Analysis of numeric conditioning of Hessian matrix of Lagrangian via singular values

This paper presents an analyze of numeric conditioning of the Hessian matrix of Lagrangian of modified barrier function Lagrangian method (MBFL) and primal-dual logarithmic barrier method (PDLB), which are obtained in the process of solution of an optimal power flow problem (OPF). This analyze is done by a comparative study through the singular values decomposition (SVD) of those matrixes. In the MBLF method the inequality constraints are treated by the modified barrier and PDLB methods. The inequality constraints are transformed into equalities by introducing positive auxiliary variables and are perturbed by the barrier parameter. The first-order necessary conditions of the Lagrangian function are solved by Newton's method. The perturbation of the auxiliary variables results in an expansion of the feasible set of the original problem, allowing the limits of the inequality constraints to be reached. The electric systems IEEE 14, 162 and 300 buses were used in the comparative analysis. ©2007 IEEE.

A singular vector perspective of 4DVAR: The spatial structure and evolution of baroclinic weather systems

The extent to which the four-dimensional variational data assimilation (4DVAR) is able to use information about the time evolution of the atmosphere to infer the vertical spatial structure of baroclinic weather systems is investigated. The singular value decomposition (SVD) of the 4DVAR observability matrix is introduced as a novel technique to examine the spatial structure of analysis increments. Specific results are illustrated using 4DVAR analyses and SVD within an idealized 2D Eady model setting. Three different aspects are investigated. The first aspect considers correcting errors that result in normal-mode growth or decay. The results show that 4DVAR performs well at correcting growing errors but not decaying errors. Although it is possible for 4DVAR to correct decaying errors, the assimilation of observations can be detrimental to a forecast because 4DVAR is likely to add growing errors instead of correcting decaying errors. The second aspect shows that the singular values of the observability matrix are a useful tool to identify the optimal spatial and temporal locations for the observations. The results show that the ability to extract the time-evolution information can be maximized by placing the observations far apart in time. The third aspect considers correcting errors that result in nonmodal rapid growth. 4DVAR is able to use the model dynamics to infer some of the vertical structure. However, the specification of the case-dependent background error variances plays a crucial role.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

Properties of singular vectors using convective available potential energy as final time norm

We study the feasibility of using the singular vector technique to create initial condition perturbations for short-range ensemble prediction systems (SREPS) focussing on predictability of severe local storms and in particular deep convection. For this a new final time semi-norm based on the convective available potential energy (CAPE) is introduced. We compare singular vectors using the CAPE-norm with SVs using the more common total energy (TE) norm for a 2-week summer period in 2007, which includes a case of mesoscale extreme rainfall in the south west of Finland. The CAPE singular vectors perturb the CAPE field by increasing the specific humidity and temperature of the parcel and increase the lapse rate above the parcel in the lower troposphere consistent with physical considerations. The CAPE-SVs are situated in the lower troposphere. This in contrast to TE-SVs with short optimization times which predominantly remain in the high troposphere. By examining the time evolution of the CAPE singular values we observe that the convective event in the south west of Finland is clearly associated with high CAPE singular values.

Singular Value Decomposition and Ligand Binding Analysis

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Singular value analyses of voltage stability on power system considering wind generation variability

Pós-graduação em Engenharia Elétrica - FEIS

Robust model-order reduction of complex biological processes

This paper addresses robust model-order reduction of a high dimensional nonlinear partial differential equation (PDE) model of a complex biological process. Based on a nonlinear, distributed parameter model of the same process which was validated against experimental data of an existing, pilot-scale BNR activated sludge plant, we developed a state-space model with 154 state variables in this work. A general algorithm for robustly reducing the nonlinear PDE model is presented and based on an investigation of five state-of-the-art model-order reduction techniques, we are able to reduce the original model to a model with only 30 states without incurring pronounced modelling errors. The Singular perturbation approximation balanced truncating technique is found to give the lowest modelling errors in low frequency ranges and hence is deemed most suitable for controller design and other real-time applications. (C) 2002 Elsevier Science Ltd. All rights reserved.

Unified min-max and interlacing theorems for linear operators

There exist striking analogies in the behaviour of eigenvalues of Hermitian compact operators, singular values of compact operators and invariant factors of homomorphisms of modules over principal ideal domains, namely diagonalization theorems, interlacing inequalities and Courant-Fischer type formulae. Carlson and Sa [D. Carlson and E.M. Sa, Generalized minimax and interlacing inequalities, Linear Multilinear Algebra 15 (1984) pp. 77-103.] introduced an abstract structure, the s-space, where they proved unified versions of these theorems in the finite-dimensional case. We show that this unification can be done using modular lattices with Goldie dimension, which have a natural structure of s-space in the finite-dimensional case, and extend the unification to the countable-dimensional case.

