Showing papers on "Linear approximation published in 1983"

An iterative scheme for variational inequalities

Abstract: In this paper we introduce and study a general iterative scheme for the numerical solution of finite dimensional variational inequalities. This iterative scheme not only contains, as special cases the projection, linear approximation and relaxation methods but also induces new algorithms. Then, we show that under appropriate assumptions the proposed iterative scheme converges by establishing contraction estimates involving a sequence of norms in En induced by symmetric positive definite matrices Gm. Thus, in contrast to the above mentioned methods, this technique allows the possibility of adjusting the norm at each step of the algorithm. This flexibility will generally yield convergence under weaker assumptions.

Two-Dimensional Radiation in Absorbing-Emitting Media Using the P-N Approximation

Abstract: Radiative energy transfer in a gray absorbing and emitting medium is considered in a two-dimensional rectangular enclosure using the P-N differential approximation approximation. The two-dimensional moment of intensity partial differential equations (PDE's) are combined to yield a single second-order PDE for the P-1 approximation and four coupled second-order PDE's for the P-3 approximation. P-1 approximation results are obtained from separation of variables solutions, and P-3 results are obtained numerically using successive-over-relaxation methods. The P-N approximation results are compared with numerical Hottel zone results and with results from an approximation method developed by Modest. The studies show that the P-3 approximation can be used to predict emissive power distributions and heat transfer rates in two-dimensional media with opacities of unity or greater. The P-1 approximation is identical to the diffusion solution and is thus applicable only if the medium is optically dense.

Evaluation of Diffraction Integrals Using Local Phase and Amplitude Approximations

Abstract: An efficient method for computing diffraction integrals is presented. It is based on an idea put forward by Hopkins [1]. The integration domain is divided into subdomains, in each of which the phase and amplitude are approximated by simple functions which make it possible to evaluate the resulting integral in terms of known functions. While Hopkins employed a linear approximation to the phase and a constant approximation to the amplitude, we here approximate both the phase and the amplitude by parabolas. A comparison of the results of our method with those of Hopkins's method shows that our method requires fewer subdomains and less computation time to yield a desired accuracy. Another advantage of our method is that it can be applied to apertures of a general shape without significant loss of accuracy.

Revisions of constraint approximations in the successive QP method for nonlinear programming problems

Abstract: In the last few years the successive quadratic programming methods proposed by Han and Powell have been widely recognized as excellent means for solving nonlinea programming problems. However, there remain some questions about their linear approximations to the constraints from both theoretical and empirical points of view. In this paper, we propose two revisions of the linear approximation to the constraints and show that the directions generated by the revisions are also descent directions of exact penalty functions of nonlinear programming problems. The new technique can cope better with bad starting points than the usual one.

A Note on the Semi-Infinite Programming Approach to Complex Approximation

Abstract: Several observations are made about a recently proposed semi-infinite programming (SIP) method for computation of linear Chebyshev approximations to complex-valued func- tions. A particular discretization of the SIP problem is shown to be equivalent to replacing the usual absolute value of a complex number with related estimates, resulting in a class of quasi-norms on the complex number field C, and consequently a class of quasi-norms on the space C(Q) consisting of all continuous functions defined on Q C C, Q compact. These quasi-norms on C(Q) are estimates of the Loo norm on C(Q) and are useful because the best approximation problem in each quasi-norm can be solved by solving (i) an ordinary linear program if Q is finite or (ii) a simplified SIP if Q is not finite. Glashoff and Roleff (1) solve a semi-infinite program (SIP) which is shown to be equivalent to the linear approximation problem for functions in C(Q), where C(Q) is the space of complex-valued continuous functions on a compact (and not necessarily finite) subset Q of the complex plane C and is equipped with the uniform (Lo) norm

A Superlinearly Convergent Algorithm for One-Dimensional Constrained Minimization Problems with Convex Functions

Abstract: This paper presents a globally and superlinearly convergent algorithm for solving one-dimensional constrained minimization problems involving (not necessarily smooth) convex functions. The constraint is handled by what can be interpreted as a new type of penalty method. The algorithm does not require the objective function to be evaluated at infeasible points and it does not use constraint function values at feasible points. The penalty parameter is automatically generated by the algorithm via linear approximation of the constraint function. As in the unconstrained case developed by Lemarechal and the author, the algorithm uses a step that is the shorter of a quadratic approximation step and a polyhedral approximation step. Here the latter is actually a “penalized” polyhedral step whose computation is well conditioned if the constraint satisfies a nondegeneracy assumption.

