Sparse Modelling and Multi-exponential Analysis

The research fields of harmonic analysis, approximation theory and computer algebra are seemingly different domains and are studied by seemingly separated research communities. However, all of these are connected to each other in many ways. The connection between harmonic analysis and approximation theory is not accidental: several constructions among which wavelets and Fourier series, provide major insights into central problems in approximation theory. And the intimate connection between approximation theory and computer algebra exists even longer: polynomial interpolation is a long-studied and important problem in both symbolic and numeric computing, in the former to counter expression swell and in the latter to construct a simple data model. A common underlying problem statement in many applications is that of determining the number of components, and for each component the value of the frequency, damping factor, amplitude and phase in a multi-exponential model. It occurs, for instance, in magnetic resonance and infrared spectroscopy, vibration analysis, seismic data analysis, electronic odour recognition, keystroke recognition, nuclear science, music signal processing, transient detection, motor fault diagnosis, electrophysiology, drug clearance monitoring and glucose tolerance testing, to name just a few. The general technique of multi-exponential modeling is closely related to what is commonly known as the Padé-Laplace method in approximation theory, and the technique of sparse interpolation in the field of computer algebra. The problem statement is also solved using a stochastic perturbation method in harmonic analysis. The problem of multi-exponential modeling is an inverse problem and therefore may be severely ill-posed, depending on the relative location of the frequencies and phases. Besides the reliability of the estimated parameters, the sparsity of the multi-exponential representation has become important. A representation is called sparse if it is a combination of only a few elements instead of all available generating elements. In sparse interpolation, the aim is to determine all the parameters from only a small amount of data samples, and with a complexity proportional to the number of terms in the representation. Despite the close connections between these fields, there is a clear lack of communication in the scientific literature. The aim of this seminar is to bring researchers together from the three mentioned fields, with scientists from the varied application domains.Output Type: Meeting Repor

Towards efficient three-dimensional wide-angle beam propagation methods and theoretical study of nanostructures for enhanced performance of photonic devices

In this dissertation, we have proposed a novel class of approximants, the so-called modified Padé approximant operators for the wide-angle beam propagation method (WA-BPM). Such new operators not only allow a more accurate approximation to the true Helmholtz equation than the conventional operators, but also give evanescent modes the desired damping. We have also demonstrated the usefulness of these new operators for the solution of time-domain beam propagation problems. We have shown this both for a wideband method, which can take reflections into account, and for a split-step method for the modeling of ultrashort unidirectional pulses. The resulting approaches achieve high-order accuracy not only in space but also in time. In addition, we have proposed an adaptation of the recently introduced complex Jacobi iterative (CJI) method for the solution of wide-angle beam propagation problems. The resulting CJI-WA-BPM is very competitive for demanding problems. For large 3D waveguide problems with refractive index profiles varying in the propagation direction, the CJI method can speed-up beam propagation up to 4 times compared to other state-of-the-art methods. For practical problems, the CJI-WA-BPM is found to be very useful to simulate a big component such as an arrayed waveguide grating (AWG) in the silicon-on-insulator platform, which our group is looking at. Apart from WA beam propagation problems for uniform waveguide structures, we have developed novel Padé approximate solutions for wave propagation in graded-index metamaterials. The resulting method offers a very promising tool for such demanding problems. On the other hand, we have carried out the study of improved performance of optical devices such as label-free optical biosensors, light-emitting diodes and solar cells by means of numerical and analytical methods. We have proposed a solution for enhanced sensitivity of a silicon-on-insulator surface plasmon interference biosensor which had been previously proposed in our group. The resulting sensitivity has been enhanced up to 5 times. Furthermore, we have developed an improved model to investigate the influence of isolated metallic nanoparticles on light emission properties of light-emitting diodes. The resulting model compares very well to experimental results. Finally, we have proposed the usefulness of core-shell nanostructures as nanoantennas to enhance light absorption of thin-film amorphous silicon solar cells. An increased absorption up to 33 % has theoretically been demonstrated

Electronic correlations in inhomogeneous model systems: numerical simulation of spectra and transmission

