the basis is defined from an initial model where none of the actual structures is known. In particular, we demonstrate that the method is efficient for the challenging reconstruction of media with salt-domes. We employ the method in two and three-dimensional experiments and show that the eigenvector representation compensates for the lack of low frequency information, it eventually serves us to extract guidelines for the implementation of the method.Comment: 45 pages, 37 figure

Electromagnetic model subdivision and iterative solvers for surface and volume double higher order numerical methods and applications

2019 Fall.Includes bibliographical references.Higher order methods have been established in the numerical analysis of electromagnetic structures decreasing the number of unknowns compared to the low order discretization. In order to decrease memory requirements even further, model subdivision in the computational analysis of electrically large structures has been used. The technique is based on clustering elements and solving/approximating subsystems separately, and it is often implemented in conjunction with iterative solvers. This thesis addresses unique theoretical and implementation details specific to model subdivision of the structures discretized by the Double Higher Order (DHO) elements analyzed by i) Finite Element Method - Mode Matching (FEM-MM) technique for closed-region (waveguide) structures and ii) Surface Integral Equation Method of Moments (SIE-MoM) in combination with (Multi-Level) Fast Multipole Method for open-region bodies. Besides standard application in decreasing the model size, DHO FEM-MM is applied to modeling communication system in tunnels by means of Standard Impedance Boundary Condition (SIBC), and excellent agreement is achieved with measurements performed in Massif Central tunnel. To increase accuracy of the SIE-MoM computation, novel method for numerical evaluation of the 2-D surface integrals in MoM matrix entries has been improved to achieve better accuracy than traditional method. To demonstrate its efficiency and practicality, SIE-MoM technique is applied to analysis of the rain event containing significant percentage of the oscillating drops recorded by 2D video disdrometer. An excellent agreement with previously-obtained radar measurements has been established providing the benefits of accurately modeling precipitation particles

Numerical Methods for Electronic Structure Calculations of Materials

This is the published version. Copyright 2010 Society for Industrial and Applied MathematicsThe goal of this article is to give an overview of numerical problems encountered when determining the electronic structure of materials and the rich variety of techniques used to solve these problems. The paper is intended for a diverse scientific computing audience. For this reason, we assume the reader does not have an extensive background in the related physics. Our overview focuses on the nature of the numerical problems to be solved, their origin, and the methods used to solve the resulting linear algebra or nonlinear optimization problems. It is common knowledge that the behavior of matter at the nanoscale is, in principle, entirely determined by the Schrödinger equation. In practice, this equation in its original form is not tractable. Successful but approximate versions of this equation, which allow one to study nontrivial systems, took about five or six decades to develop. In particular, the last two decades saw a flurry of activity in developing effective software. One of the main practical variants of the Schrödinger equation is based on what is referred to as density functional theory (DFT). The combination of DFT with pseudopotentials allows one to obtain in an efficient way the ground state configuration for many materials. This article will emphasize pseudopotential-density functional theory, but other techniques will be discussed as well

Wavelet-based image and video super-resolution reconstruction.

Super-resolution reconstruction process offers the solution to overcome the high-cost and inherent resolution limitations of current imaging systems. The wavelet transform is a powerful tool for super-resolution reconstruction. This research provides a detailed study of the wavelet-based super-resolution reconstruction process, and wavelet-based resolution enhancement process (with which it is closely associated). It was addressed to handle an explicit need for a robust wavelet-based method that guarantees efficient utilisation of the SR reconstruction problem in the wavelet-domain, which will lead to a consistent solution of this problem and improved performance. This research proposes a novel performance assessment approach to improve the performance of the existing wavelet-based image resolution enhancement techniques. The novel approach is based on identifying the factors that effectively influence on the performance of these techniques, and designing a novel optimal factor analysis (OFA) algorithm. A new wavelet-based image resolution enhancement method, based on discrete wavelet transform and new-edge directed interpolation (DWT-NEDI), and an adaptive thresholding process, has been developed. The DWT-NEDI algorithm aims to correct the geometric errors and remove the noise for degraded satellite images. A robust wavelet-based video super-resolution technique, based on global motion is developed by combining the DWT-NEDI method, with super-resolution reconstruction methods, in order to increase the spatial-resolution and remove the noise and aliasing artefacts. A new video super-resolution framework is designed using an adaptive local motion decomposition and wavelet transform reconstruction (ALMD-WTR). This is to address the challenge of the super-resolution problem for the real-world video sequences containing complex local motions. The results show that OFA approach improves the performance of the selected wavelet-based methods. The DWT-NEDI algorithm outperforms the state-of-the art wavelet-based algorithms. The global motion-based algorithm has the best performance over the super-resolution techniques, namely Keren and structure-adaptive normalised convolution methods. ALMD-WTR framework surpass the state-of-the-art wavelet-based algorithm, namely local motion-based video super-resolution.PhD in Manufacturin