Análise do sinal cardíaco ECG através da técnica SVD

Este trabalho surge no âmbito da área Electromedicina, uma componente da Engenharia Electrotécnica cada vez mais influente e em permanente desenvolvimento, existindo nela uma constante inovação e tentativa de desenvolvimento e aplicação de novas tecnologias. Este projecto possui como principal objectivo o estudo aprofundado das aplicações da técnica SVD (Singular Value Decomposition), uma poderosa ferramenta matemática que permite a manipulação de sinais através da decomposição de matrizes, ao caso específico do sinal eléctrico obtido através de um electrocardiograma (ECG). Serão discriminados os princípios da operação do sistema eléctrico cardíaco, as principais componentes do sinal ECG (a onda P, o complexo QRS e a onda T) e os fundamentos da técnica SVD. A última fase deste trabalho consistirá na aplicação, em ambiente Matlab, da técnica SVD a sinais ECG concretos, com enfase na sua filtragem, para efeitos de remoção de ruído. De modo verificar as suas vantagens e desvantagens face a outras técnicas, os resultados da filtragem por SVD serão comparados com aqueles obtidos, em condições similares, através da aplicação de um filtro FIR de coeficientes estáticos e de um filtro adaptativo iterativo.

Forecasting financial weather - can we foresee market sentiment? Spectrum of stock price behavior - NYSE case study

The desire to create a statistical or mathematical model, which would allow predicting the future changes in stock prices, was born many years ago. Economists and mathematicians are trying to solve this task by applying statistical analysis and physical laws, but there are still no satisfactory results. The main reason for this is that a stock exchange is a non-stationary, unstable and complex system, which is influenced by many factors. In this thesis the New York Stock Exchange was considered as the system to be explored. A topological analysis, basic statistical tools and singular value decomposition were conducted for understanding the behavior of the market. Two methods for normalization of initial daily closure prices by Dow Jones and S&P500 were introduced and applied for further analysis. As a result, some unexpected features were identified, such as a shape of distribution of correlation matrix, a bulk of which is shifted to the right hand side with respect to zero. Also non-ergodicity of NYSE was confirmed graphically. It was shown, that singular vectors differ from each other by a constant factor. There are for certain results no clear conclusions from this work, but it creates a good basis for the further analysis of market topology.

Classification and fault detection methods for fuel cell monitoring and quality control

In this paper, various types of fault detection methods for fuel cells are compared. For example, those that use a model based approach or a data driven approach or a combination of the two. The potential advantages and drawbacks of each method are discussed and comparisons between methods are made. In particular, classification algorithms are investigated, which separate a data set into classes or clusters based on some prior knowledge or measure of similarity. In particular, the application of classification methods to vectors of reconstructed currents by magnetic tomography or to vectors of magnetic field measurements directly is explored. Bases are simulated using the finite integration technique (FIT) and regularization techniques are employed to overcome ill-posedness. Fisher's linear discriminant is used to illustrate these concepts. Numerical experiments show that the ill-posedness of the magnetic tomography problem is a part of the classification problem on magnetic field measurements as well. This is independent of the particular working mode of the cell but influenced by the type of faulty behavior that is studied. The numerical results demonstrate the ill-posedness by the exponential decay behavior of the singular values for three examples of fault classes.

On the spectra and pseudospectra of a class of non-self-adjoint random matrices and operators

In this paper we develop and apply methods for the spectral analysis of non-selfadjoint tridiagonal infinite and finite random matrices, and for the spectral analysis of analogous deterministic matrices which are pseudo-ergodic in the sense of E. B. Davies (Commun. Math. Phys. 216 (2001), 687–704). As a major application to illustrate our methods we focus on the “hopping sign model” introduced by J. Feinberg and A. Zee (Phys. Rev. E 59 (1999), 6433–6443), in which the main objects of study are random tridiagonal matrices which have zeros on the main diagonal and random ±1’s as the other entries. We explore the relationship between spectral sets in the finite and infinite matrix cases, and between the semi-infinite and bi-infinite matrix cases, for example showing that the numerical range and p-norm ε - pseudospectra (ε > 0, p ∈ [1,∞] ) of the random finite matrices converge almost surely to their infinite matrix counterparts, and that the finite matrix spectra are contained in the infinite matrix spectrum Σ. We also propose a sequence of inclusion sets for Σ which we show is convergent to Σ, with the nth element of the sequence computable by calculating smallest singular values of (large numbers of) n×n matrices. We propose similar convergent approximations for the 2-norm ε -pseudospectra of the infinite random matrices, these approximations sandwiching the infinite matrix pseudospectra from above and below.

Optimal reactive power flow via the modified barrier Lagrangian function approach

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Global optimization for the ℋ∞-norm model reduction problem

A branch and bound algorithm is proposed to solve the [image omitted]-norm model reduction problem for continuous and discrete-time linear systems, with convergence to the global optimum in a finite time. The lower and upper bounds in the optimization procedure are described by linear matrix inequalities (LMI). Also proposed are two methods with which to reduce the convergence time of the branch and bound algorithm: the first one uses the Hankel singular values as a sufficient condition to stop the algorithm, providing to the method a fast convergence to the global optimum. The second one assumes that the reduced model is in the controllable or observable canonical form. The [image omitted]-norm of the error between the original model and the reduced model is considered. Examples illustrate the application of the proposed method.

Análise modal baseada apenas na resposta: decomposição no domínio da frequência

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)