Many fascinating features in condensed matter systems emerge due to the interaction between electrons. Magnetism is such a paramount consequence, which is explained in terms of the exchange interaction of electrons. Another prime example is the metal-to-Mott-insulator transition, where the energy cost of Coulomb repulsion competes against the kinetic energy, the latter favoring delocalization. While systems of correlated electrons are exciting and show remarkable and technologically promising physical properties, they are difficult to treat theoretically. A single-particle description is insufficient; the quantum many-body problem of interacting electrons has to be solved. In the present thesis, we study physical properties of half-metallic ferromagnets which are used in spintronic devices. Half-metals exhibit a metallic spin channel, while the other spin channel is insulating; they are characterized by a high spin polarization. This thesis contributes to the development of numerical methods and applies them to models of half-metallic ferromagnets. Throughout this work, the single-band Hubbard Hamiltonian is considered, and electronic correlations are treated within dynamical mean-field theory. Instead of directly solving the lattice model, the dynamical mean-field theory amounts to solving a local, effective impurity problem that is determined self-consistently. At finite temperatures, this impurity problem is solved employing continuous-time quantum Monte Carlo algorithms formulated in the action formalism. As these algorithms are formulated in imaginary time, an analytic continuation is required to obtain spectral functions. We formulate a version of the N-point Padé algorithm that calculates the location of the poles in a least-squares sense. To directly obtain spectra for real frequencies, we employ Hamiltonian-based tensor network methods at zero temperature. We also summarize the ideas of the density matrix renormalization group algorithm, and of the time evolution using the time-dependent variational principle, employing a diagrammatic notation. Real materials never display perfect translational symmetry. Thus, realistic models require the inclusion of disorder effects. In this work, we discuss these within a single-site approximation, the coherent potential approximation, and combine it with the dynamical mean-field theory, allowing to treat interacting electrons in multicomponent alloys on a local level. We extend this combined scheme to off-diagonal disorder, that is, disorder in the hopping amplitudes, by employing the Blackman–Esterling–Berk formalism. For this purpose, we illustrate the ideas of this formalism using tensor diagrams and provide an efficient implementation. The structure of the effective medium is discussed, and a concentration scaling is proposed that resolves some of its peculiarities. The limit of vanishing hopping between different components is discussed and solved analytically for the Bethe lattice with a general coordination number. We exemplify the combined algorithm for a Bethe lattice, showing results that exhibit alloy-band-insulator to correlated-metal to Mott-insulator transitions. We study models of half-metallic ferromagnets to elucidate the effects of local electronic correlations on the spectral function. To model half-metallicity, a static spin splitting is used to produce the half-metallic density of states. Applying the Padé analytic continuation to the self-energy instead of the Green’s function produces reliable spectral functions agreeing with the zero-temperature results obtained for real frequencies. To address transport properties, we investigate the interface of a half-metallic layer and a metallic, band insulating, or Mott insulating layer. We observe charge reconstruction which induces metallicity at the interface; quasiparticle states are present in the Mott insulating layer even for a large Hubbard interaction. The transmission through a barrier made of such a single interacting half-metallic layer sandwiched by metallic leads is studied employing the Meir–Wingreen formalism. This allows for a transparent calculation of the transmission in the presence of the Hubbard interaction. For a strong coupling of the central layer to the leads, we identify high intensity bound states which do not contribute to the transmission. For small coupling, on the other hand, we find resonant states which enhance the transmission. In particular, we demonstrate that even for a single half-metallic layer, highly polarized transmissions are achievable

The complex step method for approximating the Fréchet derivative of matrix functions in automorphism groups

We show, that the Complex Step approximation to the Fréchet derivative of matrix functions is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Padé iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure

Advances in beam propagation method for facet reflectivity analysis

Waveguide discontinuities are frequently encountered in modern photonic structures. It is important to characterize the reflection and transmission that occurs at the discontinuous during the design and analysis process of these structures. Significant effort has been focused upon the development of accurate modelling tools, and a variety of modelling techniques have been applied to solve this kind of problem. Throughout this work, a Transmission matrix based Bidirectional Beam Propagation Method (T-Bi-BPM) is proposed and applied on the uncoated facet and the single coating layer reflection problems, including both normal and angled incident situations. The T-Bi-BPM method is developed on the basis of an overview of Finite Difference Beam Propagation Method (FD-BPM) schemes frequently used in photonic modelling including paraxial FD-BPM, Imaginary Distance (ID) BPM, Wide Angle (WA) BPM and existing Bidirectional (Bi) BPM methods. The T-Bi-BPM establishes the connection between the total fields on either side of the coating layer and the incident field at the input of a single layer coated structure by a matrix system on the basis of a transmission matrix equation used in a transmission line approach. The matrix system can be algebraically preconditioned and then solved by sparse matrix multiplications. The attraction of the T-Bi-BPM method is the potential for more rapid evaluation without iterative approach. The accuracy of the T-Bi-BPM is verified by simulations and the factors that affect the accuracy are investigated

