Showing papers on "Linear approximation published in 2002"

Inverse solutions for electrical impedance tomography based on conjugate gradients methods

Abstract: A multistep inverse solution for two-dimensional electric field distribution is developed to deal with the nonlinear inverse problem of electric field distribution in relation to its boundary condition and the problem of divergence due to errors introduced by the ill-conditioned sensitivity matrix and the noise produced by electrode modelling and instruments. This solution is based on a normalized linear approximation method where the change in mutual impedance is derived from the sensitivity theorem and a method of error vector decomposition. This paper presents an algebraic solution of the linear equations at each inverse step, using a generalized conjugate gradients method. Limiting the number of iterations in the generalized conjugate gradients method controls the artificial errors introduced by the assumption of linearity and the ill-conditioned sensitivity matrix. The solution of the nonlinear problem is approached using a multistep inversion. This paper also reviews the mathematical and physical definitions of the sensitivity back-projection algorithm based on the sensitivity theorem. Simulations and discussion based on the multistep algorithm, the sensitivity coefficient back-projection method and the Newton-Raphson method are given. Examples of imaging gas-liquid mixing and a human hand in brine are presented.

Comparison of worst case errors in linear and neural network approximation

Abstract: Sets of multivariable functions are described for which worst case errors in linear approximation are larger than those in approximation by neural networks. A theoretical framework for such a description is developed in the context of nonlinear approximation by fixed versus variable basis functions. Comparisons of approximation rates are formulated in terms of certain norms tailored to sets of basis functions. The results are applied to perceptron networks.

Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function

Abstract: Since the seminal paper of Kydland and Prescott (1982) and King, Plosser and Rebelo (1988), it has become commonplace in macroeconomics to approximate the solution to nonlinear, dynamic general equilibrium models using linear methods. Linear approximation methods are useful to characterize certain aspects of the dynamic properties of complicated models. First-order approximation techniques are not however, well suited to handle questions such as welfare comparisons across alternative stochastic of policy environments. The problem with using linearized decision rules to evaluate second-order approximations to the objective function is that some second-order terms of the objective function are ignored when using a linearized decision rule. Such problems do not arise when the policy function is approximated to second-order or higher. In this paper we derive a second order approximation to the policy function of a dynamic, rational expectations model. Our approach follows the perturbation method described in Judd (1998) and developed further by Collard and Juillard (2001). We follow Collard and Juillard closely in notation and methodology. An important difference separates this Paper from the work of Collard and Juillard. Namely, Collard and Juillard apply what they call a bias reduction procedure to capture the fact that the policy function depends on the variance of the underlying shocks. Instead, we explicitly incorporate a scale parameter for the variance of the exogenous shocks as an argument of the policy function. In approximating the policy function, we take a second order Taylor expansion with respect to the state variables as well as this scale parameter. To illustrate its applicability, the method is used to solve the dynamics of a simple neoclassical model. The Paper closes with a brief description of a set of MATLAB programs designed to implement the method.

Dynamics and solutions to some control problems for water-tank systems

Abstract: We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated.

Smooth-surface reconstruction in near-linear time

Abstract: A surface reconstruction algorithm takes as input a set of sample points from an unknown closed and smooth surface in 3-d space, and produces a piece-wise linear approximation of the surface that contains the sample points. Recently, several algorithms with a correctness guarantee have been proposed. They have unfortunately a worst-case running time that is quadratic in the size of the input because they are based on the construction of 3-d Voronoi diagrams or Delaunay tetrahedrizations which can have quadratic size. In this paper, we describe a new algorithm that also has a correctness guarantee but whose worst-case running time is O(n log n) where n is the input size. This is actually optimal. As in some of the previous algorithms, the piece-wise linear approximation produced by the new algorithm is a triangulation which is a subset of the 3-d Delaunay tetrahedrization.

