Good Approximation Properties Research Articles

Comonotone Adaptive Interpolating Splines

For any given data we propose the construction of an interpolating spline of class C1, which is either a quadratic polynomial or a linear/linear rational function between the knots, and preserves the monotonicity of the data on the sections of rational intervals. We prove the uniqueness and existence of this spline. Numerical tests show good approximation properties and flexibility due to the non-coincidence of the given data arguments and the spline knots which can be chosen freely.

Parametrizations of symmetric orthogonal multifilter banks with different filter lengths

This paper is devoted to a study of parametrizations of symmetric orthogonal multifilter banks with different filter lengths. To construct symmetric orthogonal multifilter banks { H , G } which generate balanced multiwavelets of multiplicity 2, the filter lengths of the rows of H , regarded as the scalar filters, must be different. In this paper, complete factorizations of symmetric orthogonal multifilter banks with different filter lengths are obtained. Based on these factorizations, construction of balanced multiwavelets with good approximation and smoothness properties are discussed.

Orthonormal Ridgelets and Linear Singularities

We construct a new orthonormal basis for $L^2({\Bbb R}^2)$, whose elements are angularly integrated ridge functions---{\it orthonormal ridgelets}. The basis elements are smooth and of rapid decay in the spatial domain, and in the frequency domain are localized near angular wedges which, at radius $r = 2^j$, have radial extent $\Delta r \approx 2^j$ and angular extent $\Delta \theta \approx 2\pi/2^{j}$. Orthonormal ridgelet expansions expose an interesting phenomenon in nonlinear approximation: they give very efficient approximations to objects such as $1_{\{ x_1\cos\theta+ x_2\sin\theta > a\}} \ e^{-x^2_1-x^2_2}$ which are smooth away from a discontinuity along a line. The orthonormal ridgelet coefficients of such objects are sparse: they belong to every $\ell^p$, p > 0. This implies that simple thresholding in the ridgelet orthobasis is, in a certain sense, a near-ideal nonlinear approximation scheme for such objects. Orthonormal ridgelets may be viewed as L2 substitutes for approximation by sums of ridge functions, and so can perform many of the same tasks as the ridgelet systems constructed by Candès [Ph.D. Thesis, Department of Statistics, Stanford University, Stanford, CA, 1998; Appl. Comput. Harmon. Anal., 6 (1999), pp. 197--218]. Orthonormal ridgelets make available the machinery of orthogonal decompositions, which is not available for ridge functions as they are not in $L^2({\Bbb R}^2)$. The ridgelet orthobasis is constructed as the isometric image of a special wavelet basis for Radon space; as a consequence, ridgelet analysis is equivalent to a special wavelet analysis in the Radon domain. This means that questions of ridgelet analysis of linear singularities can be answered by wavelet analysis of point singularities. At the heart of our nonlinear approximation result is the study of a certain tempered distribution on ${\Bbb R}^2$ defined formally by S(u,v) = |v|^{-1/2} \sigma(u/|v|)$ with $\sigma$ a certain smooth bounded function; this is singular at (u,v) = (0,0) and $C^\infty$ elsewhere. The key point is that the analysis of this point singularity by tensor Meyer wavelets yields sparse coefficients at high frequencies; this is reflected in the sparsity of the ridgelet coefficients and the good nonlinear approximation properties of the ridgelet basis.

Anisotropic mesh refinement for problems with internal and boundary layers

The character of convection-dominated, singularly perturbed boundary value problems requires their special numerical treatment in order to guarantee stability and resolve existing layers with acceptable accuracy. In addition to discretization methods particularly developed for this aim, recently more and more attention has been directed towards adapted triangulations of the computational domain. In this paper, an adaptive strategy based on an anisotropic refinement is developed for finite element methods. Starting from some a priori information about the location of layers, the so-called hybrid meshes are constructed. By these meshes, the flexibility of unstructured meshes, good approximation properties in layers, and relatively simple rules for a posteriori anisotropic refinement are combined with each other. The efficiency of this procedure is demonstrated by selected numerical examples. Copyright © 1999 John Wiley & Sons, Ltd.

Approximation and compression of piecewise smooth functions

Wavelet or sub–band coding has been quite successful in compression applications, and this success can be attributed in part to the good approximation properties of wavelets. In this paper, we revisit rate–distortion (RD) bounds for the wavelet approximation of piecewise smooth functions, and piecewise polynomial functions in particular. We contrast these results with RD bounds achievable using an oracle–based method. We then introduce a practical dynamic programming algorithm, which achieves performance similar to the oracle method, and present experimental results.

