Error Estimates for Thin Plate Spline Approximation in the Disk

This paper is concerned with approximation properties of linear combinations of scattered translates of the thin-plate spline radial basis function |·|²log|·| where the translates are taken in the unit disk D in R². We show that the L_p approximation order for this kind of approximation is 2 + 1/p (for sufficiently smooth functions), which matches Johnson's upper bound and, thus, gives the saturation order. We also show that when one increases the density of the centers at the boundary, approximation order 4 - the best possible order in the absence of a boundary - can be obtained.     

Lattice Algorithms for Multivariate L ∞ Approximation in the Worst-Case Setting

We approximate d-variate functions from weighted Korobov spaces with the error of approximation defined in the L _∞ sense. We study lattice algorithms and consider the worst-case setting in which the error is defined by its worst-case behavior over the unit ball of the space of functions. A lattice algorithm is specified by a generating (integer) vector. We propose three choices of such vectors, each corresponding to a different search criterion in the component-by-component construction. We present worst-case error bounds that go to zero polynomially with n ^?1, where n is the number of function values used by the lattice algorithm. Under some assumptions on the weights of the function space, the worst-case error bounds are also polynomial in d, in which case we have (polynomial) tractability, or even independent of d, in which case we have strong (polynomial) tractability. We discuss the exponents of n ^?1 and stress that we do not know if these exponents can be improved.     

Numerical experiments on the condition number of the interpolation matrices for radial basis functions

Through numerical experiments, we examine the condition numbers of the interpolation matrix for many species of radial basis functions (RBFs), mostly on uniform grids. For most RBF species that give infinite order accuracy when interpolating smooth f(x)—Gaussians, sech's and Inverse Quadratics—the condition number κ(α,N) rapidly asymptotes to a limit κasymp(α) that is independent of N and depends only on α, the inverse width relative to the grid spacing. Multiquadrics are an exception in that the condition number for fixed α grows as N². For all four, there is growth proportional to an exponential of 1/α (1/α² for Gaussians). For splines and thin-plate splines, which contain no width parameter, the condition numbers grows asymptotically as a power of N—a large power as the order of the RBF increases. Random grids typically increase the condition number (for fixed RBF width) by orders of magnitude. The quasi-random, low discrepancy Halton grid may, however, have a lower condition number than a uniform grid of the same size.     

Approximation Error for Quasi-Interpolators and (Multi-)Wavelet Expansions

We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L²via an appropriate sequence of inner products. In particular, we consider integer shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets which use multiple generators. We estimate the approximation error as a function of the scale parameterTwhen the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling stepT. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multi-wavelet expansions of a given orderL.     

An error estimate for least squares approximation

Upper and lower error bounds are obtained for the error of the bestL ₂ polynomial approximation of degreen for a function belonging toC ⁿ⁺¹ [?1, 1].     

On approximations of transformed chi-squared distributions in statistical applications

We obtain some computable error bounds of order O(n ^?1) for the chi-squared approximation of transformed chi-squared random variables with n degrees of freedom. The results are applied to likelihood ratio statistics in the multivariate case.     

On the unimprovability of full-memory strategies in the risk minimization problem

We continue the study of approximation properties of local exponential splines on a uniform grid with step h > 0 corresponding to a linear differential operator L with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E.V. Strelkova and V.T. Shevaldin). We find order estimates as h → 0 for the error of approximation of certain Sobolev classes of functions by splines of the described type that are exact on the kernel of the operator L.     

Lower bounds for randomized direct search with isotropic sampling

Randomized direct-search methods for the optimization of a function f:Rⁿ→R that is given by a black box for f-evaluations are investigated. These iterative methods generate new candidate solutions by adding isotropically distributed vectors to the current candidate solution. Lower bounds on the number of f-evaluations necessary for reducing the approximation error in the search space are proved.     

On the length of Boolean functions in the class of exclusive-OR sums of pseudoproducts

Representations of Boolean functions by exclusive-OR sums (modulo 2) of pseudoproducts is studied. An ExOR-sum of pseudoproducts (ESPP) is the sum modulo 2 of products of affine (linear) Boolean functions. The length of an ESPP is defined as the number of summands in this form, and the length of a Boolean function in the class of ESPPs is defined as the minimum length of an ESPP representing this function. The Shannon function L ^ESPP(n) of the length of Boolean functions in the class of ESPPs is considered, which equals the maximum length of a Boolean function of n variables in this class. Lower and upper bounds for the Shannon function L ^ESPP(n) are found. The upper bound is proved by using an algorithm which can be applied to construct representations by ExOR-sums of pseudoproducts for particular Boolean functions.     

