ON A CLASS OF NONLINEAR UNIFORM APPROXIMATION PROBLEMS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＸｂｅａｃｏｍｐａｃｔＨａｕｓｄｏｒｆｆｓｐａｃｅ ,ｗｉｔｈＣ(Ｘ)ｔｈｅｓｐａｃｅｏｆｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎｓｄｅｆｉｎｅｄｏｎＸ ,ａｎｄｌｅｔＭ Ｃ(Ｘ)ｂｅｎｏｎｅｍｐｔｙ.ＬｅｔＦ(ｘ ,ｙ) :Ｘ×Ｒ →Ｒｂｅａｎｏｎ ｎｅｇａｔｉｖｅｆｕｎｃｔｉｏｎ ,ａｎｄｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｍｉｎｉｍｉｚａｔｉｏｎｐｒｏｂｌｅｍ :ｆｉｎｄｙ∈Ｍｔｏｍｉｎｉｍｉｚｅ‖Ｆ(.,ｙ )‖ ,(1)ｗｈｅｒｅ‖Ｆ (.,ｙ )‖ =ｓｕｐｘ∈ＸＦ(ｘ ,ｙ(ｘ) ) .Ｔｈｅｍ…     

模糊粗糙近似算子公理集的独立性

用双论域上的模糊关系定义了广义模糊粗糙近似算子,并讨论了近似算子的性质。用公理刻画了模糊集合值算子,各种公理化的近似算子可以保证找到相应的二元模糊关系,使得由模糊关系通过构造性方法定义的模糊粗糙近似算子恰好就是用公理定义的近似算子。讨论了刻画各种特殊近似算子的公理集的独立性,从而给出各种特殊模糊关系所对应的模糊粗糙近似算子的最小公理集。     

复赋范空间中复RS集的最佳同时逼近

研究了复赋范空间中具限制系数的广义多项式集G对无穷序列的最佳同时逼近问题,得到了特征定理;当G是复RS集时还得到了惟一性定理．     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.     

Convergence Rates of Adaptive Methods,Besov Spaces,and Multilevel Approximation

This paper concerns characterizations of approximation classes associated with adaptive finite element methods with isotropic h-refinements. It is known from the seminal work of Binev, Dahmen, DeVore and Petrushev that such classes are related to Besov spaces. The range of parameters for which the inverse embedding results hold is rather limited, and recently, Gaspoz and Morin have shown, among other things, that this limitation disappears if we replace Besov spaces by suitable approximation spaces associated with finite element approximation from uniformly refined triangulations. We call the latter spaces multievel approximation spaces and argue that these spaces are placed naturally halfway between adaptive approximation classes and Besov spaces, in the sense that it is more natural to relate multilevel approximation spaces with either Besov spaces or adaptive approximation classes, than to go directly from adaptive approximation classes to Besov spaces. In particular, we prove embeddings of multilevel approximation spaces into adaptive approximation classes, complementing the inverse embedding theorems of Gaspoz and Morin. Furthermore, in the present paper, we initiate a theoretical study of adaptive approximation classes that are defined using a modified notion of error, the so-called total error, which is the energy error plus an oscillation term. Such approximation classes have recently been shown to arise naturally in the analysis of adaptive algorithms. We first develop a sufficiently general approximation theory framework to handle such modifications, and then apply the abstract theory to second-order elliptic problems discretized by Lagrange finite elements, resulting in characterizations of modified approximation classes in terms of memberships of the problem solution and data into certain approximation spaces, which are in turn related to Besov spaces. Finally, it should be noted that throughout the paper we paid equal attention to both conforming and non-conforming triangulations.     

