Showing papers on "Linear approximation published in 2005"

Basis Function Adaptation in Temporal Difference Reinforcement Learning

Abstract: Reinforcement Learning (RL) is an approach for solving complex multi-stage decision problems that fall under the general framework of Markov Decision Problems (MDPs), with possibly unknown parameters. Function approximation is essential for problems with a large state space, as it facilitates compact representation and enables generalization. Linear approximation architectures (where the adjustable parameters are the weights of pre-fixed basis functions) have recently gained prominence due to efficient algorithms and convergence guarantees. Nonetheless, an appropriate choice of basis function is important for the success of the algorithm. In the present paper we examine methods for adapting the basis function during the learning process in the context of evaluating the value function under a fixed control policy. Using the Bellman approximation error as an optimization criterion, we optimize the weights of the basis function while simultaneously adapting the (non-linear) basis function parameters. We present two algorithms for this problem. The first uses a gradient-based approach and the second applies the Cross Entropy method. The performance of the proposed algorithms is evaluated and compared in simulations.

Local Geometry of Deformable Templates

Abstract: In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield efficient and robust estimations of diffeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this infinite dimensional "shape manifold" on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We finally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.

A Stream Function Method for Gradient Coil Design

Abstract: A new design approach for the construction of gradient coils for magnetic resonance imaging is presented The theoretical formulation involves a constraint cost function between the desired field in a particular region of interest in space and an almost arbitrarily defined surface that carries the current configuration based on Biot-Savart's integral equation An appropriate weight function in conjunction with linear approximation functions permits the transformation of the problem formulation into a linear matrix equation whose solution yields discrete current elements in terms of magnitude and direction within a specified coil surface Numerical predictions and comparisons with practical measurements for the Gx ,G y ,G z gradient coils underscore the success of this approach in terms of achieving highly linear fields while maintaining low parasitic fields and low inductances © 2005 Wiley Periodicals, Inc

High-speed function approximation using a minimax quadratic interpolator

Abstract: A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent p), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).

Linear models of nonlinear systems

Abstract: Linear time-invariant approximations of nonlinear systems are used in many applications and can be obtained in several ways. For example, using system identification and the prediction-error method, it is always possible to estimate a linear model without considering the fact that the input and output measurements in many cases come from a nonlinear system. One of the main objectives of this thesis is to explain some properties of such approximate models. More specifically, linear time-invariant models that are optimal approximations in the sense that they minimize a mean-square error criterion are considered. Linear models, both with and without a noise description, are studied. Some interesting, but in applications usually undesirable, properties of such optimal models are pointed out. It is shown that the optimal linear model can be very sensitive to small nonlinearities. Hence, the linear approximation of an almost linear system can be useless for some applications, such as robust control design. Furthermore, it is shown that standard validation methods, designed for identification of linear systems, cannot always be used to validate an optimal linear approximation of a nonlinear system. In order to improve the models, conditions on the input signal that imply various useful properties of the linear approximations are given. It is shown, for instance, that minimum phase filtered white noise in many senses is a good choice of input signal. Furthermore, the class of separable signals is studied in detail. This class contains Gaussian signals and it turns out that these signals are especially useful for obtaining approximations of generalized Wiener-Hammerstein systems. It is also shown that some random multisine signals are separable. In addition, some theoretical results about almost linear systems are presented. In standard methods for robust control design, the size of the model error is assumed to be known for all input signals. However, in many situations, this is not a realistic assumption when a nonlinear system is approximated with a linear model. In this thesis, it is described how robust control design of some nonlinear systems can be performed based on a discrete-time linear model and a model error model valid only for bounded inputs. It is sometimes undesirable that small nonlinearities in a system influence the linear approximation of it. In some cases, this influence can be reduced if a small nonlinearity is included in the model. In this thesis, an identification method with this option is presented for nonlinear autoregressive systems with external inputs. Using this method, models with a parametric linear part and a nonparametric Lipschitz continuous nonlinear part can be estimated by solving a convex optimization problem.

