Showing papers on "Linear approximation published in 1999"

Approximations to the Zoeppritz equations and their use in AVO analysis

Abstract: To efficiently invert seismic amplitudes for elastic parameters, pseudoquartic approximations to the Zoeppritz equations are derived to calculate P-P-wave reflection and transmission coefficients as a function of the ray parameter p. These explicit expressions have a compact form in which the coefficients of the p2 and p4 terms are given in terms of the vertical slownesses. The amplitude coefficients are also represented as a quadratic function of the elastic contrasts at an interface and are compared to the linear approximation used in conventional amplitude variation with offset (AVO) analysis, which can invert for only two elastic parameters. Numerical analysis with the second‐order approximation shows that the condition number of the Frechet matrix for three elastic parameters is improved significantly from using a linear approximation. Therefore, those quadratic approximations can be used directly with amplitude information to estimate not only two but three parameters: P-wave velocity contrast, S-wa...

On extended partially linear single-index models

Abstract: Aiming to explore the relation between the response y and the stochastic explanatory vector variable X beyond the linear approximation, we consider the single-index model, which is a well-known approach in multidimensional cases. Specifically, we extend the partially linear single-index model to take the form y = β 0 T X + o(0 0 T X) + e, where e is a random variable with Ee = 0 and var(e) = σ 2 , unknown, β 0 and θ o are unknown parametric vectors and o(.) is an unknown real function. The model is also applicable to nonlinear time series analysis. In this paper, we propose a procedure to estimate the model and prove some related asymptotic results. Both simulated and real data are used to illustrate the results.

Influence of physical processes on the tangent‐linear approximation

Abstract: A comprehensive set of linear physical parametrizations is developed for the tangent-linear andadjoint versions of ECMWF global forecast model. The following processes are described :vertical diffusion, subgrid-scale orographic effects, large scale condensation, long-wave radiationand deep cumulus convection. The accuracy of a tangent-linear model including these processesis examined by comparing 24-h forecasts obtained from an adiabatic tangent-linear version andpairs of non-linear integrations including full physics. It is shown that for finite size perturbations(analysis increments), the inclusion of physics improves the fit to the non-linear model. Theimprovement is largest for specific humidity, where combined effects of vertical diffusion (nearthe surface), large-scale condensation (in the mid-latitudes troposphere) and moist convection(in the lower tropical troposphere) contribute to a better evolution of moisture increments.Simplications have been designed with respect to the operational non-linear physics, mostly toavoid the growth of spurious unstable modes. Computation of singular vectors has revealedthat the linear package does not contain such spurious unphysical structures. This comprehensiveset of linear parametrizations is currently used in the ECMWF operational 4D-Var assimilationsystem. DOI: 10.1034/j.1600-0870.1999.00001.x

Solution to strongly nonlinear parabolic problems by a linear approximation scheme

Abstract: A nonlinear parabolic problem is solved by a linear approximation scheme using a nonstandard time discretization with a relaxation function. The relaxation function is determined by iterations. Optimal rate of convergence is proved for semi-discretization (in time). Error estimates are obtained for the full discretization scheme (in time and space). The proposed approximation scheme converges also in the case of degenerate parabolic problems.

Toeplitz Preconditioners Constructed from Linear Approximation Processes

Abstract: Preconditioned conjugate gradients (PCG) are widely and successfully used methods to solve Toeplitz linear systems An(f)x=b. Here we consider preconditioners belonging to trigonometric matrix algebras and to the band Toeplitz class and we analyze them from the viewpoint of the function theory in the case where f is supposed continuous and strictly positive. First we prove that the necessary (and sufficient) condition, in order to devise a superlinear PCG method, is that the spectrum of the preconditioners is described by a sequence of approximation operators "converging" to f . The other important information we deduce is that while the matrix algebra approach is substantially not sensitive to the approximation features of the underlying approximation operators, the band Toeplitz approach is. Therefore, the only class of methods for which we may obtain impressive evidence of superlinear convergence behavior is the one [S. Serra, Math. Comp., 66 (1997), pp. 651--665] based on band Toeplitz matrices with weakly increasing bandwidth.

