Showing papers on "Linear approximation published in 2009"

A vector radiative transfer model for coupled atmosphere and ocean systems based on successive order of scattering method

Abstract: A vector radiative transfer model has been developed for coupled atmosphere and ocean systems based on the Successive Order of Scattering (SOS) Method. The emphasis of this study is to make the model easy-to-use and computationally efficient. This model provides the full Stokes vector at arbitrary locations which can be conveniently specified by users. The model is capable of tracking and labeling different sources of the photons that are measured, e.g. water leaving radiances and reflected sky lights. This model also has the capability to separate florescence from multi-scattered sunlight. The δ - fit technique has been adopted to reduce computational time associated with the strongly forward-peaked scattering phase matrices. The exponential - linear approximation has been used to reduce the number of discretized vertical layers while maintaining the accuracy. This model is developed to serve the remote sensing community in harvesting physical parameters from multi-platform, multi-sensor measurements that target different components of the atmosphere-oceanic system.

An Adaptive Linear Approximation Algorithm for Copositive Programs

Abstract: We study linear optimization problems over the cone of copositive matrices. These problems appear in nonconvex quadratic and binary optimization; for instance, the maximum clique problem and other combinatorial problems can be reformulated as such problems. We present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit. In contrast to previous approximation schemes, our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function, yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not. Using these approximations, we derive an adaptive linear approximation algorithm for copositive programs. Numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems.

Efficient hardware implementation of the hyperbolic tangent sigmoid function

Abstract: Efficient implementation of the activation function is important in the hardware design of artificial neural networks. Sigmoid, and hyperbolic tangent sigmoid functions are the most widely used activation functions for this purpose. In this paper, we present a simple and efficient architecture for digital hardware implementation of the hyperbolic tangent sigmoid function. The proposed method employs a piecewise linear approximation as a foundation, and further improves the results using a lookup table. Our design proves to be more efficient considering area × delay as a performance metric when compared to similar proposals. VLSI implementation of the proposed design using a 0.18µm CMOS process is also presented, which shows a 35% improvement over similar recently published architectures.

Loss-Minimizing Control of PMSM With the Use of Polynomial Approximations

Abstract: Normally, lookup-table-based methods are being utilized for loss-minimizing control of permanent magnet synchronous motors (PMSMs). But numerous repetitive experiments are required to make a lookup table, and the program size becomes bulky. In this paper, analytic methods for finding the loss-minimizing solution are studied. Since the solution lies either in the interior or on the voltage limit boundary, two different cases are dealt with separately. In both cases, fourth-order polynomials are derived. To obtain approximate solutions, methods of order reduction and linear approximation are utilized. The accuracies are good enough for practical use. These approximate solutions are fused into a proposed loss-minimizing algorithm and implemented in an inverter digital signal processor. Experiments were done with a real PMSM developed for a sport utility fuel cell electric vehicle. The analytically derived minima were justified by experimental evidences, and the dynamic performances over a wide range of speed were shown to be satisfactory.

Comparison of monolithic and splitting solution schemes for dynamic porous media problems

Abstract: 5 SUMMARY Proceeding from the governing equations describing a saturated poroelastic material with intrinsically 7 incompressible solid and fluid constituents, we compare the monolithic and splitting solution of the different multi-field formulations feasible in porous media dynamics. Because of the inherent solid-fluid momentum 9 interactions, one is concerned with the class of volumetrically coupled problems involving a potentially strong coupling of the momentum equations and the algebraic incompressibility constraint. Here, the 11 resulting set of differential-algebraic equations (DAE) is solved by the finite element method (FEM) following two different strategies: (1) an implicit monolithic approach, where the equations are first 13 discretized in space using stable mixed finite elements and second in time using stiffly accurate implicit time integrators; (2) a semi-explicit-implicit splitting scheme in the sense of a fractional-step method, 15 where the DAE are first discretized in time, split using intermediate variables, and then discretized in space using linear equal-order approximations for all primary unknowns. Finally, a one- and a two-dimensional 17 wave propagation example serve to reveal the pros and cons in regard to accuracy and stability of both solution strategies. Therefore, several test cases differing in the used multi-field formulation, the 19 monolithic time-stepping method, and the approximation order of the individual unknowns are analyzed for varying degrees of coupling controlled by the permeability parameter. In the end, we provide a 21 reliable recommendation which of the presented strategies and formulations is the most suitable for which particular dynamic porous media problem. Copyright q 2009 John Wiley & Sons, Ltd. 23