Table-based polynomials for fast hardware function evaluation

Abstract: Many general table-based methods for the evaluation in hardware of elementary functions have been published. The bipartite and multipartite methods implement a first-order approximation of the function using only table lookups and additions. Recently, a single multiplier second order method of similar inspiration has also been published. This paper extends such methods to approximations of arbitrary order, using adders, small multipliers, and very small ad hoc powering units. We obtain implementations that are both smaller and faster than previously published approaches. This paper also deals with the FPGA implementation of such methods. Previous work have consistently shown that increasing the approximation degree lead to not only smaller but also faster designs, as the reduction of the table size meant a reduction of its lookup time, which compensated for the addition and multiplication time. The experiments in this paper suggest that this still holds when going from order 2 to order 3, but no longer when using higher order approximations, where a tradeoff appears.

Instability of pedestrian flow and phase structure in a two-dimensional optimal velocity model

Abstract: A two-dimensional optimal velocity model was proposed for the study of pedestrian and granular flow. We investigate the stability of homogeneous flow in the linear approximation and show the phase diagram of the model. We also investigate the property of the model by numerical simulation in the cases of unidirectional and counter flow. From these results, we present a unified understanding of the behavior of pedestrians and other related systems.

On stability by the first approximation for discrete systems

Abstract: The problem of stability by the first approximation for discrete systems is considered. The Malkin-Massera-Chetaev theorem on stability by the first approximation is extended to discrete systems. The stability conditions by the first approximation for cascade are obtained

Post-Einsteinian tests of linearized gravitation

Abstract: The general relativistic treatment of gravitation can be extended by preserving the geometrical nature of the theory but modifying the form of the coupling between curvature and stress tensors. The gravitation constant is thus replaced by two running coupling constants which depend on scale and differ in the sectors of traceless and traced tensors. When calculated in the solar system in a linearized approximation, the metric is described by two gravitation potentials. This extends the parametrized post-Newtonian (PPN) phenomenological framework while allowing one to preserve compatibility with gravity tests performed in the solar system. Consequences of this extension are drawn here for phenomena correctly treated in the linear approximation. We obtain a Pioneer-like anomaly for probes with an eccentric motion as well as a range dependence of Eddington parameter γ to be seen in light deflection experiments.

Taylor expansion of the differential range for monostatic SAR

Abstract: The polar format algorithm (PFA) for spotlight synthetic aperture radar (SAR) is based on a linear approximation for the differential range to a scatterer. We derive a second-order Taylor series approximation of the differential range. We provide a simple and concise derivation of both the far-field linear approximation of the differential range, which forms the basis of the PFA, and the corresponding approximation limits based on the second-order terms of the approximation.

Quasi-Norm interpolation error estimates for the piecewise linear finite element approximation of p -Laplacian problems

Abstract: In this work, new interpolation error estimates have been derived for some well-known interpolators in the quasi-norms. The estimates are found to be essential to obtain the optimal a priori error bounds under the weakened regularity conditions for the piecewise linear finite element approximation of a class of degenerate equations. In particular, by using these estimates, we can close the existing gap between the regularity required for deriving the optimal error bounds and the regularity achievable for the smooth data for the 2-d and 3-d p-Laplacian.

Sub-Optimum H2 Rational Approximations to Fractional Order Linear Systems

Abstract: In this paper, we propose a procedure to achieve rational approximation to arbitrary fractional order linear time invariant (FO-LTI) systems with sub-optimum H2 -norm. Through illustrations, we show that the rational approximation is simple and effective. It is also demonstrated that this sub-optimum approximation method is effective in designing integer order controllers for FO-LTI systems in general form. Useful Matlab codes are also given in the appendices.Copyright © 2005 by ASME

Flow over heated terrain. Part I. Linear theory and idealized numerical simulations