A unified approach to form error evaluation

Abstract: Evaluation of form error is a critical aspect of many manufacturing processes. Machines such as the coordinate measuring machine (CMM) often employ the technique of the least squares form fitting algorithms. While based on sound mathematical principles, it is well known that the method of least squares often overestimates the tolerance zone, causing good parts to be rejected. Many methods have been proposed in efforts to improve upon results obtained via least squares, including those, which result in the minimum zone tolerance value. However, these methods are mathematically complex and often computationally slow for cases where a large number of data points are to be evaluated. Extensive amount of data is generated where measurement equipment such as laser scanners are used for inspection, as well as in reverse engineering applications. In this report, a unified linear approximation technique is introduced for use in evaluating the forms of straightness, flatness, circularity, and cylindricity. Non-linear equation for each form is linearized using Taylor expansion, then solved as a linear program using software written in C++ language. Examples are taken from the literature as well as from data collected on a coordinate measuring machine for comparison with least squares and minimum zone results. For all examples, the new formulations are found to equal or better than the least squares results and provide a good approximation to the minimum zone tolerance.

Linearized radiation and cloud schemes in the ECMWF model: Development and evaluation

Abstract: A proper consideration of cloud–radiation interactions in linearized models is required for variational assimilation of cloud properties. Therefore, both a linearized cloud scheme and a linearized radiation scheme have been developed for the European Centre for Medium-Range Weather Forecasts (ECMWF) data assimilation system. The tangent-linear and adjoint versions of the ECMWF short-wave radiation scheme are prepared without a priori modifications. The complexity of the radiation scheme for the long-wave part of the spectrum makes accurate computations expensive. To reduce its computational cost, a combination of artificial neural networks and Jacobian matrices is defined for the linearized long-wave radiation scheme. The linearized cloud scheme is diagnostic and has been adapted for this study. The accuracy of the linearization of both radiation and diagnostic cloud schemes is examined. The inclusion of a more sophisticated radiation scheme within the existing linearized parametrizations improves the accuracy of the tangent-linear approximation. However, the impact of the linearized diagnostic cloud scheme is small, suggesting that the linearized model will require further developments of cloud parametrization. The adjoint technique is used to investigate the sensitivity of the radiation schemes to changes in temperature, humidity and cloud properties. This study shows which variables can be adjusted when certain observations of the surface and/or of the top-of-atmosphere radiation fluxes are used in data assimilation. It also indicates the vertical extent of the influence of such observations. Copyright © 2002 Royal Meteorological Society.

Controllability of Nonlinear Discrete Systems

Abstract: In the present paper local constrained controllability problems for nonlinear finite-dimensional discrete system with constant coefficients are formulated and discussed. Using some mapping theorems taken from functional analysis and linear approximation methods sufficient conditions for constrained controllability are derived and proved. The present paper extends the controllability conditions with unconstrained controls given in the literature to cover the nonlinear discrete systems with constrained controls.

Greedy Approximation with Regard to Bases and General Minimal Systems

Abstract: This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying non- linear approximation. This was motivated by numerous applications. Non- linear approximation is important in applications because of its increased efficiency. Two types of nonlinear approximation are employed frequently in applications. Adaptive methods are used in PDE solvers. The m-term approximation considered here is used in image and signal processing as well as the design of neural networks. The basic idea behind nonlinear approx- imation is that the elements used in the approximation do not come from a fixed linear space but are allowed to depend on the function being ap- proximated. The fundamental question of nonlinear approximation is how to construct good methods (algorithms) of nonlinear approximation. In this paper we discuss greedy type and thresholding type algorithms.

Amplification of small perturbations in a Hartmann layer

Abstract: We investigate in this paper the stability of the laminar Hartmann layer to small perturbations. Previous works have shown that the critical Reynolds number for linear stability is Rc≈50 000. It is much higher than the experimental threshold values for laminarization of magnetohydrodynamic flows (150

Sampling of periodic signals: a quantitative error analysis

Abstract: We present an exact expression for the L/sub 2/ error that occurs when one approximates a periodic signal in a basis of shifted and scaled versions of a generating function. This formulation is applicable to a wide variety of linear approximation schemes including wavelets, splines, and bandlimited signal expansions. The formula takes the simple form of a Parseval's-like relation, where the Fourier coefficients of the signal are weighted against a frequency kernel that characterizes the approximation operator. We use this expression to analyze the behavior of the error as the sampling step approaches zero. We also experimentally verify the expression of the error in the context of the interpolation of closed curves.