Recovery-Based Error Estimator for Interface Problems: Conforming Linear Elements

Abstract: This paper studies a new recovery-based a posteriori error estimator for the conforming linear finite element approximation to elliptic interface problems. Instead of recovering the gradient in the continuous finite element space, the flux is recovered through a weighted $L^2$ projection onto $H(\mathrm{div})$ conforming finite element spaces. The resulting error estimator is analyzed by establishing the reliability and efficiency bounds and is supported by numerical results. This paper also proposes an adaptive finite element method based on either the recovery-based estimators or the edge estimator through local mesh refinement and establishes its convergence. In particular, it is shown that the reliability and efficiency constants as well as the convergence rate of the adaptive method are independent of the size of jumps.

A New Look at Projected Gradient Method for Equilibrium Assignment

Abstract: A new adaptation of Rosen's projected gradient algorithm for solving fixed-demand equilibrium traffic assignments is developed. It is based on a Gauss-Seidel decomposition scheme in which origin-destination pairs are considered sequentially. The method operates in the space of path flows and shares this approach with earlier work on adapting the gradient projection method, the restricted simplicial decomposition, and the projected gradient adapted for solving equilibrium traffic assignments with explicit capacity constraints. The details of the algorithm are nevertheless quite different and are intended to solve large-scale problem instances. The development of the method is provided, and then computational experiments are performed with an implementation done with the Emme software package. Performance comparisons are carried out against the linear approximation method and the origin base algorithm code of Bar-Gera. The algorithm compares well with these methods and achieves relative gaps of the order of...

Robustness Issues of the Best Linear Approximation of a Nonlinear System

Abstract: In many engineering applications, linear models are preferred, even if it is known that the system is disturbed by nonlinear distortions. A large class of nonlinear systems, which are excited with a ldquoGaussianrdquo random excitation, can be represented as a linear system G BLA plus a nonlinear noise source Y S . The nonlinear noise source represents that part of the output that is not captured by the linear approximation. In this paper, it is shown that the best linear approximation G BLA and the power spectrum S Y S of the nonlinear noise source Y S are invariants for a wide class of excitations with a user-specified power spectrum. This shows that the alternative ldquolinear representationrdquo of a nonlinear system is robust, making its use in the daily engineering practice very attractive. This result also opens perspectives to a new generation of dynamic system analyzers that also provide information on the nonlinear behavior of the tested system without increasing the measurement time.

Mean-field approximations for coupled populations of generalized linear model spiking neurons with markov refractoriness

Abstract: There has recently been a great deal of interest in inferring network connectivity from the spike trains in populations of neurons. One class of useful models that can be fit easily to spiking data is based on generalized linear point process models from statistics. Once the parameters for these models are fit, the analyst is left with a nonlinear spiking network model with delays, which in general may be very difficult to understand analytically. Here we develop mean-field methods for approximating the stimulus-driven firing rates (in both the time-varying and steady-state cases), auto- and cross-correlations, and stimulus-dependent filtering properties of these networks. These approximations are valid when the contributions of individual network coupling terms are small and, hence, the total input to a neuron is approximately gaussian. These approximations lead to deterministic ordinary differential equations that are much easier to solve and analyze than direct Monte Carlo simulation of the network activity. These approximations also provide an analytical way to evaluate the linear input-output filter of neurons and how the filters are modulated by network interactions and some stimulus feature. Finally, in the case of strong refractory effects, the mean-field approximations in the generalized linear model become inaccurate; therefore, we introduce a model that captures strong refractoriness, retains all of the easy fitting properties of the standard generalized linear model, and leads to much more accurate approximations of mean firing rates and cross-correlations that retain fine temporal behaviors.

A Posteriori Error Estimates in the Maximum Norm for Parabolic Problems

Abstract: We derive a posteriori error estimates in the $L_\infty((0,T];L_\infty(\Omega))$ norm for approximations of solutions to linear parabolic equations. Using the elliptic reconstruction technique introduced by Makridakis and Nochetto and heat kernel estimates for linear parabolic problems, we first prove a posteriori bounds in the maximum norm for semidiscrete finite element approximations. We then establish a posteriori bounds for a fully discrete backward Euler finite element approximation. The elliptic reconstruction technique greatly simplifies our development by allowing the straightforward combination of heat kernel estimates with existing elliptic maximum norm error estimators.