Abstract: The flow past heated topography is examined with both linear and nonlinear models. It is first shown that the forcing of an obstacle with horizontally homogenous surface heating can be approximated by the forcing of an obstacle with surface heating isolated over the obstacle. The small-amplitude flow past an obstacle with isolated heating is then examined with a linear model. Under the linear approximation, the flow response to heated topography is simply the addition of the separate responses to thermal and orographic forcing. These separate responses are first considered individually and then the combined response is examined. Nondimensional parameters are developed that measure the relative importance of thermal and orographic forcing. Nonaxisymmetric forcing is then considered by examining the flow along and across a heated elliptically shaped obstacle. It is shown that the low-level lifting is maximized when the flow is along the major axis of the obstacle. The linear solutions are then test...

A coupled zigzag theory for the dynamics of piezoelectric hybrid cross-ply plates

Abstract: A recently developed coupled third-order zigzag theory for the statics of piezoelectric hybrid cross-ply plates is extended to dynamics. The theory combines a third-order zigzag approximation for the in-plane displacements and a sub-layerwise linear approximation for the electric potential, considering all components of the electric field. The nonuniform variation of the transverse displacement due to the piezoelectric field is accounted for. The conditions for the absence of shear traction at the top and bottom surfaces and continuity of transverse shear stresses in the presence of electromechanical loading are satisfied exactly, thereby reducing the number of displacement variables to five, which is the same as in a first- or third-order equivalent single-layer theory. The governing equations of motion are derived from the extended Hamilton's principle. The theory is assessed by comparing the Navier solutions for the free and forced harmonic vibration response of simply supported plates with the exact three-dimensional piezoelasticity solutions. Comparisons for hybrid test, composite and sandwich plates establish that the present theory is quite accurate for the dynamic response of moderately thick plates.

A least-squares approximation method for the time-harmonic Maxwell equations

Abstract: In this paper we introduce and analyze a new approach for the numerical approximation of Maxwell's equations in the frequency domain. Our method belongs to the recently proposed family of negative-norm least-squares algorithms for electromagnetic problems which have already been applied to the electrostatic and magnetostatic problems as well as the Maxwell eigenvalue problem (see [4,5]). The scheme is based on a natural weak variational formulation and does not employ potentials or 'gauge conditions'. The discretization involves only simple, piecewise polynomial, finite element spaces, avoiding the use of the complicated Nédélec elements. An interesting feature of this approach is that it leads to simultaneous approximation of the magnetic and electric fields, in contrast to other methods where one of the unknowns is eliminated and is later computed by differentiation. More importantly, the resulting discrete linear system is well-conditioned, symmetric and positive definite. We demonstrate that the overall numerical algorithm can be efficiently implemented and has an optimal convergence rate, even for problems with low regularity.

Model quality in identification of nonlinear systems

Abstract: In this note, the problem of the quality of identified models of nonlinear systems, measured by the errors in simulating the system behavior for future inputs, is investigated. Models identified by classical methods minimizing the prediction error, do not necessary give "small" simulation error on future inputs and even boundedness of this error is not guaranteed. In order to investigate the simulation error boundedness (SEB) property of identified models, a Nonlinear Set Membership (NSM) method recently proposed by the authors is taken, assuming that the nonlinear regression function, representing the difference between the system to be identified and a linear approximation, has gradient norm bounded by a constant /spl gamma/. Moreover, the noise sequence is assumed unknown but bounded by a constant /spl epsiv/. The NSM method allows to obtain validation conditions, useful to derive "validated regions" within which to suitably choose the bounding constants /spl gamma/ and /spl epsiv/. Moreover, the method allows to derive an "optimal" estimate of the true system. If the chosen linear approximation is asymptotically stable (a necessary condition for the SEB property), in the present note a sufficient condition on /spl gamma/ is derived, guaranteeing that the identified optimal NSM model has the SEB property. If values of /spl gamma/ in the validated region exist, satisfying the sufficient condition, the previous results can be used to give guidelines for choosing the bounding constants /spl gamma/ and /spl epsiv/, additional to the ones required for assumptions validation and useful for obtaining models with "low" simulation errors. The numerical example, representing a mass-spring-damper system with nonlinear damper and input saturation, demonstrates the effectiveness of the presented approach.

