Showing papers on "Linear approximation published in 2014"

Convex relaxations and linear approximation for optimal power flow in multiphase radial networks

Abstract: Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123-bus networks and a real-world 2065-bus network.

Strong oracle optimality of folded concave penalized estimation

Abstract: Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamental issue still remains that it is not clear whether the local optimum computed by a given optimization algorithm possesses those nice theoretical properties. To close this important theoretical gap in over a decade, we provide a unified theory to show explicitly how to obtain the oracle solution via the local linear approximation algorithm. For a folded concave penalized estimation problem, we show that as long as the problem is localizable and the oracle estimator is well behaved, we can obtain the oracle estimator by using the one-step local linear approximation. In addition, once the oracle estimator is obtained, the local linear approximation algorithm converges, namely it produces the same estimator in the next iteration. The general theory is demonstrated by using four classical sparse estimation problems, i.e., sparse linear regression, sparse logistic regression, sparse precision matrix estimation and sparse quantile regression.

A fast and accurate approximation for planar pose graph optimization

Abstract: This work investigates the pose graph optimization problem, which arises in maximum likelihood approaches to simultaneous localization and mapping SLAM. State-of-the-art approaches have been demonstrated to be very efficient in medium- and large-sized scenarios; however, their convergence to the maximum likelihood estimate heavily relies on the quality of the initial guess. We show that, in planar scenarios, pose graph optimization has a very peculiar structure. The problem of estimating robot orientations from relative orientation measurements is a quadratic optimization problem after computing suitable regularization terms; moreover, given robot orientations, the overall optimization problem becomes quadratic. We exploit these observations to design an approximation of the maximum likelihood estimate, which does not require the availability of an initial guess. The approximation, named LAGO Linear Approximation for pose Graph Optimization, can be used as a stand-alone tool or can bootstrap state-of-the-art techniques, reducing the risk of being trapped in local minima. We provide analytical results on existence and sub-optimality of LAGO, and we discuss the factors influencing its quality. Experimental results demonstrate that LAGO is accurate in common SLAM problems. Moreover, it is remarkably faster than state-of-the-art techniques, and is able to solve very large-scale problems in a few seconds.

Applied Optimization Methods for Wireless Networks

Abstract: 1. Introduction Part I. Methods for Optimal Solutions: 2. Linear programming and applications 3. Convex programming and applications 4. Design of polynomial-time exact algorithm Part II. Methods for Near-Optimal and Approximation Solutions: 5. Branch-and-bound framework and application 6. Reformulation-linearization technique and applications 7. Linear approximation 8. Approximation algorithm and its applications - part 1 9. Approximation algorithm and its applications - part 2 Part III. Methods for Efficient Heuristic Solutions: 10. An efficient technique for mixed-integer optimization 11. Metaheuristic methods Part IV. Other Topics: 12. Asymptotic capacity analysis.

Theory of U(1) gauged Q-balls revisited

Abstract: In this paper, the main properties of ($3+1$)-dimensional $U(1)$ gauged $Q$ balls are examined. In particular, it is shown that the relation $\frac{dE}{dQ}=\ensuremath{\omega}$ holds for such gauged $Q$ balls in the general case. As a consequence, it is shown that the well-known estimate for the maximal charge of stable gauged $Q$ balls was derived by means of an inconsistent procedure and cannot be considered as correct. A simple method for obtaining the main characteristics of gauged $Q$ balls using only the nongauged background solution for the scalar field in the case, when the backreaction of the gauge field on the scalar field is small and the linearized theory can be used, is proposed. The criteria of applicability of the linearized theory, which do not reduce to the demand of the smallness of the coupling constant, are established. Some interesting properties of gauged $Q$ balls, as well as the advantages of the proposed method, are demonstrated by the example of two models, admitting, in the linear approximation in the perturbations, exact analytic solutions for gauged $Q$ balls.

A linearized approach to worst-case design in parametric and geometric shape optimization

Abstract: The purpose of this paper is to propose a deterministic method for optimizing a structure with respect to its worst possible behavior when a "small" uncertainty exists over some of its features. The main idea of the method is to linearize the considered cost function with respect to the uncertain parameters, then to consider the supremum function of the obtained linear approximation, which can be rewritten as a more "classical" function of the design, owing to standard adjoint techniques from optimal control theory. The resulting "linearized worst-case" objective function turns out to be the sum of the initial cost function and of a norm of an adjoint state function, which is dual with respect to the considered norm over perturbations. This formal approach is very general, and can be justified in some special cases. In particular, it allows to address several problems of considerable importance in both parametric and shape optimization of elastic structures, in a unified framework.