Automatic linearization of nonlinear skinning

Abstract: Linear blending is a very popular skinning technique for virtual characters, even though it does not always generate realistic deformations. Recently, nonlinear blending techniques (such as dual quaternions) have been proposed in order to improve upon the deformation quality of linear skinning. The trade-off consists of the increased vertex deformation time and the necessity to redesign parts of the 3D engine. In this paper, we demonstrate that any nonlinear skinning technique can be approximated to an arbitrary degree of accuracy by linear skinning, using just a few samples of the nonlinear blending function (virtual bones). We propose an algorithm to compute this linear approximation in an automatic fashion, requiring little or no interaction with the user. This enables us to retain linear skinning at the core of our 3D engine without compromising the visual quality or character setup costs.

Limitations of a linear model for the hurricane: boundary layer

Abstract: The linear model for the steady boundary layer of a rapidly rotating axisymmetric vortex is derived from a detailed scale analysis of the full equations of motion. The previously known analytic solution is re-appraised for vortices of hurricane scale and strength. The internal consistency of the linear approximation is investigated for such a vortex by calculating from the solution the magnitude of the nonlinear terms that are neglected in the approximation compared with the terms retained. It is shown that the nonlinear terms are not negligibly small in a large region of the vortex, a feature that is consistent with the scale analysis. We argue that the boundary-layer problem is well-posed only at outer radii where there is subsidence into the layer. At inner radii, where there is ascent, only the radial pressure gradient may be prescribed and not the wind components at the top of the boundary layer, but the linear problem cannot be solved in these circumstances. We examine the radius at which the vertical flow at the top of the boundary layer changes sign for different tangential wind profiles relevant to hurricanes and show that this is several hundred kilometres from the vortex centre. This feature represents a further limitation of the linear model applied to hurricanes. While the present analysis assumes axial symmetry, the same limitations presumably apply to non-axisymmetric extensions to the linear model. Copyright c � 2009 Royal Meteorological Society

Approximation Schemes for Functional Optimization Problems

Abstract: Approximation schemes for functional optimization problems with admissible solutions dependent on a large number d of variables are investigated. Suboptimal solutions are considered, expressed as linear combinations of n-tuples from a basis set of simple computational units with adjustable parameters. Different choices of basis sets are compared, which allow one to obtain suboptimal solutions using a number n of basis functions that does not grow "fast" with the number d of variables in the admissible decision functions for a fixed desired accuracy. In these cases, one mitigates the "curse of dimensionality," which often makes unfeasible traditional linear approximation techniques for functional optimization problems, when admissible solutions depend on a large number d of variables.

Non‐linear non‐local Cosmology

Abstract: Non‐local equations of motion contain an infinite number of derivatives and commonly appear in a number of string theory models. We review how these equations can be rewritten in the form of a diffusion‐like equation with non‐linear boundary conditions. Moreover, we show that this equation can be solved as an initial value problem once a set of non‐trivial initial conditions that satisfy the boundary conditions is found. We find these initial conditions by looking at the linear approximation to the boundary conditions. We then numerically solve the diffusion‐like equation, and hence the non‐local equations, as an initial value problem for the full non‐linear potential and subsequently identify the cases when inflation is attained.

Bounded Linear Stability Analysis - A Time Delay Margin Estimation Approach for Adaptive Control

Abstract: This paper presents a method for estimating time delay margin for model-reference adaptive control of systems with almost linear structured uncertainty. The bounded linear stability analysis method seeks to represent the conventional model-reference adaptive law by a locally bounded linear approximation within a small time window using the comparison lemma. The locally bounded linear approximation of the combined adaptive system is cast in a form of an input-time-delay differential equation over a small time window. The time delay margin of this system represents a local stability measure and is computed analytically by a matrix measure method, which provides a simple analytical technique for estimating an upper bound of time delay margin. Based on simulation results for a scalar model-reference adaptive control system, both the bounded linear stability method and the matrix measure method are seen to provide a reasonably accurate and yet not too conservative time delay margin estimation.