Directional wavelet transforms and frames

Abstract: The application of the wavelet transform in image processing is most frequently based on a separable transform. Lines and columns in an image are treated independently and the basis functions are simply products of corresponding one-dimensional functions. Such a method keeps simplicity in design and computation. A new two-dimensional approach is proposed, which retains the simplicity of separable processing, but allows more directionalities. The method can be applied in many areas like denoising, nonlinear approximation and compression. The results on nonlinear approximation and denoising show interesting gains compared to the standard two-dimensional analysis.

Clustered linear regression

Abstract: Clustered linear regression (CLR) is a new machine learning algorithm that improves the accuracy of classical linear regression by partitioning training space into subspaces. CLR makes some assumptions about the domain and the data set. Firstly, target value is assumed to be a function of feature values. Second assumption is that there are some linear approximations for this function in each subspace. Finally, there are enough training instances to determine subspaces and their linear approximations successfully. Tests indicate that if these approximations hold, CLR outperforms all other well-known machine-learning algorithms. Partitioning may continue until linear approximation fits all the instances in the training set — that generally occurs when the number of instances in the subspace is less than or equal to the number of features plus one. In other case, each new subspace will have a better fitting linear approximation. However, this will cause over fitting and gives less accurate results for the test instances. The stopping situation can be determined as no significant decrease or an increase in relative error. CLR uses a small portion of the training instances to determine the number of subspaces. The necessity of high number of training instances makes this algorithm suitable for data mining applications.

A Chosen Plaintext Linear Attack on Block Cipher CIKS-1

Abstract: In this paper, we firstly evaluate the resistance of the reduced 5-round version of the block cipher CIKS-1 against linear cryptanalysis (LC). A feature of the CIKS-1 is the use of both Data-Dependent permutations(DDP) and internal key scheduing which consist in datadapendent transformation of the round subkeys. Taking into account the structure of CIKS-1 we investigate linear approximation. That is, we consider 16 linear approximations with p = 3/4 for 16 parallel modulo 22 additions to construct one-round linear approximation and derive one-round linear approximation with the probability of P = 1/2 + 2-17 by Piling-Up lemma. Also we estimate that the P is a valid probability of one-round approximation and achieve that the probability P for oneround approximation is better than 1/2 +2-17 through experiments. Then we construct 3-round linear approximation with P = 1/2 +2-17 using this one-round approximation and can attack the reduced 5-round CIKS-1 with 64-bit block by LC. In conclusion, we present that our attack requires about 236 chosen plaintexts with a probability of success of 78.5 % and 1/5 × 232 × 236 ? 265.7 encryption times to recover last round(5-round) key. In addition, we discuss a few improvements of the cipher CIKS-1.

Minimum linear span approximation of binary sequences

Abstract: The determination of the minimum linear span sequence that differs from a given binary sequence, of period N=2/sup n/-1, by at most one digit is discussed and three methods are presented: the sequential divisions method, the congruential equations method, and the phase synchronization method. High-level algorithm organizations are provided. Finally, guidelines on sequence characterization and design via the notion of robustness are given.

Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

Abstract: We consider an Allen–Cahn/Cahn–Hilliard system with a non-degenerate mobility and (i) a logarithmic free energy and (ii) a non-smooth free energy (the deep quench limit). This system arises in the modelling of phase separation and ordering in binary alloys. In particular we prove in each case that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear finite element approximation of (i) and (ii) in one and two space dimensions (and three space dimensions for constant mobility). The error bound being optimal in the deep quench limit. In addition an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments are presented.

Approximate Moving Least-Squares Approximation: A Fast and Accurate Multivariate Approximation Method

Abstract: We propose a fast and accurate approximation method for large sets of multivariate data using radial functions. In the tradi- tional radial basis function approach this task is usually accomplished by solving a large system of linear equations stemming from an inter- polation formulation. In the traditional moving least-squares method one needs to solve a small linear system for each evaluation of the ap- proximant. We present an approximation scheme { based on the work on approximate approximation by Maz'ya and Schmidt { that has ap- proximation properties similar to the moving least-squares method, but completely avoids the solution of linear systems. Moreover, the sums required for the evaluation of the approximant can be processed quickly. We establish a connection to traditional radial basis func- tion approximation by using appropriate radial generating functions. Examples of locally supported as well as globally supported functions with arbitrary approximation orders are given.