The Finite Elements for Design of Frame of Thin-Walled Beams

Abstract: Recent years there have been observed a wide application of a metalware in industrial and civil engineering. Special place in the building industry is belonged to light steel thin-walled constructions having a lot of technological advantages. In the article the first development cycle of a numerical method creating is considered – creating of the stiffness matrixes of thin-walled finite elements of various types using the semisheared theory (by V.I.Slivker) – depending on a way of approximation of functions of deformations (torsion and warping): 1. Linear approximation of torsional functions with a 2-central finite element having 4 transitions; 2. Quadratic approximation of torsional functions and linear approximation of warping function with a 3-central finite element having 5 transitions; 3. Cubical approximation of functions of torsional and warping functions with a 3-central finite element having 6 transitions. Thus deformation functions (torsional angle and warping) are approximated as mutually independent functions.

Nonlocal Gravity: The General Linear Approximation

Abstract: The recent classical nonlocal generalization of Einstein's theory of gravitation is presented within the framework of general relativity via the introduction of a preferred frame field. The nonlocal generalization of Einstein's field equations is derived. The linear approximation of nonlocal gravity is thoroughly examined and the solutions of the corresponding field equations are discussed. It is shown that nonlocality, with a characteristic length scale of order 1 kpc, simulates dark matter in the linear regime while preserving causality. Light deflection in linearized nonlocal gravity is studied in connection with gravitational lensing; in particular, the propagation of light in the weak gravitational field of a uniformly moving source is investigated. The astrophysical implications of the results are briefly mentioned.

Linear control of the flywheel inverted pendulum

Abstract: The flywheel inverted pendulum is an underactuated mechanical system with a nonlinear model but admitting a linear approximation around the unstable equilibrium point in the upper position. Although underactuated systems usually require nonlinear controllers, the easy tuning and understanding of linear controllers make them more attractive for designers and final users. In a recent paper, a simple PID controller was proposed by the authors, leading to an internally unstable controlled plant. To achieve global stability, two options are developed here: first by introducing an internal stabilizing controller and second by replacing the PID controller by an observer-based state feedback control. Simulation and experimental results show the effectiveness of the design.

Quadratic Basis Pursuit

Abstract: In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately characterize nonlinear models has the potential to improve the results in both existing compressive sensing applications and those where a linear approximation does not suffice, e.g., phase retrieval. In this paper, we extend the classical compressive sensing framework to a second-order Taylor expansion of the nonlinearity. Using a lifting technique and a method we call quadratic basis pursuit, we show that the sparse signal can be recovered exactly when the sampling rate is sufficiently high. We further present efficient numerical algorithms to recover sparse signals in second-order nonlinear systems, which are considerably more difficult to solve than their linear counterparts in sparse optimization.

Global Stability of a Discrete Mutualism Model

Abstract: A discrete mutualism model is studied in this paper. By using the linear approximation method, the local stability of the interior equilibrium of the system is investigated. By using the iterative method and the comparison principle of difference equations, sufficient conditions which ensure the global asymptotical stability of the interior equilibrium of the system are obtained. The conditions which ensure the local stability of the positive equilibrium is enough to ensure the global attractivity are proved.

On uniqueness and proportionality in multi-class equilibrium assignment

Abstract: Over the past few years, much attention has been paid to computing flows for multi-class network equilibrium models that exhibit uniqueness of the class flows and proportionality (Bar-Gera et al., 2012). Several new algorithms have been developed such as bush based methods of Bar-Gera (2002), Dial (2006), and Gentile (2012) that are able to obtain very fine solutions of network equilibrium models. These solutions can be post processed (Bar-Gera, 2006) in order to ensure proportionality and class uniqueness of the flows. Recently developed, the TAPAS, algorithm (Bar Gera, 2010) is able to produce solutions that have proportionality embedded, without requiring post processing. It was generally accepted that these methods for solving UE traffic assignment are the only way to obtain unique path and class link flows. The purpose of this paper is to show that the linear approximation method and some of its variants satisfy these conditions as well. In addition, some analytical results regarding the relation between steps of the linear approximation algorithm and the path flows entropy are presented.