Atomic transport via point defects in crystals. I. General kinetic theory of diffusion

Abstract: This paper presents a general formulation of the kinetic theory of solid state diffusion applicable to systems showing small degrees of vacancy and interstitial disorder. These equations are solved in the linear approximation, specifically for steady state conditions. The results verify the correctness of the phenomenological equations provided by non-equilibrium thermodynamics including the Onsager reciprocal relation. This theory includes previous kinetic theories made for a variety of systems as special cases and shows how systematic improvements in accuracy can be made within this framework. The relations between this theory and alternative theories, those derived from the Einstein-Smoluchowski equation and from Kubo linear response theory, are briefly touched upon.

A characterization of an element of best simultaneous approximation

Abstract: Deutsch [4] has suggested that some problems of best simultaneous approximation might profitably be viewed as problems of best approximation in an appropriate product space. A few authors have touched upon this approach; none, however, have pursued it consistently or developed a complete problem along such a line, even in the simplest of cases. In this paper, we show that Deutsch's suggestion can easily be carried out using known results from approximation theory to establish existence, uniqueness, and characterization results. An algorithm guaranteed to converge strongly to the element of best simultaneous approximation under certain circumstances is also proposed.

An optimal output feedback solution to the strip shape multivariable control problem

Abstract: The design of shape control systems for producing flat metal strip products is discussed. Static and dynamic models for a Sendzimir mill are described briefly. Optimal dynamic output feedback solutions are presented for the shape control system design. The optimal control solutions provide guidance on the best structure to be used for shape control. It is shown that by judicious choice of the performance criterion weighting matrices particularly simple controllers may be derived; dimension reduction by parameterisation is also shown to result in a simplification to the controller structure. The effect of nonlinearities in the actuators is discussed, a linear approximation being used for design purposes. A variety of simulation results are presented showing the transient response and the shape control performance of the multivariable system. The effect of mismatch is also demonstrated, that is, using the controller for a mill schedule other than the one for which it was designed.

Linear approximations of nonlinear systems

Abstract: A method for designing an automatic flight controller for short and vertical take off aircraft is presently being developed at NASA Ames Research Center. This technique involves transformations of nonlinear systems to controllable linear systems and takes into account the nonlinearities of the aircraft. In general, the transformations cannot always be given in closed form. Using partial differential equations, an approximate linear system called the modified tangent model, was recently introduced. A linear transformation of this tangent model to Brunovsky canonical form can be constructed, and from this an approximation (about a state space point x0) of an exact transformation for the nonlinear system can be found. Here we show that a canonical expansion in Lie brackets about the point x0 yields the same modified tangent model.

A note on the effects of linear approximation on hypothesis testing

Abstract: In this note we demonstrate that the use of a linear approximation model may effect the usual F -test for linear hypothesis when the true model is non-linear.

Uniform non-linear constrained approximation in normed linear spaces

Abstract: The theory of H-sets as originally defined by L. Collatz has been developed in recent papers and shown to be a unifying concept for linear approximation problems. We here extend the theory of H-sets for non-linear uniform approximation with functional constraints where the approximating functions have a compact domain and range in a Banach space. With the theory of H-sets as here developed the characterization of global and local best approximations follow, and conditions for local to be global approximations are given. A convergence theorem for a descent algorithm is given.

Efficient Model-Based Sequential Designs for Sensitivity Experiments.

Abstract: : A sequential design for estimating the percentiles of a quantal response curve is proposed. Its updating rule is based on an efficient summary of all the data available via a parametric model. Its efficiency in terms of saving the number of runs and its robustness against the distributional assumption are demonstrated heuristically and in a simulation study. A linear approximation to the logit-MLE version of the proposed sequential design is shown to be equivalent to an asymptotically optimal stochastic approximation method, thereby providing a large sample justification. For sample size between 12 and 35, the simulation study shows that the logit-MLE version of the general sequential procedure substantially outperforms an adaptive (and asymptotically optimal) version of the Robbins-Monro method, which in turn outperforms the nonadaptive Robbins-Munro and Up-and-Down methods. A nonparametric sequential design, via the Spearman-Karber estimator, for estimating the median is also proposed. (Author)

The Nonlinearity Factors in Analysis of a Conductive Arc Track

Abstract: The corrections introduced by an accurate numerical analysis of actual nonlinear thermal conditions in a conductive track under an arc are compared to those for the linear approximation up to the ablation stage. The relevant nonlinearity factors are: the actual arc-current and power distribution, temperature dependences of insulation properties and their stepwise change at the phase transition, and voltage gradient feedback from the track temperature. The plot of temperature distribution in the track shows that the error introduced by the use of a linear approach can be considered as moderate when area of the track which are important for the recovery dielectric strength are considered.