Near-field and far-field approximations by the Adomian and asymptotic decomposition methods

The Adomian decomposition method and the asymptotic decomposition method give the near-field approximate solution and far-field approximate solution, respectively, for linear and nonlinear differential equations. The Padé approximants give solution continuation of series solutions, but the continuation is usually effective only on some finite domain, and it can not always give the asymptotic behavior as the independent variables approach infinity. We investigate the global approximate solution by matching the near-field approximation derived from the Adomian decomposition method with the far-field approximation derived from the asymptotic decomposition method for linear and nonlinear differential equations. For several examples we find that there exists an overlap between the near-field approximation and the far-field approximation, so we can match them to obtain a global approximate solution. For other nonlinear examples where the series solution from the Adomian decomposition method has a finite convergent domain, we can match the Padé approximant of the near-field approximation with the far-field approximation to obtain a global approximate solution representing the true, entire solution over an infinite domain.     

Nonlinear Approximation with Dictionaries I. Direct Estimates

We study various approximation classes associated with m-term approximation by elements from a (possibly redundant) dictionary in a Banach space. The standard approximation class associated with the best m-term approximation is compared to new classes defined by considering m-term approximation with algorithmic constraints: thresholding and Chebychev approximation classes are studied, respectively. We consider embeddings of the Jackson type (direct estimates) of sparsity spaces into the mentioned approximation classes. General direct estimates are based on the geometry of the Banach space, and we prove that assuming a certain structure of the dictionary is sufficient and (almost) necessary to obtain stronger results. We give examples of classical dictionaries in L^p spaces and modulation spaces where our results recover some known Jackson type estimates, and discuss some new estimates they provide.     

A RAPID APPROXIMATION FOR PREDICTING THE HYDROCARBON DISCOVERY RATE: PART 2. IMPROVING THE ACCURACY

In Part 1 of this paper, we noted the systematic errors in the estimates of means and standard deviations produced by a rapid approximation applied to a model of hydrocarbon discovery. In Part 2, we apply regression to predict the approximation errors, as functions of model parameters and approximation output. With the regression model, we can correct much of the error in the approximation, as we illustrate with data from the Nisku-Shelf play of western Canada.     

Fast one-sided approximation with spline functions

For the approximation of functions, interpolation compromises approximation error for computational convenience. For a bounded interpolation operator the Lebesque inequality bounds the factor by which the interpolation differs from the best approximation available in the range of the operator. A comparable process for one-sided approximation is not readily apparent. Methods are suggested for the computationally economical construction of one-sided spline approximation to large classes of functions, and criteria for comparing such approximation operators are investigated. Since the operators are generally nonlinear the Lebesque inequality is invalidated as an aid for comparing with the best one-sided approximation in the range of the operator, but comparable inequalities are shown to exist in some cases.     

General characterization of best approximation and its application to uniform approximation with restricted ranges

Let R be a normed linear space, K be an arbitrary convex subset of an n-dimensional subspace Φ _n ⊂R. This paper first gives a general charactaerization for a best approximation from K in form of “zero in the convex hull”. Applying it to the uniform approximation by generalized polynomials with restricted ranges, we get further an alternation characterization. Our results ocntains the special cases of interpolatory approximation, positive approximation, copositive approximation, and the classical characterizations in forms of convex hull and alternation in approximation without restriction.     

The weak metric approximation property

We introduce and investigate the weak metric approximation property of Banach spaces which is strictly stronger than the approximation property and at least formally weaker than the metric approximation property. Among others, we show that if a Banach space has the approximation property and is 1-complemented in its bidual, then it has the weak metric approximation property. We also study the lifting of the weak metric approximation property from Banach spaces to their dual spaces. This enables us, in particular, to show that the subspace of c₀, constructed by Johnson and Schechtman, does not have the weak metric approximation property. The research of the second-named author was partially supported by Estonian Science Foundation Grant 5704 and the Norwegian Academy of Science and Letters.     