Parametric yield formulation of MOS IC's affected by mismatch effect

Abstract: A rigorous formulation of the parametric yield for very large scale integration (VLSI) designs including the mismatch effect is proposed. The theory has been carried out starting from a general statistical model relating random variations of device parameters to the stochastic behavior of process parameters. The model predicts a dependence of correlation, between devices fabricated in the same die, on their dimensions and mutual distances so that mismatch between equally designed devices can be considered as a particular case of such a model. As an application example, a new model for the autocorrelation function is proposed from which the covariance matrix of the parameters is derived. By assuming a linear approximation, a suitable formulation of the parametric yield for VLSI circuit design is obtained in terms of the covariance matrix of parameters.

Moving embedded solitons

Abstract: The first theoretical results are reported predicting moving solitons residing inside (embedded into) the continuous spectrum of radiation modes. The model taken is a Bragg-grating medium with Kerr nonlinearity and additional second-derivative (wave) terms. The moving embedded solitons (ESs) are doubly isolated (of codimension two), but, nevertheless, structurally stable. Like quiescent ESs, moving ESs are argued to be stable to linear approximation, and semi-stable nonlinearly. Estimates show that moving ESs may be experimentally observed as ~10 fs pulses with velocity 1/10th that of light.

Fresnel coefficients of a rough interface

Abstract: Second-order perturbation theory has been used to compute the optical transfer coefficients of a rough dielectric interface. The derivation proceeds through classical solution of Maxwell’s equations at normal incidence. Reflection and transmission coefficients were first obtained for an interface represented by a sine profile of wave vector G interacting with a planar wave polarized perpendicular to G. Second, the expressions have been generalized to a real rough interface, i.e., the sum of sine profiles, and arbitrary polarization. We discuss the validity of the linear approximation, comparing the reflection calculated by the more general Davies–Bennett formula to that calculated by our formula. It appears that, for most real multilayer systems, this approximation is valid.

Bilinear Driving Force Approximation in the Modeling of a Simulated Moving Bed Using Bidisperse Adsorbents

Abstract: The linear driving force (LDF) approximation has been extensively used to represent the intraparticle diffusion in adsorptive processes regardless of its real nature This work introduces a bilinear driving force (bi-LDF) approximation to account for intraparticle diffusion and to obtain concentration profiles in the macropores and microparticles of the adsorbent The bi-LDF approximation is applied to the simulation of glucose/fructose breakthrough curves in a fixed bed and to their separation in a simulated moving bed (SMB) Equivalence with the simple LDF model is well-established for both fixed-bed operation and the SMB mode The flow rate constraints that define a region of separation (both product purities higher than 995%) for a SMB configuration are shown to be greatly affected by relatively small deviations in the macro- and microparticle time constants, which illustrate the relevance of the proposed approximation

Secular gravitational instability of a compressible viscoelastic sphere

Abstract: For a self-gravitating viscoelastic compressible sphere we have shown that unstable modes can exist by means of the linear viscoelastic theory by both initial-value and normal-mode approaches. For a uniform sphere we have derived analytical expressions for the roots of the secular determinant based on the asymptotic expansion of the spherical Bessel functions. From the two expressions, both the destabilizing nature of gravitational forces and the stabilizing influences of increasing elastic strength are revealed. Fastest growth times on the order of ten thousand years are developed for the longest wavelength. In contrast, a self-gravitating incompressible viscoelastic model is found to be stable. This result of linear approximation suggests that a more general approach, e.g., non-Maxwellian rheology or a non-linear finite-amplitude theory, should be considered in global geodynamics.

