Showing papers on "Linear approximation published in 1997"
01 Jan 1997
TL;DR: After demonstrating the gain of nonlinear approximation over linear approximation measured in a Sobolev scale, it is reviewed some recent results on the SoboleV and Besov regularity of solutions to elliptic boundary value problems.
Abstract: This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions, the best possible approximation rate which a function can have for a given number of degrees of freedom is characterized in terms of its regularity in a certain scale of Besov spaces. Therefore, after demonstrating the gain of nonlinear approximation over linear approximation measured in a Sobolev scale, we review some recent results on the Sobolev and Besov regularity of solutions to elliptic boundary value problems. On the other hand, nonlinear approximation requires information that is generally not available in practice. Instead one has to resort to the concept of adaptive approximation. We briefly summarize some recent results on wavelet based adaptive schemes for elliptic operator equations. In contrast to more conventional approaches one can show that these schemes converge without prior assumptions on the solution, such as the saturation property. One central objective of this paper is to contribute to interrelating nonlinear approximation and adaptive methods in the context of elliptic operator equations.
72 citations
TL;DR: It is shown that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method.
Abstract: Given an orthonormal basis and a certain class X of vectors in a Hilbert space H, consider the following nonlinear approximation process: approach a vector $x\in X$ by keeping only its N largest coordinates, and let N go to infinity. In this paper, we study the accuracy of this process in the case where $H=L^2(I)$, and we use either the trigonometric system or a wavelet basis to expand this space. The class of function that we are interested in is described by a stochastic process. We focus on the case of "piecewise stationary processes" that describe functions which are smooth except at isolated points. We show that the nonlinear wavelet approximation is optimal in terms of mean square error and that this optimality is lost either by using the trigonometric system or by using any type of linear approximation method, i.e., keeping the N first coordinates. The main motivation of this work is the search for a suitable mathematical model to study the compression of images and of certain types of signals.
61 citations
TL;DR: Any nonlinear programming problem, including non-convex ones, with an objective function that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.
Abstract: The goal of increasing computational efficiency is one of the fundamental challenges of both theoretical and applied research in mathematical modeling. The pursuit of this goal has lead to wide diversity of efforts to transform a specific mathematical problem into one that can be solved efficiently. Recent years have seen the emergence of highly efficient methods and software for solving Mixed Integer Programming Problems, such as those embodied in the packages CPLEX, MINTO, XPRESS-MP. The paper presents a method to develop a piece-wise linear approximation of an any desired accuracy to an arbitrary continuous function of two variables. The approximation generalizes the widely known model for approximating single variable functions, and significantly expands the set of nonlinear problems that can be efficiently solved by reducing them to Mixed Integer Programming Problems. By our development, any nonlinear programming problem, including non-convex ones, with an objective function (and/or constraints) that can be expressed as sums of component nonlinear functions of no more than two variables, can be efficiently approximated by a corresponding Mixed Integer Programming Problem.
53 citations
TL;DR: An efficient initial approximation method for multiplicative division and square root is proposed, a modification of the piecewise linear approximation that requires only a bit-wise inversion and a one-bit shift.
Abstract: An efficient initial approximation method for multiplicative division and square root is proposed. It is a modification of the piecewise linear approximation. The multiplication and the addition required for the linear approximation are replaced by only one multiplication with a slight modification of the operand. The same accuracy is achieved. The modification of the operand requires only a bit-wise inversion and a one-bit shift, and can be implemented by a very simple circuit. One clock cycle may be saved, because the addition is removed. The required table size is also reduced, because only one coefficient instead of two has to be stored.
49 citations
01 Oct 1997
TL;DR: Presents an algorithm for the visualization of vector field topology based on Clifford algebra that allows the detection of higher-order singularities by first analysing the possible critical points and then choosing a suitable polynomial approximation.
Abstract: Presents an algorithm for the visualization of vector field topology based on Clifford algebra. It allows the detection of higher-order singularities. This is accomplished by first analysing the possible critical points and then choosing a suitable polynomial approximation, because conventional methods based on piecewise linear or bilinear approximation do not allow higher-order critical points and destroy the topology in such cases. The algorithm is still very fast, because of using linear approximation outside the areas with several critical points.
