Showing papers on "Linear approximation published in 2007"

Simplified Spatial Correlation Models for Clustered MIMO Channels With Different Array Configurations

Abstract: An approximate spatial correlation model for clustered multiple-input multiple-output (MIMO) channels is proposed in this paper. The two ingredients for the model are an approximation for uniform linear and circular arrays to avoid numerical integrals and a closed-form expression for the correlation coefficients that is derived for the Laplacian azimuth angle distribution. A new performance metric to compare parametric and nonparametric channel models is proposed and used to show that the proposed model is a good fit to the existing parametric models for low angle spreads (i.e., smaller than 10deg). A computational-complexity analysis shows that the proposed method is a numerically efficient way of generating the spatially correlated MIMO channels.

Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems

Abstract: In a recent paper the authors developed and tested two novel computational algorithms for predicting the mean linear response of a chaotic dynamical system to small changes in external forcing via the fluctuation-dissipation theorem (FDT): the short-time FDT (ST-FDT), and the hybrid Axiom A FDT (hA-FDT). Unlike the earlier work in developing fluctuation-dissipation theorem-type computational strategies for chaotic nonlinear systems with forcing and dissipation, these two new methods are based on the theory of Sinai–Ruelle–Bowen probability measures, which commonly describe the equilibrium state of such dynamical systems. These two algorithms take into account the fact that the dynamics of chaotic nonlinear forced-dissipative systems often reside on chaotic fractal attractors, where the classical quasi-Gaussian (qG-FDT) approximation of the fluctuation-dissipation theorem often fails to produce satisfactory response prediction, especially in dynamical regimes with weak and moderate degrees of chaos. It has been discovered that the ST-FDT algorithm is an extremely precise linear response approximation for short response times, but numerically unstable for longer response times. On the other hand, the hA-FDT method is numerically stable for all times, but is less accurate for short times. Here we develop blended linear response algorithms, by combining accurate prediction of the ST-FDT method at short response times with numerical stability of qG-FDT and hA-FDT methods at longer response times. The new blended linear response algorithms are tested on the nonlinear Lorenz 96 model with 40 degrees of freedom, chaotic behaviour, forcing, dissipation, and mimicking large-scale features of real-world geophysical models in a wide range of dynamical regimes varying from weakly to strongly chaotic, and to fully turbulent. The results below for the blended response algorithms have a high level of accuracy for the linear response of both mean state and variance throughout all the different chaotic regimes of the 40-mode model. These results point the way towards the potential use of the blended response algorithms in operational long-term climate change projection.

Through-Wall Imaging via a Linear Inverse Scattering Algorithm

Abstract: A through-wall imaging problem for a 2-D scalar geometry is addressed. It is cast as an inverse scattering problem and tackled under the linear model of the electromagnetic scattering that is provided by the Born approximation. A truncated singular value decomposition inversion scheme is exploited, and the performances that are achievable by such an inversion scheme are assessed by exploiting synthetic data. The cases of weakly and strongly scattering objects are both considered. Finally, an example of reconstruction that is obtained by exploiting experimental data is presented.

Localized Light Waves: Paraxial and Exact Solutions of the Wave Equation (a Review)

Abstract: Simple explicit localized solutions are systematized over the whole space of a linear wave equation, which models the propagation of optical radiation in a linear approximation. Much attention has been paid to exact solutions (which date back to the Bateman findings) that describe wave beams (including Bessel-Gauss beams) and wave packets with a Gaussian localization with respect to the spatial variables and time. Their asymptotics with respect to free parameters and at large distances are presented. A similarity between these exact solutions and harmonic in time fields obtained in the paraxial approximation based on the Leontovich-Fock parabolic equation has been studied. Higher-order modes are considered systematically using the separation of variables method. The application of the Bateman solutions of the wave equation to the construction of solutions to equations with dispersion and nonlinearity and their use in wavelet analysis, as well as the summation of Gaussian beams, are discussed. In addition, solutions localized at infinity known as the Moses-Prosser “acoustic bullets”, as well as their harmonic in time counterparts, “X waves”, waves from complex sources, etc., have been considered. Everywhere possible, the most elementary mathematical formalism is used.