On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form

In the approximation of linear elliptic operators in mixed form, it is well known that the so-called inf-sup and ellipticity in the kernel properties are sufficient (and, in a sense to be made precise, necessary) in order to have good approximation properties and optimal error bounds. One might think, in the spirit of Mercier-Osborn-Rappaz-Raviart and in consideration of the good behavior of commonly used mixed elements (like Raviart-Thomas or Brezzi-Douglas-Marini elements), that these conditions are also sufficient to ensure good convergence properties for eigenvalues. In this paper we show that this is not the case. In particular we present examples of mixed finite element approximations that satisfy the above properties but exhibit spurious eigenvalues. Such bad behavior is proved analytically and demonstrated in numerical experiments. We also present additional assumptions (fulfilled by the commonly used mixed methods already mentioned) which guarantee optimal error bounds for eigenvalue approximations as well.

Risk bounds for model selection via penalization

Performance bounds for criteria for model selection are developed using recent theory for sieves. The model selection criteria are based on an empirical loss or contrast function with an added penalty term motivated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of observations. Most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function. We show that the quadratic risk of the minimum penalized empirical contrast estimator is bounded by an index of the accuracy of the sieve. This accuracy index quantifies the trade-off among the candidate models between the approximation error and parameter dimension relative to sample size.If we choose a list of models which exhibit good approximation properties with respect to different classes of smoothness, the estimator can be simultaneously minimax rate optimal in each of those classes. This is what is usually called adaptation. The type of classes of smoothness in which one gets adaptation depends heavily on the list of models. If too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously, it may happen that the estimator is only approximately adaptive (typically up to a slowly varying function of the sample size).We shall provide various illustrations of our method such as penalized maximum likelihood, projection or least squares estimation. The models will involve commonly used finite dimensional expansions such as piecewise polynomials with fixed or variable knots, trigonometric polynomials, wavelets, neural nets and related nonlinear expansions defined by superposition of ridge functions.

Multinode finite element based on boundary integral equations

A finite element constructed on the basis of boundary integral equations is proposed. This element has a flexible shape and arbitrary number of nodes. It also has good approximation properties. A procedure of constructing an element stiffness matrix is demonstrated first for one-dimensional case and then for two-dimensional steady-state heat conduction problem. Numerical examples demonstrate applicability and advantages of the method. © 1998 John Wiley & Sons, Ltd.

Weighted NP Optimization Problems: Logical Definability and Approximation Properties

Extending a well-known property of NP optimization problems in which the value of the optimum is guaranteed to be polynomially bounded in the length of the input, it is observed that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a logical characterization. It is shown that any NP optimization problem can be stated as a problem in which the constraint conditions can be expressed by a $\Pi_2$ first-order formula. The paper analyzes the weighted analogue of all syntactically defined classes of optimization problems that are known to have good approximation properties in the nonweighted case. Dramatic changes occur when negative weights are allowed.

Low degree rational spline interpolation

For a strictly monotone functionf on [a,b] we describe the possibility of finding an interpolating rational splineS of the formS(x)=c0+c1x/(1+d1x) on each subinterval of the grida=x0<x1<...<xn=b. This leads to a nonlinear system for which we get the local existence and uniqueness of a solution. We prove that ‖S−f‖∞=O(h3). Numerical test shows good approximation properties of these splines.

Haptic interaction with multiresolution image curves

We describe a system for interacting with shapes in two-dimensional images with force feedback, using haptic devices. We construct multiresolution models of boundary curves in images, using wavelet multiresolution with good approximation properties. The boundary curves are treated as solid objects that will produce a force when pushed with a mouse-like haptic interface. This interaction force is then to be rendered through a 2-D haptic interface, providing kinesthetic feedback to the user.

Image compression by means of wavelet transform - application to solar observations

Images and spectra contain a large amount of data. Therefore the question arises, how this data can be decreased or compressed without losing important detail. The discrete wavelet transform is a tool which can be used to compress data because of its good approximation properties. It is very easy to implement and requires approximately the same amount of calculation as the fast Fourier transform. It has the advantage of giving information both in time and frequency. Since most of the coefficients in the transformed data are very small compared to the maximum values, which means that they do not contribute much to the transform, a large number of them can be neglected. Although some data get lost, the physical results deduced from the data remain the same, as is demonstrated by various examples. By this method it is even possible to compress data containing much noise to high-compression ratios.

Constrained Lp-Approximation by Generalized n-Convex Functions Induced by ECT-Systems

The problem of finding a best Lp-approximation (1 ≤ p < ∞) to a function in Lp from a special subcone of generalized n-convex functions induced by an ECT-system is considered. Tchebycheff splines with a countably infinite number of knots are introduced and best approximations are characterized in terms of local best approximations by these splines. Various properties of best approximations and their uniqueness in L1 are investigated. Some special results for generalized monotone and convex cases are obtained.