Range identification for perspective dynamic systems with 3D imaging surfaces

Abstract: Range identification using image sequence via observations from a traditional camera-type vision system has been discussed in the literature. In this paper, the camera-type planar imaging surface is extended to several well-known 3D surfaces, such as an arbitrary plane, a sphere, an ellipsoid, and a paraboloid. For a general imaging surface, the resulting perspective dynamic system is not guaranteed to preserve an affine form. In this case, most-existing nonlinear observers that are applicable to the perspective dynamic system observed via a camera can not be applied for the range identification problem directly. We show via simulations that our recently proposed linear approximation observer can perform the state estimation.

Capacitor placement in radial distribution networks through a linear deterministic optimization model

Abstract: The classical optimization problem for re- ducing investment costs and energy loss in radial d istribu- tion systems is addressed through a strategy of lin ear approximations and new variable representation. The linear approximation of the problem is performed in two main aspects: one is by using Support Hipper-planes (SH) allowing the approximation of the loss function in each branch of the distribution system. The other is rel ated to the treatment given to the product of integer varia bles and continuous variables. The classical problem is then trans- formed into a new mixed linear integer optimization prob- lem, easy to solve by traditional large scale robus t solvers. Rigid upper and lower limits to the voltage variabl es, instead of using penalty functions to prevent viola tion of limits, allow evaluating voltage profile improvemen t against corresponding cost savings. Examples on several distribution networks and planning periods show the robustness and performance indexes which indicate this method as an appealing alternative to utilities int erested in planning radial distribution networks.

An accurate error measure for adaptive subdivision surfaces

Abstract: A tight estimate on the maximum distance between a subdivision surface and its linear approximation is introduced to guide adaptive subdivision with guaranteed accuracy.

A new and simpler approach for the solution of the electrostatic potential differential equation. Enhanced solution for planar, cylindrical and annular geometries.

Abstract: The motivation of the study performed in this project is focused on deriving a more effective, accurate, and mathematically friendly solution for the prediction of the electrostatic potential, commonly used on electrokinetic research and its related applications. In this contribution, based on the Debye-Huckel approximation, a new solution strategy for the differential equations of the electrostatic potential is proposed. In fact, a simple predictor-corrector calculation is developed to achieve more accurate predictions of electrostatic potential profiles. Furthermore, in this study the authors introduce the correction function f(AO) to the inverse Debye length, lambda. The f(AO) function improves the Debye-Huckel approximation and it is a recursive function of the electrical potential. Once the inverse Debye length, lambda, has been corrected by the f(AO) function and introduced in the simplified solution of the Poisson-Boltzmann equation (i.e., the linear approximation, due to Debye and Huckel), the electrostatic potential outcome little differs from the numerical solution of the complete (nonlinear) differential equation. This new approach embraces different geometries of interest, such as planar, cylindrical, and annular, with excellent results in all the cases and for a wide range of electrostatic potential values. This new predicting semi-analytical technique can be a useful tool on electrical field applications such as the separation of a mixture of macromolecules and the removal of contaminants in soil cleaning processes. Illustrative results are presented for the geometries identified above.

Observability quadratic normal form for discrete-time systems

Abstract: This note deals with quadratic observability normal form for nonlinear discrete-time single-input-single-output (SISO) system. First of all, the main concept of quadratic equivalence with respect to the observability property, is introduced for discrete-time systems. Subsequently, normal form structure for discrete time system is developed for system with unobservable linear approximation in one direction. Finally, the effect of the so-called resonant terms on the observer design and synchronization of chaotic systems is pointed out in an illustrative example.