A Nearly Linear-Time Approximation Scheme for the Euclidean $k$-Median Problem

Abstract: This paper provides a randomized approximation scheme for the $k$-median problem when the input points lie in the $d$-dimensional Euclidean space. The worst-case running time is $O(2^{O((\log(1/\epsilon) / \varepsilon)^{d-1})} n \log^{d+6} n ),$ which is nearly linear for any fixed $\varepsilon$ and $d$. Moreover, our method provides the first polynomial-time approximation scheme for and uncapacitated facility location instances in $d$-dimensional Euclidean space for any fixed $d > 2.$ Our work extends techniques introduced originally by Arora for the Euclidean traveling salesman problem (TSP). To obtain the improvement we develop a structure theorem to describe hierarchical decomposition of solutions. The theorem is based on an adaptive decomposition scheme, which guesses at every level of the hierarchy the structure of the optimal solution and accordingly modifies the parameters of the decomposition. We believe that our methodology is of independent interest and may find applications to further geometric problems.

Q-learning with linear function approximation

Abstract: In this paper, we analyze the convergence of Q-learning with linear function approximation. We identify a set of conditions that implies the convergence of this method with probability 1, when a fixed learning policy is used. We discuss the differences and similarities between our results and those obtained in several related works. We also discuss the applicability of this method when a changing policy is used. Finally, we describe the applicability of this approximate method in partially observable scenarios.

An Upwind-Biased Conservative Advection Scheme for Spherical Hexagonal–Pentagonal Grids

Abstract: A discrete form of the flux-divergence operator is developed to compute advection of tracers on spherical hexagonal–pentagonal grids. An upwind-biased advection scheme based on a piecewise linear approximation for one-dimensional regular grids is extended simply for spherical hexagonal–pentagonal grids. The distribution of a tracer over the upwind side of a cell face is linearly approximated using a nodal value and a gradient at a computational node on the upwind side. A piecewise linear approximation is relaxed to a local linear approximation, and the relaxation precludes the complicated conditional branching present in remapping schemes. Results from a cosine bell advection test show that the new scheme compares favorably with other upwind-biased schemes for spherical hexagonal–pentagonal grids.

Filtering of Stochastic Nonlinear Differential Systems via a Carleman Approximation Approach

Abstract: This paper deals with the state estimation problem for stochastic nonlinear differential systems, driven by standard Wiener processes, and presents a filter that is a generalization of the classical extended kalman-bucy filter (EKBF). While the EKBF is designed on the basis of a first order approximation of the system around the current estimate, the proposed filter exploits a Carleman-like approximation of a chosen degree v ges 1. The approximation procedure, applied to both the state and the measurement equations, allows to define an approximate representation of the system by means of a bilinear system, for which a filtering algorithm is available from the literature. Numerical simulations on an example show the improvement, in terms of sample error covariance, of the filter based on the first-order, second-order and third-order system approximations (v = 1,2,3).

Explicit Reformulation of the Colebrook−White Equation for Turbulent Flow Friction Factor Calculation

Abstract: In this paper, we present an improvement of a mathematically equivalent representation of the Colebrook−White (CW) equation to compute friction factors for turbulent flow in rough pipes. This new form is simple and very well-suited for accurately estimating the friction factor, because no iterative calculations are necessary. Specifically, the friction factor is expressed as the sum of known simple functions and an unknown correction term. This correction term satisfies an auxiliary equation that can be accurately and easily solved with predictable error bounds over the complete range of pipe roughness and Reynolds number values encountered in practice. The simplest case, with the unknown correction term set to zero, resulted in friction factor estimates with errors of <1%. A simple linear approximation of the correction term resulted in a maximum error of 3.64 × 10-4%, whereas friction factor estimates from a continued-fractions-based approximation had a maximum error of 1.04 × 10-10%. These maximum erro...

A Note on Linear Time Algorithms for Maximum Error Histograms

Abstract: Histograms and Wavelet synopses provide useful tools in query optimization and approximate query answering. Traditional histogram construction algorithms, e.g., V-Optimal, use error measures which are the sums of a suitable function, e.g., square, of the error at each point. Although the best-known algorithms for solving these problems run in quadratic time, a sequence of results have given us a linear time approximation scheme for these algorithms. In recent years, there have been many emerging applications where we are interested in measuring the maximum (absolute or relative) error at a point. We show that this problem is fundamentally different from the other traditional {\rm{non}}{\hbox{-}}\ell_\infty error measures and provide an optimal algorithm that runs in linear time for a small number of buckets. We also present results which work for arbitrary weighted maximum error measures.