Uncertainty quantification for unsteady fluid flow using adjoint-based approaches

Abstract: Uncertainty quantification of numerical simulations has raised significant interest in recent years. One of the main challenges remains the efficiency in propagating uncertainties from the sources to the quantities of interest, especially when there are many sources of uncertainties. The traditional Monte Carlo methods converge slowly and are undesirable when the required accuracy is high. Most modern uncertainty propagation methods such as polynomial chaos and collocation methods, although extremely efficient, suffer from the so called "curse of dimensionality". The computational resources required for these methods grow exponentially as the number of uncertainty sources increases. The aim of this work is to address the challenge of efficiently propagating uncertainties in numerical simulations with many sources of uncertainties. Because of the large amount of information that can be obtained from adjoint solutions, we focus on using adjoint equations to propagate uncertainties more efficiently. Unsteady fluid flow simulations are the main application of this work, although the uncertainty propagation methods we discuss are applicable to other numerical simulations. We first discuss how to solve the adjoint equations for time-dependent fluid flow equations. We specifically address the challenge associated with the backward time advance of the adjoint equation, requiring the solution of the primal equation in backward order. Two methods are proposed to address this challenge. The first method solves the adjoint equation forward in time, completely eliminating the need for storing the solution of the primal equation. The other method is a checkpointing algorithm specifically designed for dynamic time-stepping. The adjoint equation is still solved backward in time, but the present scheme retrieves the primal solution in reverse order. This checkpointing method is applied to an incompressible Navier-Stokes adjoint solver on unstructured mesh. With the adjoint equation solved, we obtain a linear approximation of the quantities of interest as functions of the random variables describing the uncertainty sources in a probabilistic setting. We use this linear approximation to accelerate the convergence of the Monte Carlo method in calculating tail probabilities for estimating margins and risk. In addition, we developed a multivariate interpolation scheme that uses multiple adjoint solutions to construct an interpolant of the quantities of interest as functions of the uncertainty sources. This interpolation scheme converge exponentially to the true function, thus providing very accurate and efficient means of propagating of uncertainties and remains accurate independently of the locations of the available data.

On spin-2 electromagnetic interactions

Abstract: In this paper we (re)consider the problem of electromagnetic interactions for massless spin-2 particles and show that in (A)dS spaces with nonzero cosmological constant it is indeed possible (at least in linear approximation) to switch on minimal electromagnetic interactions supplemented by third derivative non-minimal ones which are necessary to restore gauge invariance.

Approximation by certain linear operators preserving x^2

Abstract: We investigate certain positive linear operators Ln preserving the functions ek(x) = x k , k = 0, 1, and modified operators Ln which preserve e0 and e2 . We show that the error of approximation of f by L ∗ n(f) is smaller than for Ln(f) .

A Method of Successive Corrections of the Control Subspace in the Reduced-Order Variational Data Assimilation*

Abstract: A version of the reduced control space four-dimensional variational method (R4DVAR) of data assimilation into numerical models is proposed. In contrast to the conventional 4DVAR schemes, the method does not require development of the tangent linear and adjoint codes for implementation. The proposed R4DVAR technique is based on minimization of the cost function in a sequence of low-dimensional subspaces of the control space. Performance of the method is demonstrated in a series of twin-data assimilation experiments into a nonlinear quasigeostrophic model utilized as a strong constraint. When the adjoint code is stable, R4DVAR’s convergence rate is comparable to that of the standard 4DVAR algorithm. In the presence of strong instabilities in the direct model, R4DVAR works better than 4DVAR whose performance is deteriorated because of the breakdown of the tangent linear approximation. Comparison of the 4DVAR and R4DVAR also shows that R4DVAR becomes advantageous when observations are sparse and noisy.

Near Optimum Low Complexity Smoothing Loops for Dynamical Phase Estimation—Application to BPSK Modulated Signals

Abstract: This correspondence provides and analyzes a low complexity, near optimum, fixed-interval smoothing algorithm that approaches the performance of an optimal smoother for the price of two low complexity sequential estimators, i.e., two phase-locked loops (PLLs). Based on a linear approximation of the problem, a theoretical performance evaluation is given. The theoretical results are compared to some simulation results and to the Bayesian and hybrid Cramer-Rao bounds. They illustrate the good performance of the proposed smoothing PLL (S-PLL) algorithm.

Semi-Implicit Approximation of Wave Maps into Smooth or Convex Surfaces

Abstract: A semi-implicit scheme using linear finite elements for the approximation of wave maps into smooth or convex surfaces is devised, and its stability is analyzed. Convergence is established for the case of the unit sphere as a target manifold, which is unconditional in the case of $(2+1)$ Minkowski space. Numerical experiments illustrate the theoretical results.