48 citations
TL;DR: This work describes the construction of diagonally implicit multistage integration methods of order and stage order p = q = 7 and p =q = 8 for ordinary differential equations using state-of-the-art optimization methods, particularly variable-model trust-region least-squares algorithms.
41 citations
01 Oct 1997
TL;DR: The paper presents a framework for multiresolution compression and geometric reconstruction of arbitrarily dimensioned data designed for distributed applications and discusses the problem of oracles in semiorthogonal settings and proposes sophisticated oracles to remove unimportant coefficients.
Abstract: The paper presents a framework for multiresolution compression and geometric reconstruction of arbitrarily dimensioned data designed for distributed applications. Although being restricted to uniform sampled data, the versatile approach enables the handling of a large variety of real world elements. Examples include nonparametric, parametric and implicit lines, surfaces or volumes, all of which are common to large scale data sets. The framework is based on two fundamental steps: compression is carried out by a remote server and generates a bit-stream transmitted over the underlying network. Geometric reconstruction is performed by the local client and renders a piecewise linear approximation of the data. More precisely, the compression scheme consists of a newly developed pipeline starting from an initial B-spline wavelet precoding. The fundamental properties of wavelets allow progressive transmission and interactive control of the compression gain by means of global and local oracles. In particular the authors discuss the problem of oracles in semiorthogonal settings and propose sophisticated oracles to remove unimportant coefficients. In addition, geometric constraints such as boundary lines can be compressed in a lossless manner and are incorporated into the resulting bit-stream. The reconstruction pipeline performs a piecewise adaptive linear approximation of data using a fast and easy to use point removal strategy which works with any subsequent triangulation technique.
38 citations
TL;DR: In this article, the authors compared three methods for computing diffusive mass transfer in non-reactive high temperature flows and showed that the widely used effective binary approximation may lead to large errors (up to 70%) in the evaluation of the mass fluxes, reflecting, therefore, on the solution of the species conservation equations.
30 citations
TL;DR: In this article, a semi-analytical extension of the Sobolev approximation for the line radiative transfer problem in molecular clouds and outflows is developed, which is applied to test the range of the validity of the ordinary Soboleve approximation and to solve problems beyond its limits.
29 citations
TL;DR: In this paper, an automatic inverse method was developed for generating layered earth models from electrical sounding data, which has the minimum number of layers required to fit a resistivity sounding curve or a combined resistivity and induced polarization sounding.
Abstract: An automatic inverse method has been developed for generating layered earth models from electrical sounding data. The models have the minimum number of layers required to fit a resistivity sounding curve or a combined resistivity and induced polarization sounding. The ground is modeled using a very large number of thin layers to accommodate arbitrary variations. The properties of the layers are optimized using as a constraint the L1 norm of the vertical derivative of the resistivity distribution. The use of linear programming leads to piecewise smooth distributions that simulate traditional models made up of a few uniform layers. The process considers from the simplest model of a uniform half‐space to models of many layers, without fixing a priori the number of discontinuities. The solution is sought by iterating a new linear approximation, similar to the classical process of linearization, except that a reference model is not present in either the data vector or the unknown function. For induced polariza...
TL;DR: In this paper, the validity of the moist tangent linear model (TLM) derived from a time-dependent 1D Eulerian cloud model is investigated by comparing TLM solutions to differences between results from a nonlinear model identically perturbed.
Abstract: The validity of the moist tangent linear model (TLM) derived from a time-dependent 1D Eulerian cloud model is investigated by comparing TLM solutions to differences between results from a nonlinear model identically perturbed. The TLM solutions are found to be highly sensitive to the amplitude of the applied perturbation, and thus the linear approximation is valid only for a specific range of perturbations. The TLM fails to describe the evolution of perturbations when the initial variation is given on a parameter in the vicinity of a nonlinear regime change, a result that has important implications for many cloud-scale processes. Uncertainty imposed on certain aspects of microphysical processes can have a significant influence on the behavior of the TLM solutions, and in some cases this behavior can be explained by the particular discretizations used to solve the equations. The frequency at which the nonlinear basic state is updated in the TLM can also have a profound effect on the TLM validity, ...