Scaling full seismic waveform inversions

The main goal of this research study is to scale full seismic waveform inversions using the adjoint-state method to the data volumes that are nowadays available in seismology. Practical issues hinder the routine application of this, to a certain extent theoretically well understood, method. To a large part this comes down to outdated or flat out missing tools and ways to automate the highly iterative procedure in a reliable way. This thesis tackles these issues in three successive stages. It first introduces a modern and properly designed data processing framework sitting at the very core of all the consecutive developments. The ObsPy toolkit is a Python library providing a bridge for seismology into the scientific Python ecosystem and bestowing seismologists with effortless I/O and a powerful signal processing library, amongst other things. The following chapter deals with a framework designed to handle the specific data management and organization issues arising in full seismic waveform inversions, the Large-scale Seismic Inversion Framework. It has been created to orchestrate the various pieces of data accruing in the course of an iterative waveform inversion. Then, the Adaptable Seismic Data Format, a new, self-describing, and scalable data format for seismology is introduced along with the rationale why it is needed for full waveform inversions in particular and seismology in general. Finally, these developments are put into service to construct a novel full seismic waveform inversion model for elastic subsurface structure beneath the North American continent and the Northern Atlantic well into Europe. The spectral element method is used for the forward and adjoint simulations coupled with windowed time-frequency phase misfit measurements. Later iterations use 72 events, all happening after the USArray project has commenced, resulting in approximately 150`000 three components recordings that are inverted for. 20 L-BFGS iterations yield a model that can produce complete seismograms at a period range between 30 and 120 seconds while comparing favorably to observed data

Custom optimization algorithms for efficient hardware implementation

The focus is on real-time optimal decision making with application in advanced control systems. These computationally intensive schemes, which involve the repeated solution of (convex) optimization problems within a sampling interval, require more efficient computational methods than currently available for extending their application to highly dynamical systems and setups with resource-constrained embedded computing platforms. A range of techniques are proposed to exploit synergies between digital hardware, numerical analysis and algorithm design. These techniques build on top of parameterisable hardware code generation tools that generate VHDL code describing custom computing architectures for interior-point methods and a range of first-order constrained optimization methods. Since memory limitations are often important in embedded implementations we develop a custom storage scheme for KKT matrices arising in interior-point methods for control, which reduces memory requirements significantly and prevents I/O bandwidth limitations from affecting the performance in our implementations. To take advantage of the trend towards parallel computing architectures and to exploit the special characteristics of our custom architectures we propose several high-level parallel optimal control schemes that can reduce computation time. A novel optimization formulation was devised for reducing the computational effort in solving certain problems independent of the computing platform used. In order to be able to solve optimization problems in fixed-point arithmetic, which is significantly more resource-efficient than floating-point, tailored linear algebra algorithms were developed for solving the linear systems that form the computational bottleneck in many optimization methods. These methods come with guarantees for reliable operation. We also provide finite-precision error analysis for fixed-point implementations of first-order methods that can be used to minimize the use of resources while meeting accuracy specifications. The suggested techniques are demonstrated on several practical examples, including a hardware-in-the-loop setup for optimization-based control of a large airliner.Open Acces