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

Nonlinear Analysis and Optimization with Applications

Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world

ESPRIT for multidimensional general grids

We present a new method for complex frequency estimation in several variables, extending the classical (1d) ESPRIT-algorithm. We also consider how to work with data sampled on non-standard domains (i.e going beyond multi-rectangles)

Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected O(n^{2.2131}) time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory

Faster Sparse Matrix Inversion and Rank Computation in Finite Fields

We improve the current best running time value to invert sparse matrices over finite fields, lowering it to an expected $O\big(n^{2.2131}\big)$ time for the current values of fast rectangular matrix multiplication. We achieve the same running time for the computation of the rank and nullspace of a sparse matrix over a finite field. This improvement relies on two key techniques. First, we adopt the decomposition of an arbitrary matrix into block Krylov and Hankel matrices from Eberly et al. (ISSAC 2007). Second, we show how to recover the explicit inverse of a block Hankel matrix using low displacement rank techniques for structured matrices and fast rectangular matrix multiplication algorithms. We generalize our inversion method to block structured matrices with other displacement operators and strengthen the best known upper bounds for explicit inversion of block Toeplitz-like and block Hankel-like matrices, as well as for explicit inversion of block Vandermonde-like matrices with structured blocks. As a further application, we improve the complexity of several algorithms in topological data analysis and in finite group theory

Representaciones racionales de series matriciales con aplicación a la especificación de modelos multivariantes

Se caracterizan funciones racionales matriciales de dimensión arbitraria, a partir de series formales de potencias cuyos coeficientes son matrices, trasladando posteriormente los resultados a la especificación de modelos racionales de series temporales, en particular a modelos Varma. Los aspectos de minimalidad y unicidad de representación o, en terminología de series temporales, la intercambiabilidad de modelos y la identificabilidad de los parámetros han sido considerados también desde la aproximación de Padé matricial para su tratamiento y posterior aplicación a series temporales. Al mismo tiempo se aportan resultados sobre la estructura de la tabla de Padé en el caso de funciones matriciales de dimensión arbitraria. por otro lado, la forma que se indica para estudiar los parámetros nulos y/o redundantes de una representación racional responde también a ciertos problemas de sobreparametrización en la estimación de modelos racionales de series temporales

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

Using Real Time Statistical Data To Improve Long Term Voltage Stability In Stochastic Power Systems

In order to optimize limited infrastructure, many power systems are frequently operated close to critical, or bifurcation, points. While operating close to such critical points can be economically advantageous, doing so increases the probability of a blackout. With the continued deployment of Phasor Measurement Units (PMUs), high sample rate data are dramatically increasing the real time observability of the power grids. Prior research has shown that the statistics of these data can provide useful information regarding network stability and associated bifurcation proximity. Currently, it is not common practice for transmission and distribution control centers to leverage the higher order statistical properties of PMU data. If grid operators have the tools to determine when these statistics warrant control action, though, then the otherwise unused statistical data present in PMU streams can be transformed into actionable information. In order to address this problem, we present two methods that aim to gauge and improve system stability using the statistics of PMU data. The first method shows how sensitivity factors associated with the spectral analysis of the reduced power flow Jacobian can be used to weight and filter incoming PMU data. We do so by demonstrating how the derived participation factors directly predict the relative strength of bus voltage variances throughout a system. The second method leverages an analytical solver to determine a range of critical bus voltage variances. The monitoring and testing of raw statistical data in a highly observable load pocket of a large system are then used to reveal when control actions are needed to mitigate the risk of voltage collapse. A simple reactive power controller is then implemented that pushes the stability of the system back to a stable operating paradigm. Full order dynamic time domain simulations are used in order to test this method on both the IEEE 39 bus system and the 2383 bus Polish system. We also compare this method to two other, more conventional, controllers. The first relies on voltage magnitude signals, and the second depends only on local control of a reactive power resource. This comparison illustrates how the use of statistical information from PMU measurements can substantially improve the performance of voltage collapse mitigation methods