Wavelet Galerkin method for eigenvalue problem of a compact integral operator

We consider the approximation of eigenfunctions of a compact integral operator with a smooth kernel by the Galerkin method using wavelet bases. By truncating the Galerkin operator, we obtain a sparse representation of a matrix eigenvalue problem. We prove that the error bounds for the eigenvalues and for the distance between the spectral subspaces are of the orders O(nμ^-2nr) and O(μ^-nr), respectively, where μ⁻ⁿ denotes the norm of the partition and r denotes the order of the wavelet basis functions. By iterating the eigenvectors, we show that the error bounds for the eigenvectors are of the order O(nμ^-2nr). We illustrate our results with numerical results.     

Error Estimates for Multilevel Approximation Using Polyharmonic Splines

Polyharmonic splines are used to interpolate data in a stationary multilevel iterative refinement scheme. By using such functions the necessary tools are provided to obtain simple pointwise error bounds on the approximation. Linear convergence between levels is shown for regular data on a scaled multiinteger grid, and a multilevel domain decomposition method.     

Dynamic Programming and Value-Function Approximation in Sequential Decision Problems: Error Analysis and Numerical Results

Value-function approximation is investigated for the solution via Dynamic Programming (DP) of continuous-state sequential N-stage decision problems, in which the reward to be maximized has an additive structure over a finite number of stages. Conditions that guarantee smoothness properties of the value function at each stage are derived. These properties are exploited to approximate such functions by means of certain nonlinear approximation schemes, which include splines of suitable order and Gaussian radial-basis networks with variable centers and widths. The accuracies of suboptimal solutions obtained by combining DP with these approximation tools are estimated. The results provide insights into the successful performances appeared in the literature about the use of value-function approximators in DP. The theoretical analysis is applied to a problem of optimal consumption, with simulation results illustrating the use of the proposed solution methodology. Numerical comparisons with classical linear approximators are presented.     

Error bounds for derivative estimates based on spline smoothing of exact or noisy data

Estimates are found for the L₂ error in approximating the jth derivative of a given smooth function f by the corresponding derivative of the 2mth order smoothing spline based on an n-point sample from the function. The results cover both the case of an exact sample from f and the case when the sample is subject to some random noise. In the noisy case, the estimates are for the expected value of the approximation error. These bounds show that, even in the presence of noise, the derivatives of the smoothing splines of order less than m can be expected to converge to those of f as the number of (uniform) sample points increases, and the smoothing parameter approaches zero at a rate appropriately related to m, n, and the order of differentiability of f.     

Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceR d is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.     

Some extremal properties of multivariate polynomial splines in the metric Lp(ℝd)

We constructed a kind of continuous multivariate spline operators as the approximation tools of the multivariate functions on the (?^d instead of the usual multivariate cardinal interpolation operators of splines, and obtained the approximation error by this kind of spline operators. Meantime, by the results, we also obtained that the spaces of multivariate polynomial splines are weakly asyrnptotically optimal for the Kolrnogorov widths and the linear widths of some anisotropic Sobolev classes of smooth functions on (?^d in the metric L_p((?^d).     

On the Lagrange interpolation for a subset ofC k functions

We study the optimal order of approximation forC ^k piecewise analytic functions (cf. Definition 1.2) by Lagrange interpolation associated with the Chebyshev extremal points. It is proved that the Jackson order of approximation is attained, and moreover, ifx is away from the singular points, the local order of approximation atx can be improved byO(n ^?1). Such improvement of the local order of approximation is also shown to be sharp. These results extend earlier results of Mastroianni and Szabados on the order of approximation for continuous piecewise polynomial functions (splines) by the Lagrange interpolation, and thus solve a problem of theirs (about the order of approximation for |x|³) in a much more general form.     

Approximation by fuzzy B-spline series

We study properties concerning approximation of fuzzy-number-valued functions by fuzzy B-spline series. Error bounds in approximation by fuzzy B-spine series are obtained in terms of the modulus of continuity. Particularly simple error bounds are obtained for fuzzy splines of Schoenberg type. We compare fuzzy B-spline series with existing fuzzy concepts of splines.     

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C², represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.     

Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree structure. This makes their positions easy to encode. Previous work [4,6] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. This paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. We accomplish this in the case that the approximation error is measured in L ₂, or in the case p2, in the Besov spaces B _p ⁰(L _p ), which are close to (but not the same as) L _p . Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients.     

On the Approximation Order of Splines on Spherical Triangulations

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a C ^r function whose pieces are the restrictions of homogeneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms defined with the help of radial projection, and are obtained using appropriate quasi-interpolation operators.