Least Squares Approximations to Lognormal Sum Distributions

Abstract: In this paper, the least squares (LS) approximation approach is applied to solve the approximation problem of a sum of lognormal random variables (RVs). The LS linear approximation is based on the widely accepted assumption that the sum of lognormal RVs can be approximated by a lognormal RV. We further derive the solution for the LS quadratic (LSQ) approximation, and our results show that the LSQ approximation exhibits an excellent match with the simulation results in a wide range of the distributions of the summands. Using the coefficients obtained from the LSQ method, we present the explicit closed-form expressions of the coefficients as a function of the decibel spread and the number of the summands by applying an LS curve fitting technique. Closed-form expressions for the cumulative distribution function and the probability density function for the sum RV, in both the linear and logarithm domains, are presented

Comparison between a model-based and a conventional pyramid sensor reconstructor.

Abstract: A model of a nonmodulated pyramid wavefront sensor (P-WFS) based on Fourier optics has been presented. Linearizations of the model represented as Jacobian matrices are used to improve the P-WFS phase estimates. It has been shown in simulations that a linear approximation of the P-WFS is sufficient in closed-loop adaptive optics. Also a method to compute model-based synthetic P-WFS command matrices is shown, and its performance is compared to the conventional calibration. It was observed that in poor visibility the new calibration is better than the conventional.

Two-Sided A Posteriori Error Estimates for Mixed Formulations of Elliptic Problems

Abstract: The present work is devoted to the a posteriori error estimation for mixed approximations of linear self-adjoint elliptic problems. New guaranteed upper and lower bounds for the error measured in the natural product norm are derived, and individual sharp upper bounds are obtained for approximation errors in each of the physical variables. All estimates are reliable and valid for any approximate solution from the class of admissible functions. The estimates contain only global constants depending solely on the domain geometry and the given operators. Moreover, it is shown that, after an appropriate scaling of the coordinates and the equation, the ratio of the upper and lower bounds for the error in the product norm never exceeds 3. The possible methods of finding the approximate mixed solution in the class of admissible functions are discussed. The estimates are computationally very cheap and can also be used for the indication of the local error distribution. As applications, the diffusion problem as well as the problem of linear elasticity are considered.

Confidence region estimation techniques for nonlinear regression in groundwater flow: Three case studies

Abstract: [1] This work focuses on different methods to generate confidence regions for nonlinear parameter identification problems. Three methods for confidence region estimation are considered: a linear approximation method, an F test method, and a log likelihood method. Each of these methods are applied to three case studies. One case study is a problem with synthetic data, and the other two case studies identify hydraulic parameters in groundwater flow problems based on experimental well test results. The confidence regions for each case study are analyzed and compared. Although the F test and log likelihood methods result in similar regions, there are differences between these regions and the regions generated by the linear approximation method for nonlinear problems. The differing results, capabilities, and drawbacks of all three methods are discussed.

Linear stability of ring systems

Abstract: We give a self-contained modern linear stability analysis of a system of n equal-mass bodies in circular orbit about a single more massive body. Starting with the mathematical description of the dynamics of the system, we form the linear approximation, compute all of the eigenvalues of the linear stability matrix, and finally derive inequalities that guarantee that none of these eigenvalues have a positive real part. In the end we rederive the result that Maxwell found for large n in his seminal paper on the nature and stability of Saturn's rings, which was published 150 years ago. In addition, we identify the exact matrix that defines the linearized system even when n is not large. This matrix is then investigated numerically (by computer) to find stability inequalities. Furthermore, using properties of circulant matrices, the eigenvalues of the large 4n × 4n matrix are computed by solving n quartic equations, which further facilitates the investigation of stability. Finally, we implement an n-body simulator, and we verify that the threshold mass ratios that we derive mathematically or numerically do indeed identify the threshold between stability and instability. Throughout the paper we consider only the planar n-body problem so that the analysis can be carried out purely in complex notation, which makes the equations and derivations more compact, more elegant, and therefore, we hope, more transparent. The result is a fresh analysis that shows that these systems are always unstable for 2 ≤ n ≤ 6, and for n > 6 they are stable provided that the central mass is massive enough. We give an explicit formula for this mass-ratio threshold.

Linear Approximation of Weakly Nonlinear MIMO Systems

Abstract: A nonlinear multiple-input-multiple-output (MIMO) Volterra system, measured with random multisine signals, can be replaced by an approximating linear MIMO system followed by nonlinear-noise sources. This approximation is a straightforward extension of the approximation to single-input-single-output Volterra systems. For a two-input-two-output cubic system, an optimized measurement strategy is also proposed to reduce as much as possible the influence of nonlinear noise on the measurement of the nonparametric frequency-response function

Exact solutions of Regge-Wheeler equation

Abstract: The Regge-Wheeler equation describes the axial perturbations of Schwarzschild metric in linear approximation. We present its exact solutions in terms of the confluent Heun's functions, the basic properties of the general solution, novel analytical approach and numerical techniques for study of different boundary problems which correspond to quasi-normal modes of black holes and other simple models of compact objects. We depict in more detail the exact solutions of Regge-Wheeler equation in the Schwarzschild black hole interior and on Kruscal-Szekeres manifold.