On the brownian movement of a rotator and its application to nonlinear dielectric relaxation

Abstract: The Debye theory of dielectric relaxation as corrected for inertial effects has as yet been only considered in the linear approximation. There, the rise and decay transients are identical. Here a method recently developed for the treatment of a rotator in a periodic potential is applied to calculate the transient behaviour when the linear approximation is discarded. The Kramers equation for the problem is expanded in a set of orthogonal functions which lead to a set of linear differential difference equations giving the relaxation behaviour. It is shown that the Mori formalism for the problem leads to the same set of differential difference equations as the Kramers equation.

Resonance phenomena in the nonlinear equations of a proper semiconductor h2Δu=shu

Abstract: One considers a Steklov-type boundary-value problem for the nonlinear equation of a semiconductor. Under the assumption of the existence on the surface of the semiconductor of a closed geodesic, stable in a linear approximation, one constructs asymptotic solutions which are concentrated in the neighborhood of this geodesic. The obtained solutions are expressed in terms of the known asymptotic eigenfunctions of the Laplace operator on a Riemann manifold and in terms of the multisoliton solutions of the Sine-Gordon equation. Similar solutions are obtained for the mixed boundary-value problem.

Relativistic derivation of the Maxwell-Lorentz equations for a medium

Abstract: In the transition from Maxwell's equation in vacuum to the equations for a medium, the problem of calculating the average total current density in the medium arises. Until now, this problem has only been solved in the linear approximation. Here a rigorous derivation is given. We also consider the physical meaning of the magnetization vector and its expression in terms of the parameters of the medium in the general (not only static) case.

Polarization radiation in the planetary atmosphere delimited by a heterogeneous diffusely reflecting surface

Abstract: Spatial frequency characteristics (SFC) and the scattering functions were studied in the two cases of a uniform horizontal layer with absolutely black bottom, and an isolated layer. The mathematical model for these examples describes the horizontal heterogeneities in a light field with regard to radiation polarization in a three dimensional planar atmosphere, delimited by a heterogeneous surface with diffuse reflection. The perturbation method was used to obtain vector transfer equations which correspond to the linear and nonlinear systems of polarization radiation transfer. The boundary value tasks for the vector transfer equation that is a parametric set and one dimensional are satisfied by the SFC of the nonlinear system, and are expressed through the SFC of linear approximation. As a consequence of the developed theory, formulas were obtained for analytical calculation of albedo in solving the task of dissemination of polarization radiation in the planetary atmosphere with uniform Lambert bottom.

Nonlinear interaction of longitudinal—Transverse acoustic waves in a circular cylinder

Abstract: The limiting amplitudes of acoustic oscillations in a cylindrical volume of a heat releasing medium in which one or several modes are unstable in the linear approximation are determined. One of the mechanisms limiting the amplitudes of unstable acoustic modes is the transfer of energy from them to damped modes by nonlinear interaction. The nonlinear interactions of plane acoustic waves in a long channel have been considered by Artamonov and Vorob'ev [1]; in the present paper, the interaction of mixed longitudinal—transverse acoustic modes in a closed cylindrical volume is considered. The equations describing the interaction of two and three longitudinal—transverse modes are derived and investigated in the quadratic approximation by the method of slowly varying amplitudes and phases of the oscillations [2]. The treatment is applicable to a high-temperature gas, for which general stability conditions in the linear approximation have been formulated by Artamonov [3].

Reply to Comments on 'A Matched Delay Approach to Subtractive Linear Phase High-Pass Filtering'

Abstract: passband best preserves the waveform of the signal, except for a Fig. 1. Block diagram of a matched delay subtractive (MDS) high-pass filter. time delay proport ional to the slope of the phase response. Though techniques for linear phase low-pass filtering are well known, analog methods for constructing linear phase high-pass [l-~H2(j,)(]c-isw. Th e magni tude factor [ 1 ) H2( jw) I] is the filters are not general ly d iscussed in the standard references on amplitude response and is clearly that of a high-pass filter, while filter design, except for the case in which the passband is small the phase factor c-jso signifies a l inear-phase response. compared to the high-pass cutoff f requency (i.e., the bandpass case). This paper presents a semianalog method for linear phase II. MDS FILTER USING A BESSEL LOW-PASS ELEMENT