Advances in beam propagation method for facet reflectivity analysis

Waveguide discontinuities are frequently encountered in modern photonic structures. It is important to characterize the reflection and transmission that occurs at the discontinuous during the design and analysis process of these structures. Significant effort has been focused upon the development of accurate modelling tools, and a variety of modelling techniques have been applied to solve this kind of problem. Throughout this work, a Transmission matrix based Bidirectional Beam Propagation Method (T-Bi-BPM) is proposed and applied on the uncoated facet and the single coating layer reflection problems, including both normal and angled incident situations. The T-Bi-BPM method is developed on the basis of an overview of Finite Difference Beam Propagation Method (FD-BPM) schemes frequently used in photonic modelling including paraxial FD-BPM, Imaginary Distance (ID) BPM, Wide Angle (WA) BPM and existing Bidirectional (Bi) BPM methods. The T-Bi-BPM establishes the connection between the total fields on either side of the coating layer and the incident field at the input of a single layer coated structure by a matrix system on the basis of a transmission matrix equation used in a transmission line approach. The matrix system can be algebraically preconditioned and then solved by sparse matrix multiplications. The attraction of the T-Bi-BPM method is the potential for more rapid evaluation without iterative approach. The accuracy of the T-Bi-BPM is verified by simulations and the factors that affect the accuracy are investigated

The approximation of functions with branch points

In recent years Pade approximants have proved to be one of the most useful computational tools in many areas of theoretical physics, most notably in statistical mechanics and strong interaction physics. The underlying reason for this is that very often the equations describing a physical process are so complicated that the simplest (if not the only) way of obtaining their solution is to perform a power series expansion in some parameters of the problem. Furthermore, the physical values of the para­meters are often such that this perturbation expansion does not converge and is therefore only a formal solution to the problem; as such it cannot be used quantitatively. However, the relevant information is contained in the coefficients of the perturbation series and the Fade approximants provide a convenient mathematical technique for extracting this information in a convergent way. A major difficulty with these approximants is that their convergence is restricted to regions of the complex plane free from any branch cuts; for example, the (N/N+j) Pade approximants to a series of Stieltjes converge to an analytic function in the complex plane cut along the negative real axis. The central idea of the present work is to obtain convergence along these branch cuts by using approximants which themselves have branch points. The ideas presented in this thesis are expected to be only the beginning of a large investigation into the use of multi-valued approximants as a practical method of approximation. In Chapter 1 we shall see that such approximants arise as natural generalisations of Pade approximants and possess many of the properties of Pade approximants; in particular, the very important property of homographic covariance. We term these approximants ’algebraic' approximants (since they satisfy an algebraic equation) and we are mainly concerned with the 'simplest' of these approximants, the quadratic approximants of Shafer. Chapter 2 considers some of the known convergence results for Pade approximants to indicate the type of results we nay reasonably expect to hold (and to be able to prove) for quadratic (and higher order) approximants. A discussion of various numerical examples is then given to illustrate the possible practical usefulness of these latter approximants. A major application of all these approximants is discussed in Chapter 3, where the problem of evaluating Feynman matrix elements in the physical region is considered; in this case, the physical region is along branch cuts. Several simple Feynman diagrams are considered to illustrate (a) the potential usefulness of the calculational scheme presented and (b) the relative merits of rational (Pade), quadratic and cubic approximation schemes. The success of these general approximation schemes in one variable (as exhibited by the results of Chapters 2 and 3) leads, in Chapter 4 to a consideration of the corresponding approximants in two variables. We shall see that the two variable scheme developed for rational approximants can be extended in a very natural way to define two variable "t-power" approximants. Numerical results are presented to indicate the usefulness of these schemes in practice. A final application to strong interaction physics is given in Chapter 5, where the analytic continuation of Legendre series is considered. Such series arise in partial wave expansions of the scattering amplitude. We shall see that the Pade Legendre approximants of Fleischer and Common can be generalised to produce corresponding quadratic Legendre approximants: various examples are considered to illustrate the relative merits of these schemes

Dynamical Systems in Spiking Neuromorphic Hardware

Dynamical systems are universal computers. They can perceive stimuli, remember, learn from feedback, plan sequences of actions, and coordinate complex behavioural responses. The Neural Engineering Framework (NEF) provides a general recipe to formulate models of such systems as coupled sets of nonlinear differential equations and compile them onto recurrently connected spiking neural networks – akin to a programming language for spiking models of computation. The Nengo software ecosystem supports the NEF and compiles such models onto neuromorphic hardware. In this thesis, we analyze the theory driving the success of the NEF, and expose several core principles underpinning its correctness, scalability, completeness, robustness, and extensibility. We also derive novel theoretical extensions to the framework that enable it to far more effectively leverage a wide variety of dynamics in digital hardware, and to exploit the device-level physics in analog hardware. At the same time, we propose a novel set of spiking algorithms that recruit an optimal nonlinear encoding of time, which we call the Delay Network (DN). Backpropagation across stacked layers of DNs dramatically outperforms stacked Long Short-Term Memory (LSTM) networks—a state-of-the-art deep recurrent architecture—in accuracy and training time, on a continuous-time memory task, and a chaotic time-series prediction benchmark. The basic component of this network is shown to function on state-of-the-art spiking neuromorphic hardware including Braindrop and Loihi. This implementation approaches the energy-efficiency of the human brain in the former case, and the precision of conventional computation in the latter case