Majorization minimization by coordinate descent for concave penalized generalized linear models

Abstract: Recent studies have demonstrated theoretical attractiveness of a class of concave penalties in variable selection, including the smoothly clipped absolute deviation and minimax concave penalties. The computation of the concave penalized solutions in high-dimensional models, however, is a difficult task. We propose a majorization minimization by coordinate descent (MMCD) algorithm for computing the concave penalized solutions in generalized linear models. In contrast to the existing algorithms that use local quadratic or local linear approximation to the penalty function, the MMCD seeks to majorize the negative log-likelihood by a quadratic loss, but does not use any approximation to the penalty. This strategy makes it possible to avoid the computation of a scaling factor in each update of the solutions, which improves the efficiency of coordinate descent. Under certain regularity conditions, we establish theoretical convergence property of the MMCD. We implement this algorithm for a penalized logistic regression model using the SCAD and MCP penalties. Simulation studies and a data example demonstrate that the MMCD works sufficiently fast for the penalized logistic regression in high-dimensional settings where the number of covariates is much larger than the sample size.

A Cross Weighted-Residual Time Integration Scheme for Structural Dynamics

Abstract: In this article, we develop a novel stable time integration scheme for the transient analysis of structural dynamics problems. A second-order (in time) differential operator equation (e.g. obtained after finite element discretization in space) is written as a pair of first-order equations in terms of displacements and velocities. Then the solution is sought by minimizing the inner product of the residuals in the two equations (an unconventional approach) over typical time interval to obtain a symmetric set of algebraic equations involving displacements and velocities at two subsequent intervals. The new time integration scheme is termed the cross weighted-residual (CWR) time integration scheme because each of the two residuals takes the other one as a weight function in the minimization. The CWR time integration scheme is developed by using a uniform linear time approximation of the displacement and velocity fields to yield only a single step time integration scheme, which is comparable to the Newmark family of time integration scheme. A reduced integration technique is used to prevent velocity locking, which is caused by linear approximation of both the displacement and velocity fields. For the verification of the consistency and the stability, the CWR time integration scheme is tested with single-degree as well as multi-degree of freedom problems. The scheme performs extremely well compared with those of the well-known Newmark family of time integration schemes.

A note on upscaling retardation factor in hierarchical porous media with multimodal reactive mineral facies

Abstract: We present a model for upscaling the time-dependent effective retardation factor in hierarchical porous media with multimodal reactive mineral facies. The model extends the approach by Deng et al. (2013) in which they expanded a Lagrangian-based stochastic theory presented by Rajaram (1997) in order to describe the scaling effect of retardation factor. They used a first-order linear approximation in deriving their model to make the derivation tractable. Importantly, the linear approximation is known to be valid only to variances of 0.2. In this article we show that the model can be derived with a higher-order approximation, which allows for representing variances from 0.2 to 1.0. We present the derivation, and use the resulting model to recalculate the time-dependent effective retardation for the scenario examined by Deng et al. (2013).

Successive Linear Approximation for Large Deformation—Instability of Salt Migration

Abstract: In continuum mechanics, concerning the motion of a body, besides the Lagrangian and the Eulerian descriptions, the description relative to the present configuration of the body instead of the fixed reference configuration has been known as the relative motion description. Although the relative motion description is mostly ignored in the formulation of boundary value problems, it is interesting to consider such a formulation for problems in general for solid bodies. In doing so, there is an advantage that when the time increment from the present state is small enough, the nonlinear constitutive equation can be linearized relative to the present configuration, so that the resulting boundary value problem becomes linear. We can then propose a linear algorithm for large deformation, by building up successive small incremental deformation problem at every time step in the deformation process. In fact, the proposed method is a process of repeated applications of the well-known “small deformation superposed on finite deformation” in the literature. As an application of the proposed numerical method, we consider instability of a two-layered solid body of a denser material on top of a lighter one. This problem is widely known to geoscientist in sediment-salt migration as salt diapirism. In the literature, this problem has often been treated as Rayleigh–Taylor instability in viscous fluids instead of solid bodies. As an example, we propose a viscoelastic solid material model from constitutive theories of continuum mechanics, and present results of numerical simulations of sediment-salt migration which exhibit some main characteristics of salt diapirism as observed by geophysicists.

A Simplified Flexible Multibody Dynamics for a Main Landing Gear with Flexible Leaf Spring

Abstract: The dynamics of multibody systems with deformable components has been a subject of interest in many different fields such as machine design and aerospace. Traditional rigid-flexible systems often take a lot of computer resources to get accurate results. Accuracy and efficiency of computation have been the focus of this research in satisfying the coupling of rigid body and flex body. The method is based on modal analysis and linear theory of elastodynamics: reduced modal datum was used to describe the elastic deformation which was a linear approximate of the flexible part. Then rigid-flexible multibody system was built and the highly nonlinearity of the mass matrix caused by the limited rotation of the deformation part was approximated using the linear theory of elastodynamics. The above methods were used to establish the drop system of the leaf spring type landing gear of a small UAV. Comparisons of the drop test and simulation were applied. Results show that the errors caused by the linear approximation are acceptable, and the simulation process is fast and stable.