Neural networks, approximation theory, and finite precision computation

The concepts of universal approximation and best approximation are discussed in relation to artificial neural networks explicitly addressing the fact that networks are typically simulated on computers. In relation to the property of universal approximation, it is shown that networks can be considered as producing a polynomial approximation to the training data, and only a finite number of coefficients of this polynomial can be manipulated. Thus, they are not capable of universal approximation. In relation to the property of best approximation, a short discussion shows that possession, or not, of this property is irrelevant when comparing the approximation abilities of different networks. In summary, existence proofs derived from approximation theory prove to be irrelevant when the numerical limitations imposed by computer simulation are taken into account.

An approach to the numerical solution of the problem of electron scattering in thin crystals, based on the Galerkin method

The problem of electron scattering in a thin crystal (sample), arising in high-resolution transmission electron microscopy (HRTEM), is considered. In contrast to the theory of electron diffraction, the sample is supposed to be of arbitrary structure. The crystal potential is supposed to be, in general, complex; this allows inelastic scattering to be taken into account phenomenologically. The high-energy approximation to the three-dimensional time-independent Schrödinger equation, represented by the two-dimensional time-dependent Schrödinger equation, is used. To solve the problem numerically, the latter is reduced, by the semidiscrete Galerkin method, to an initial-value problem for a system of ordinary differential equations. Orthonormed families of functions, taking into account the features of the problem and having good approximation properties, are introduced for this. In contrast to the system of ordinary differential equation (ODE) for the complex amplitudes of diffracted beams, describing electron diffraction in the dynamical theory, the system derived in this paper corresponds to the discretization of the problem in the coordinate space. This allows the problem of electron scattering in a thin crystal (sample) having arbitrary structure to be solved.

One-sided non-linear mean approximation

This note examines one-sided (from above) nonlinear approximation with respect to a general integral (mean) “norm” on an interval. Necessary conditions for local minima are given. Interpolating properties of best approximations are given. When degenerate approximations can be best is studied for the L1 and L2k cases. Some comparable aspects of the discrete case are examined.

Networks and the best approximation property

Networks can be considered as approximation schemes. Multilayer networks of the perceptron type can approximate arbitrarily well continuous functions (Cybenko 1988, 1989; Funahashi 1989; Stinchcombe and White 1989). We prove that networks derived from regularization theory and including Radial Basis Functions (Poggio and Girosi 1989), have a similar property. From the point of view of approximation theory, however, the property of approximating continuous functions arbitrarily well is not sufficient for characterizing good approximation schemes. More critical is the property ofbest approximation. The main result of this paper is that multilayer perceptron networks, of the type used in backpropagation, do not have the best approximation property. For regularization networks (in particular Radial Basis Function networks) we prove existence and uniqueness of best approximation.

The Hamiltonian system and the spline function

This paper examines the inherent connection between the generalized Hamiltonian system and the spline function. By considering the generalized canonical multipoint boundary value (GCMBV) problem, a multipoint complementary variational principle for generalized Hamiltonian system is provided, and the sufficient and necessary condition for solvability of the GCMBV problem is obtained. With the aid of Hamiltonian equations, a new class of vector spline functions with good variational and approximation properties called H-spline are defined, it includes the general vector spline as a special one. Finally, the H-spline interpolation problems are also discussed.

Semi-discreteness and dilation theory for nest algebras

In this paper it is shown that a contractive σ-weakly continuous Hilbert space representation of a nest algebra admits a σ-weakly continuous dilation to the containing algebra of all operators. This is accomplished by first establishing the complete contractivity of contractive representations through a semi-discreteness property for nest algebras relative to finite-dimensional nest algebras (Theorem 2.1). The semi-discreteness property is obtained by an examination of the order type, spectral type, and multiplicity of the nest, and by the construction of subalgebras that are completely isometric copies of finite-dimensional nest algebras, with good approximation properties. With complete contractivity at hand, the desired dilation follows from Arveson's dilation theorem and auxiliary arguments.

Further asymptotic properties of best approximation by splines

Let S n k, r denote the collection of polynomial splines of order k with at most ( n − 1) free knots, each of multiplicity r. This paper explicitly finds the constants C k, r, p so that lim n→∞ n k dist L p[0, 1] {ƒ, S n k,r} = C k,r,p ∥ƒ (k)∥ L σ[0,1] , where σ = p (kp + 1) and ƒ is sufficiently smooth. This completely fills the gap between previously known results for simple knots ( r = 1) and for piecewise polynomials ( r = k). We also consider similar asymptotic properties pertaining to approximation of vector functions by vector splines. We define the latter to be families of vector functions whose components are splines of order k with common knots.