01 Sep 1997
TL;DR: The inversion algorithm coupled with the method of Meier becomes a powerful tool to the random number based on one-dimensional cellular showing that these random number generators provide less security than their state size would imply.
Abstract: An algorithm for inverting an iteration of the one-dimensional cellular automaton is presented. The algorithm is based on the linear approximation of the updating function, and requires less than exponential time for particular classes of updating functions and seed values. For example, an n-cell cellular automaton based on the updating function CA30 can be inverted in O(n) time for certain seed values, and, at most, 2/sup n/2/ trials are required for arbitrary seed values. The inversion algorithm requires at most 2(/sup q-1/)(/sup 1-a/)/sup n/ trials for arbitrary nonlinear functions and seed values, where q is the number of variables of the updating function, and a is the probability of agreement between the function and its best affine approximation. The inversion algorithm coupled with the method of Meier and becomes a powerful tool to the random number based on one-dimensional cellular showing that these random number generators provide less security than their state size would imply.
01 Jan 1997
TL;DR: This paper considers certain problems of nonlinear approximation which arise in image processing, this includes approximation using m terms from a dictionary of functions and greedy algorithms for approximation from such a dictionary.
TL;DR: In this paper, the Cauchy-Schwarz condition for the invertibility of a kriging system is linked to a simple situation of coregionalization.
Abstract: Myers developed a matrix form of the cokriging equations, but one that entails the solution of a large system of linear equations. Large systems are troublesome because of memory requirements and a general increase in the matrix condition number. We transform Myers’s system into a set of smaller systems, whose solution gives the classical kriging results, and provides simultaneously a nested set of lower dimensional cokriging results. In the course of developing the new formulation we make an interesting link to the Cauchy-Schwarz condition for the invertibility of a system, and another to a simple situation of coregionalization. In addition, we proceed from these new equations to a linear approximation to the cokriging results in the event that the crossvariograms are small, allowing one to take advantage of a recent results of Xie and others which proceeds by diagonalizing the variogram matrix function over the lag classes.
01 Jul 1997
TL;DR: In this article, a simple state-space representation for fractional linear filters, in the form of a system made up of an infinite number of ordinary differential equations, can be derived in a straightforward manner from the Taylor expansion of (1 − z)d.
Abstract: This paper shows that a simple state-space representation for the so-called fractional linear filters, in the form of a system made up of an infinite number of ordinary differential equations, can be derived in a straightforward manner from the Taylor expansion of (1 − z)d. As an immediate corollary, we also obtain a finite-dimensional approximation of this representation, corresponding to a rational approximation of the fractional filter. These results are applied to continuous time filter with transfer function ((s + b)/(s + a))d.
22 Jun 1997
TL;DR: In this paper, a nonlinear discrete-time model derived under the assumption of piecewise linearity of the system but without the usual linear-ripple approximation is presented for analysis of subharmonic oscillations under large parameter and large signal variations in a converter with current-mode programming.
Abstract: This paper presents analysis of subharmonic oscillations under large parameter and large signal variations in a converter with current-mode programming. Onset of oscillations is investigated through analysis of period-doubling bifurcation. The analysis is based on a nonlinear discrete-time model derived under the assumption of piecewise linearity of the system but without the usual linear-ripple approximation. Results are confirmed by measurements on an experimental converter and by simulations. It is also shown that a model based on the linear-ripple approximation gives erroneous results both quantitatively and qualitatively.
TL;DR: In this article, the evolution of the spectrum of isotropic uniform radiation in an infinite space filled with a homogeneous, nonrelativistic electron gas is calculated by solving the Kompaneets equation.