一般模糊近似算子的公理化刻画

给出无限双论域上一般模糊近似算子的构造性定义,叙述一般模糊近似算子的基本性质。引入邻域有限模糊关系的概念,利用上、下模糊粗糙近似的截集性质,给出一个刻画模糊近似算子的新公理,得到不同于以往的刻画模糊近似算子的公理集。     

On variational approximations in quantum molecular dynamics

The Dirac-Frenkel-McLachlan variational principle is the basic tool for obtaining computationally accessible approximations in quantum molecular dynamics. It determines equations of motion for an approximate time-dependent wave function on an approximation manifold of reduced dimension. This paper gives a near-optimality result for variational approximations. It bounds the error in terms of the distance of the exact wave function to the approximation manifold and identifies the parameters that control the deviation of the variational approximation from the best approximation on the manifold.

     

Approximation bounds for sparse principal component analysis

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. The sparse maximum eigenvalue problem cannot be efficiently approximated up to a constant approximation ratio, so our bounds depend on the optimum value of the semidefinite relaxation: the higher this value, the better the approximation. In particular, these bounds allow us to control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component.     

A Markov Embedding Approximation for a Stochastic Population Model with Exogenous Disturbances

We present and study an approximation scheme for the mean of a stochastic simulation that models a population subject to nonlinear birth and exogenous disturbances. We use the information from the probability distribution for the disturbance times to construct a method that improves upon the mean-field approximation. We show through two example systems the effectiveness of the Markov embedding approximation and discuss the contexts in which it is an appropriate method.     

Exact and approximation algorithms for a soft rectangle packing problem

We consider the problem of finding an arrangement of rectangles with given areas that minimizes the total length of all inner and outer border lines. We present a polynomial time approximation algorithm and derive an upper bound estimation on its approximation ratio. Furthermore, we give a formulation of the problem as mixed-integer nonlinear program and show that it can be approximatively reformulated as linear mixed-integer program. On a test set of problem instances, we compare our approximation algorithm with another one from the literature. Using a standard numerical mixed-integer linear solver, we show that adding the solutions from the approximation algorithm as advanced starter helps to reduce the overall solution time for proven global optimality, or gives better primal and dual bounds if a certain time-limit is reached before.     

Adaptive radial basis function interpolation using an error indicator

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.     

Upper and lower approximation models in interval regression using regression quantile techniques

In this paper, we propose new interval regression analysis based on the regression quantile techniques. To analyze a phenomenon in a fuzzy environment, we propose two interval approximation models. Without using all data, we first identify the main trend from the designated proportion of the given data. To select the main part of data to be analyzed, we introduce the regression quantile techniques. The obtained model is not influenced by extreme points since it is formulated from the center-located main proportion of the given data. After that, the interval regression model including all data can be identified based on the acquired main trend. The obtained interval regression model by the main proportion of the given data is called the lower approximation model, while interval regression model by all data is called the upper approximation model for the given phenomenon. Also it is shown that, from the lower approximation model (main trend) and the upper approximation model, we can construct a trapezoidal fuzzy model. The membership function of this fuzzy model is useful to obtain the locational information for each observation. The characteristic of our approach can be described as obtaining the upper and lower approximation models and combining them to be a fuzzy model for representing the given phenomenon in a fuzzy environment.     

Local approximation on surfaces with discontinuities,given limited order Fourier coefficients

We present a spline approximation method for a piece of a surface where jump discontinuities occur along curves. The data for the surface is assumed to be Fourier coefficients which are limited in order and possibly contaminated with noise. The support of the approximation is bounded by three sides of a rectangle with a fourth boundary possibly curved. Discontinuities of the surface may occur across the curved side and linear sides adjacent to it. The approximation uses a small number of lines through the support and parallel to the straight boundary lines that are adjacent to the curve. Along each line a one-dimensional spline approximation is done for a section of the surface over the line. This approximation uses two-dimensional Fourier coefficient data, localizing spline functions, and a technique which we developed earlier for one-dimensional analogues of the problem. We use a spline quasi-interpolation scheme to create a surface approximation from the section approximations. The result is accurate even when the surface is discontinuous across the curved boundary and adjacent side boundaries.