Alternative subcell discretisations for viscoelastic flow: Stress interpolation

Abstract: This study is concerned with the investigation of the associated properties of subcell discretisations for viscoelastic flows, where aspects of compatibility of solution function spaces are paramount. We introduce one new scheme, through a subcell finite element approximation fe(sc), and compare and contrast this against two precursor schemes—one with finite element discretisation in common, but at the parent element level quad-fe; the other, at the subcell level appealing to hybrid finite element/finite volume discretisation fe/fv(sc). To conduct our comparative study, we consider Oldroyd modelling and two classical steady benchmark flow problems to assess issues of numerical accuracy and stability—cavity flow and contraction flow. We are able to point to specific advantages of the finite element subcell discretisation and appreciate the characteristic properties of each discretisation, by analysing stress and flow field structure up to critical states of Weissenberg number. Findings reveal that the subcell linear approximation for stress within the constitutive equation (either fe or fv) yields a more stable scheme, than that for its quadratic counterpart (quad-fe), whilst still maintaining second-third order accuracy. The more compatible form of stress interpolation within the momentum equation is found to be via the subcell elements under fe(sc); yet, this makes no difference under fe/fv(sc). Furthermore, improvements in solution representation are gathered through enhanced upwinding forms, which may be coupled to stability gains with strain-rate stabilisation.

Convergence of Q-learning with linear function approximation

Abstract: In this paper, we analyze the convergence properties of Q-learning using linear function approximation. This algorithm can be seen as an extension to stochastic control settings of TD-learning using linear function approximation, as described in [1]. We derive a set of conditions that implies the convergence of this approximation method with probability 1, when a fixed learning policy is used. We provide an interpretation of the obtained approximation as a fixed point of a Bellman-like operator. We then discuss the relation of our result with several related works as well as its general applicability.

System and method for designing multi-channel RF pulses for MR imaging

Abstract: A system and method are provided for designing RF pulses for multi-channel and/or multi-dimensional spatially-selective applications using a linear approximation. Embodiments of the system and method may use a generalized linear-class large tip angle approximation to design RF pulses for multi-channel and parallel transmission. Further, some of these approximations allow for the design of arbitrarily large flip angles, irregularly-shaped flip angle profiles, or arbitrary initial magnetization values. Embodiments of the system and method may also provide for the design of k-space trajectories which aid in maintaining assumptions of the various linear class approximations.

Controlling a Semiconductor Optical Amplifier Using a State-Space Model

Abstract: We derive nonlinear and linear state-space control models for a multichannel semiconductor optical amplifier. Verified against the governing partial differential equations through simulation, the linear model tracks modulations up to 20% qualitatively well. Linear feedback control is then employed to design two interchannel crosstalk suppressing systems, one using state feedback into the electronic drive current and the other using optical output feedback into an optical control channel; the controller designed with the linear model is seen to work well even with 100% modulations of the nonlinear system. This linear state-space model opens the way for further robust analysis, design and control of integrated active photonic circuits

Tracking value function dynamics to improve reinforcement learning with piecewise linear function approximation

Abstract: Reinforcement learning algorithms can become unstable when combined with linear function approximation. Algorithms that minimize the mean-square Bellman error are guaranteed to converge, but often do so slowly or are computationally expensive. In this paper, we propose to improve the convergence speed of piecewise linear function approximation by tracking the dynamics of the value function with the Kalman filter using a random-walk model. We cast this as a general framework in which we implement the TD, Q-Learning and MAXQ algorithms for different domains, and report empirical results demonstrating improved learning speed over previous methods.

Locally Weighted Online Approximation-Based Control for Nonaffine Systems

Abstract: This paper is concerned with tracking control problems for nonlinear systems that are not affine in the control signal and that contain unknown nonlinearities in the system dynamic equations. This paper develops a piecewise linear approximation to the unknown functions during the system operation. New control and parameter adaptation algorithms are designed and analyzed using Lyapunov-like methods. The objectives are to achieve semiglobal stability of the state, accurate tracking of bounded reference signals contained within a known domain , and at least boundedness of the function approximator parameter estimates. Numerical simulations are included to illustrate the effectiveness of the learning algorithm.