Abstract: The evolution of the spectrum of isotropic uniform radiation in an infinite space filled with a homogeneous, nonrelativistic electron gas is calculated by solving the Kompaneets equation. For an infinitely narrow initial spectrum, the time dependence of the average frequency and frequency dispersion is determined in a linear approximation of the equation. Characteristic times corresponding to changes in the character of this dependence are introduced. Two schemes are proposed for the numerical solution of the nonlinear equation: a nonconservative scheme with a grid that is uniform in frequency and a conservative scheme with automatic selection of an adaptive grid in frequency and time. For the linear equation the method yields results consistent with calculations of its solutions in terms of an eigenfunction expansion of the Kompaneets operator calculated in [D. I. Nagirner and V. M. Loskutov, Astrofizika, 40, 97 (1977)]. The influence of nonlinearity on the evolution of the spectrum of initially monochromatic radiation of various intensities is traced as an example of the application of the method.
TL;DR: In this paper, a nonlinear programming (NLP) model is developed to obtain approximate solutions for the problem of minimizing the variance of a non-linear function of several random variables, where the decision variables are the mean values of these random variables.
TL;DR: An accurate and efficient parameter extraction methodology, utilizing a new technique called recursive inverse approximation (RIA), is proposed for statistical modeling of integrated circuits.
Abstract: An accurate and efficient parameter extraction methodology, utilizing a new technique called recursive inverse approximation (RIA), is proposed for statistical modeling of integrated circuits. The main features of RIA are (1) linear approximation is used to obtain initial model parameter estimates, (2) reverse verification performs accuracy checking, and (3) error correction functions are constructed in the extracted parameter space to recursively refine the previously extracted parameter values. As a result, an approximate inverse mapping from the measured performance space to the model parameter space is established for statistical parameter extraction. Examples of parameter extraction for MOS transistors and IC multiplier block demonstrate high efficiency and accuracy of the new method.
TL;DR: The tangent linear model (TLM) as discussed by the authors is obtained by linearizing the governing equations around a space and time-dependent basic state referred to as the trajectory, which describes to first-order the evolution of perturbations in a nonlinear model and is now used widely in many applications including four-dimensional data assimilation.
Abstract: The tangent linear model (TLM) is obtained by linearizing the governing equations around a space- and time-dependent basic state referred to as the trajectory. The TLM describes to first-order the evolution of perturbations in a nonlinear model and it is now used widely in many applications including four-dimensional data assimilation. This paper is concerned with the difficulties that arise when developing the tangent linear model for a semi-Lagrangian integration scheme. By permitting larger time steps than those of Eulerian advection schemes, the semi-Lagrangian treatment of advection improves the model efficiency. However, a potential difficulty in linearizing the interpolation algorithms commonly used in semi-Lagrangian advection schemes has been described by Polavarapu et al, who showed that for infinitesimal perturbations, the tangent linear approximation of an interpolation scheme is correct if and only if the first derivative of the interpolator is continuous at every grid point. Here, t...
02 Jul 1997
TL;DR: Two estimation methods based on state space representation of chirp signal parameters are presented, based on an exact nonlinear state model of the signal but the state estimation is issued from extended Kalman filtering, which is an approximate state estimator as it is based on a linearized model.
Abstract: We consider the problem of estimating chirp signal parameters. We present two estimation methods based on state space representation of such signals. The first one uses an improvement of Tretter's approximation to obtain an approximate linear state model. Then the state estimation is performed by Kalman filtering, which is an optimal linear state estimator. The second method is based on an exact nonlinear state model of the signal but the state estimation is issued from extended Kalman filtering, which is an approximate state estimator as it is based on a linearized model. The performance of the two methods is compared by simulation. Finally we extend the second algorithm to multi-component chirp signals, which is impossible for the first method.
TL;DR: In this paper, a feedback system for aircraft nonlinear equations of motion is designed without using linear approximation, which is based on inverse dynamics transformation and singular perturbation theory, and enables us to control the nonlinear dynamics directly.