Approximation Bias in Linearized Euler Equations

Abstract: A wide range of empirical applications rely on linear approximations to dynamic Euler equations. Among the most notable of these is the large and growing literature on precautionary saving that examines how consumption growth and saving behavior are affected by uncertainty and prudence. Linear approximations to Euler equations imply a linear relationship between expected consumption growth and uncertainty in consumption growth, with a slope coefficient that is a function of the coefficient of relative prudence. This literature has produced puzzling results: Estimates of the coefficient of relative prudence (and the coefficient of relative risk aversion) from regressions of consumption growth on uncertainty in consumption growth imply estimates of prudence and risk aversion that are unrealistically low. Using numerical solutions to a fairly standard intertemporal optimization problem, our results show that the actual relationship between expected consumption growth and uncertainty in consumption growth differs substantially from the relationship implied by a linear approximation. We also present Monte Carlo evidence that shows that the instrumental variables methods commonly used to estimate the parameters correct some, but not all, of the approximation bias.

Hydrogen atom in a spherical well: linear approximation

Abstract: We discuss the boundary effects on a quantum system by examining the problem of a hydrogen atom in a spherical well. By using an approximation method which is linear in energy we calculate the boundary corrections to the ground-state energy and wave function. We obtain the asymptotic dependence of the ground-state energy on the radius of the well.

Convergence of the Equations for a Maxwell Fluid

Abstract: The equations for the flow of a viscoelastic fluid of the Maxwell type are analyzed in a linear approximation. First, we establish that the solution depends continuously on changes in the relaxation time. Next, we investigate how the solution to the linearized Maxwell system converges to the solution to Stokes flow as the relaxation time tends to zero. Convergence in different measures is examined and specific a priori bounds are derived.

Contour dynamics, waves, and solitons in the quantum Hall effect

Abstract: We present a theoretical study of the excitations on the edge of a two-dimensional electron system in a perpendicular magnetic field in terms of a contour dynamics formalism. In particular, we focus on edge excitations in the quantum Hall effect. Beyond the usual linear approximation, a nonlinear analysis of the shape deformations of an incompressible droplet yields soliton solutions which correspond to shapes that propagate without distortion. A perturbative analysis is used and the results are compared to analogous systems, like vortex patches in ideal hydrodynamics. Under a local induction approximation, we find that the contour dynamics is described by a nonlinear partial differential equation for the curvature: the modified Korteweg‐de Vries equation. @S0163-1829~99!13339-3#

Nonlinear distortions and multisine signals. I. Measuring the best linear approximation

Abstract: This paper examines the effects of nonlinearities on periodic multifrequency signals. A class of broadband pilot test signals is proposed, termed sparse odd multisines, which can be used to establish the system bandwidth and detect nonlinearities. Signals are then defined within this class which allow the measurement of the best linear approximation to a nonlinear system. A comparison is made with related work in this area.

Solving Bilevel Network Design Problem Using a Linear Reaction Function Without Nondegeneracy Assumption

Abstract: A sensitivity analysis-based heuristic is proposed for application to solving the continuous equilibrium network design problem. The heuristic uses a minimum distance approach to generate sensitivity information and release the nondegeneracy assumption. The trip maker's behavior is then represented by a linear approximation of the reaction function that is generated from the sensitivity information. Two numerical examples are given.

Optimal design problem of system reliability with interval coefficient using improved genetic algorithms

Abstract: In this paper, we first formulate a De Novo nonlinear integer programming (NIP-I(DN)) problem of system reliability with interval coefficients. It is used for estimating and designing optimal reliability of an incomplete fault detecting and switching (FDS) system. Because of this we are able to use a linear approximation for monotonically increasing reliability function (i.e., objective function) of the original NIP problem. We then transformed it into Knapsack problem with interval coefficients. Last, the problem is solved directly using improved genetic algorithms (IGA) by keeping the nonlinear constraints. We discuss and compare the efficiency between the proposed method and the former one.

A linear approximation algorithm using BSDE

Abstract: This note demonstrates that a linear approximation algorithm based on Chow’s Lagrangean one-dimensional backward stochastic differential equations (BSDEs) converges under reasonable assumptions.