Dynamic approximation of complex graphical constraints by linear constraints

Abstract: Current constraint solving techniques for interactive graphical applications cannot satisfactorily handle constraints such as non-overlap, or containment within non-convex shapes or shapes with smooth edges We present a generic new technique for efficiently handling such kinds of constraints based on trust regions and linear arithmetic constraint solving Our approach is to model these more complex constraints by a dynamically changing conjunction of linear constraints At each stage, these give a local approximation to the complex constraints During direct manipulation, linear constraints in the current local approximation can become active indicating that the current solution is on the boundary of the trust region for the approximation The associated complex constraint is notified and it may choose to modify the current linear approximation Empirical evaluation demonstrates that it is possible to (re-)solve systems of linear constraints that are dynamically approximating complex constraints such as non-overlap sufficiently quickly to support direct manipulation in interactive graphical applications

Method and system for estimating a base-2 logarithm of a number

Abstract: The present invention is directed to methods and systems for estimating the log base-2 of a fixed point binary number using a single polynomial for an entire possible range of input numbers. An estimation of the log base-2 of a fixed-point binary number in either hardware or software may be implemented using a minimal number of parameters. In particular, a single 2 nd order or greater polynomial may be sufficient to cover an entire range of input values for any arbitrary input word precision. The present invention provides a method and system for estimating a logarithm of a number where a linear approximation of a fractional part is determined and the linear approximation is implemented in a single polynomial function for estimating the fractional part for a range of input values. A circuit for generating an integer part and an estimate of a fractional part of a logarithm may include a shift register for loading a valid input data and for generating an estimate of a fractional part and a counter for loading a total number of bits in an input data and for generating an integer part, wherein the circuit implements a single polynomial for generating an improved estimate of the fractional part.

Probability density function for frequency response function measurements using periodic signals

Abstract: Frequency response function (FRF) measurements are often used to characterize linear dynamic systems. Nowadays uncertainty bounds on FRF measurements still are based on linear approximations, which are valid for sufficiently large input signal-to-noise-ratios (SNR) only. In this paper exact uncertainty bounds are calculated, which are valid for any input SNR. These bounds are obtained via an analytic expression of the probability density function (PDF) of the FRF measurements. The results are valid for open- and closed-loop measurements, and the theory is illustrated on a real measurement example.

Exact Values of Widths of Classes of Analytic Functions on the Disk and Best Linear Approximation Methods

Abstract: In the Hardy space Hp,ρ (p≥1, 0<ρ≤ 1, Hp,1≡ Hp) we develop best linear approximation methods (previously studied by Taikov and Ainulloev) for the classes W(r,Φ,μ) of analytic functions on the unit disk and calculate the exact values of linear, Gelfand, and informational n-widths of these classes.

Optimal control of nonlinear systems

Abstract: The construction of optimal closed loop and open loop solutions to a terminal optimal control problem for nonlinear dynamical systems with a linear objective functional and linear terminal constraints is considered. Locally optimal controls are constructed by correcting the solution to a linearized problem by the small parameter technique. Global optimization is also carried out in two steps. First, a solution to a piecewise linear approximation of the original problem is constructed, and then it is improved using an asymptotic technique. The results are illustrated by solving the problem of damping a simple pendulum and the problem of control of a nonlinear system governed by the Duffing equation.

Integrated circuit having an FM demodulator and method therefor

Abstract: A digital FM demodulator employs a baseband phase lock loop (BBPLL), which is particularly effective for long range reception, for combining and demodulating a pair of signals represented by the mathematical expression A(t)e jθ(t) to result in an approximation of dθ/dt. This approximation is then subjected to an inverse of the linear approximation of the frequency response of the BBPLL that produces a very accurate θ. This is conveniently achieved with a IIR filter whose transfer function happens to be the same as the inverse of the linear approximation of the frequency response of the BBPLL. The derivative is then taken of θ to produce a very accurate dθ/dt, the desired result for the output of an FM demodulator. To aid operation of the BBPLL, the incoming digital intermediate frequency is upsampled by a combination of sample and hold and FIR filtering prior to being processed by the BBPLL.