Abstract: A feedback system for aircraft nonlinear equations of motion is designed in this paper without using linear approximation. The design method is based on inverse dynamics transformation and singular perturbation theory, and enables us to control the nonlinear dynamics directly. This leads to an advantage that the dynamics can be controlled by a single control law through the whole flight envelope. To evaluate and substantiate the design method, ALFLEX (AutoLanding FLight EXperiment) plane is selected, and simulations are given regarding on the time separation between inner and outer loops and on the robustness of modeling errors.
TL;DR: In this article, a semi-analytical Transfer Function Model (TFM) is proposed to describe the thermosphere, which is based on the work of C.O. Hines.
TL;DR: In this paper, the existence of a real analytic stabilizing feedback control of this form is established, and necessary and sufficient conditions for such a stabilizing control of real analytic feedback control are given.
Abstract: If a smooth nonlinear affine control system has a controllable linear approximation, a standard technique for constructing a smooth (linear) asymptotically stabilizing feedbackcontrol is via the
LQR (linear, quadratic, regulator) method. The nonlinear system may not have a controllable linear approximation, but instead may be shown to be small (or large) time locally controllable via a high order, homogeneous approximation. In this case one can attempt to construct an asymptotically stabilizing feedback control as the optimal control, relative to a cost functional with homogeneous integrand, for the approximating system. Necessary, and some sufficient, conditions for the existence of a smooth (real analytic), stabilizing feedback control of this form are given. For some systems which satisfy these necessary conditions, the specific form of a stabilizing control is established.
TL;DR: The problem of adaptive polygonization of regular surfaces of the euclidean 3D space is discussed, and effective algorithms for computing optimal polygonizations of surfaces described in parametric or implicit form are presented.
Abstract: We discuss the problem of adaptive polygonization of regular surfaces of the euclidean 3D space, and present effective algorithms for computing optimal polygonizations of surfaces described in parametric or implicit form
Patent•
07 Jan 1997TL;DR: In this paper, a fuzzy logic processor is used to process a measurement of the voltage and provide an information element representing the current, which can be combined with a Gaussian interpolation.
Abstract: A circuit for the measurement of current in a load includes a non-linear element for transforming a current into a voltage and includes a fuzzy logic processor to process a measurement of the voltage and provide an information element representing the current. The device permits the precise use of semi-conductor or other components for the measurement of current in making a linear approximation of the current-voltage characteristics of the semi-conductors used by using triangular type membership functions. The linear approximation may also be combined with a Gaussian interpolation.
Proceedings Article•
01 Jan 1997
TL;DR: This workigate this problem for B6zier-curves in two or higher dimensions, i.e. parameterized curves of the form n C(t) = ~ Bi,n( t)pi, t E [0,1], to efficiently approximate object of the first type by objects of the second type.
TL;DR: In this paper, a triple-deck approach is used to reduce the Cauchy problem for a thin near-wall region to unsteady three-dimensional nonlinear boundary-layer equations subject to an interaction law.
Abstract: The problem for a thin near-wall region is reduced, within the triple-deck approach, to unsteady three-dimensional nonlinear boundary-layer equations subject to an interaction law. A linear version of the boundary-value problem describes eigenmodes of different nature (including crossflow vortices) coupled together. The frequency ω of the eigenmodes is connected with the components k and m of the wavenumber vector through a dispersion relation. This relation exhibits two singular properties. One of them is of basic importance since it makes the imaginary part Im(ω) of the frequency increase without bound as k and m tend to infnity along some curves in the real (k, m)-plane. The singularity turns out to be strong, rendering the Cauchy problem ill posed for linear equations.Accounting for the second-order approximation in asymptotic expansions for the upper and main decks brings about significant alterations in the interaction law. A new mathematical model leans upon a set of composite equations without rescaling the original independent variables and desired functions. As a result, the right-hand side of a modified dispersion relation involves an additional term multiplied by a small parameter e=R−1/8, R being the reference Reynolds number. The aforementioned strong singularity is missing from solutions of the modified dispersion relation. Thus, the range of validity of a linear approximation becomes far more extended in ω, k and m, but the incorporation of the higher-order term into the interaction law means in essence that the Reynolds number is retained in the formulation of a key problem for the lower deck.