Convex Relaxations and Linear Approximation for Optimal Power Flow in Multiphase Radial Networks

Abstract: Distribution networks are usually multiphase and radial. To facilitate power flow computation and optimization, two semidefinite programming (SDP) relaxations of the optimal power flow problem and a linear approximation of the power flow are proposed. We prove that the first SDP relaxation is exact if and only if the second one is exact. Case studies show that the second SDP relaxation is numerically exact and that the linear approximation obtains voltages within 0.0016 per unit of their true values for the IEEE 13, 34, 37, 123-bus networks and a real-world 2065-bus network.

On Shapiro's lethargy theorem and some applications

Abstract: Shapiro’s lethargy theorem [48] states that if {An} is any non-trivial linear approximation scheme on a Banach space X ,t hen the sequences of errors of best approximation E(x,An )=i nfa∈An ∥x − an∥X may decay almost arbitrarily slowly. Recently, Almira and Oikhberg [11, 12] investigated this kind of result for general approximation schemes in the quasi-Banach setting. In this paper, we consider the same question for F-spaces with non decreasing metric d .W e also provide applications to the rate of decay of s-numbers, entropy numbers and slow convergence of sequences of operators.

Sensorless Control of Synchronous Machines by Linear Approximation of Oversampled Current

Abstract: The present work examines the potential of current oversampling for the sensorless control of synchronous machines. Using linear regression, the current evolution during a switching state is approximated by a straight line. The approximated offset and slope represent low-noise current and current derivative values. Upon this basis, three position estimation techniques are developed and merged in two oppositional hybrid sensorless control methods which demonstrate the advantages of the oversampling.

Non-classical solutions of a continuum model for rock descriptions

Abstract: The strain-gradient and non-Euclidean continuum theories are employed for construction of non-classical solutions of continuum models. The linear approximation of both models' results in identical structures in terms of their kinematic and stress characteristics. The solutions obtained in this study exhibit a critical behaviour with respect to the external loading parameter. The conclusions are obtained based on an investigation of the solution for the scalar curvature in the non-Euclidean continuum theory. The proposed analysis enables us to use different theoretical approaches for description of rock critical behaviour under different loading conditions.

Local Linear Approximation of the Jacobian Matrix Better Captures Phase Resetting of Neural Limit Cycle Oscillators

Abstract: One effect of any external perturbations, such as presynaptic inputs, received by limit cycle oscillators when they are part of larger neural networks is a transient change in their firing rate, or phase resetting. A brief external perturbation moves the figurative point outside the limit cycle, a geometric perturbation that we mapped into a transient change in the firing rate, or a temporal phase resetting. In order to gain a better qualitative understanding of the link between the geometry of the limit cycle and the phase resetting curve (PRC), we used a moving reference frame with one axis tangent and the others normal to the limit cycle. We found that the stability coefficients associated with the unperturbed limit cycle provided good quantitative predictions of both the tangent and the normal geometric displacements induced by external perturbations. A geometric-to-temporal mapping allowed us to correctly predict the PRC while preserving the intuitive nature of this geometric approach.

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

Abstract: This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.

River flow forecasting through nonlinear local approximation in a fuzzy model

Abstract: This study investigates the potential of nonlinear local function approximation in a Takagi---Sugeno (TS) fuzzy model for river flow forecasting. Generally, in a TS framework, the local approximation is performed by a linear model, while in this approach, linear function approximation is substituted using a nonlinear function approximation. The primary hypothesis herein is that the process being modeled (rainfall---runoff in this study) is highly nonlinear, and a linear approximation at the local domain might still leave a lot of unexplained variance by the model. In this study, subtractive clustering technique is used for domain partition, and neural network is used for function approximation. The modeling approach has been tested on two case studies: Kolar basin in India and Kentucky basin in USA. The results of fuzzy nonlinear local approximation (FNLLA) model are highly promising. The performance of the FNLLA is compared with that of a pure fuzzy inference system (FIS), and it is observed that both the models perform similar at 1-step-ahead forecasts. However, the FNLLA performs much better than FIS at higher lead times. It is also observed that FNLLA forecasts the river flow with lesser error compared to FIS. In the case of Kolar River, more than 40 % of the total data are forecasted with <2 % error by FNLLA at 1 h ahead, while the corresponding value for FIS is only 20 %. In the case of 3-h-ahead forecasts, these values are 25 % for FNLLA and 15 % for FIS. Performance of FNLLA in the case of Kentucky River basin was also better compared to FIS. It is also found that FNLLA simulates the peak flow better than FIS, which is certainly an improvement over the existing models.