Solution of Parabolic Equations by Backward Euler-Mixed Finite Element Methods on a Dynamically Changing Mesh

Abstract: We develop and analyze methods based on combining the lowest-order mixed finite element method with backward Euler time discretization for the solution of diffusion problems on dynamically changing meshes. The methods developed are shown to preserve the optimal rate error estimates that are well known for static meshes. The novel aspect of the scheme is the construction of a linear approximation to the solution, which is used in projecting the solution from one mesh to another. Extensions to advection-diffusion equations are discussed, where the advection is handled by upwinding. Numerical results validating the theory are also presented.

Linear Stability and Pressure-Driven Response Function of Solid Propellants with Phase Transition

Abstract: An extension of the Zel’dovich–Novozhilov approach to adiabatic burning of solid energetic materials subjected to a concentrated phase transition is presented. The pressure-driven frequency response function and intrinsic stability boundary are obtained in the linear approximation of the problem. The intrinsic stability boundary is portrayed as a parametric representation of oscillatory burning frequency. The corresponding previous results are recovered as a special case for no phase transition. The surface-temperature sensitivity parameter r is deduced by assuming the Arrhenius surface pyrolysis law. It is shown that phase transition may strongly affect the frequency response function, notwithstanding its limited thermal effect, if the operating point moves closer to the stability boundary. Some typical results are discussed. To validate these theoretical expectations, accurate error estimates of experimental results are needed.

Accuracy of stochastic perturbuation methods: the case of asset pricing models

Abstract: This paper investigates the accuracy of a perturbation method in approximating the solution to stochastic equilibrium models under rational expectations. As a benchmark model, we use a version of asset pricing models proposed by Burnside [1988] which admits a closed-form solution while not making the assumptions of certainty equivalence. We then check the accuracy of perturbation methods -extended to a stochastic environment- against the closed form solution. Second an especially fourth order expansions are then found to be more efficient than standard linear approximation, as they are able to account for higher order moments of the distribution.

On the best linear approximation methods and the widths of certain classes of analytic functions

Abstract: We discuss the best linear approximation methods in the Hardy spaceH q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained.

Reinforcement Learning From State and Temporal Differences

Abstract: TD( ) with function approximation has proved empirically successful for some complex reinforcement learning problems. For linear approximation, TD( ) has been shown to minimise the squared error between the approximate value of each state and the true value. However, as far as policy is concerned, it is error in the relative ordering of states that is critical, rather than error in the state values. We illustrate this point, both in simple two-state and three-state systems in which TD( )—starting from an optimal policy—converges to a sub-optimal policy, and also in backgammon. We then present a modified form of TD( ), called STD( ), in which function approximators are trained with respect to relative state values on binary decision problems. A theoretical analysis, including a proof of monotonic policy improvement for STD( ) in the context of the two-state system, is presented, along with a comparison with Bertsekas’ differential training method [1]. This is followed by successful demonstrations of STD( ) on the two-state system and a variation on the well known acrobot problem.

An approximation tour

Abstract: In signal processing, orthogonal bases are of interest because they can efficiently approximate certain types of signals with just a few vectors. Two examples of such applications are image compression and the estimation of noisy signals. Approximation theory studies the error produced by different approximation schemes in an orthonormal basis. A linear approximation projects the signal over M vectors chosen a priori. In Fourier or wavelet bases, this linear approximation is particularly precise for uniformly regular signals. However, better approximations are obtained by choosing the M basis vectors depending on the signal. Signals with isolated singularities are well approximated in a wavelet basis with this non-linear procedure. A further degree of freedom is introduced by choosing the basis adaptively, depending on the signal properties. From families of wavelet packet bases and local cosine bases, a fast dynamical programming algorithm is used to select the best basis that minimizes a Schur concave cost function. The approximation vectors chosen from this best basis outline the important signal structures, and characterize their time-frequency properties. Pursuit algorithms generalize these adaptive approximations by selecting the approximation vectors from redundant dictionaries of time-frequency atoms, with no orthogonality constraint.