Related papers: Some Sharp Error Bounds for Multivariate Linear In…
We determine the pointwise error in Hermite interpolation by numerically solving an appropriate differential equation, derived from the error term itself. We use this knowledge to approximate the error term by means of a polynomial, which…
Sensitivity analysis for measurement error can be applied in the absence of validation data by means of regression calibration and simulation-extrapolation. These have not been compared for this purpose. A simulation study was conducted…
In this paper, we develop approximation error estimates as well as corresponding inverse inequalities for B-splines of maximum smoothness, where both the function to be approximated and the approximation error are measured in standard…
In recent years, sharp or quantitative weighted inequalities have attracted considerable attention on account of $A_2$ conjecture solved by Hyt\"{o}nen. Advances have greatly improved conceptual understanding of classical objects such as…
This paper gives some results related to the research problem about infinite-dimensional affine variational inequalities raised by N.D. Yen and X. Yang [Affine variational inequalities on normed spaces, J. Optim. Theory Appl., 178 (2018),…
We consider bounds on the generalization performance of the least-norm linear regressor, in the over-parameterized regime where it can interpolate the data. We describe a sense in which any generalization bound of a type that is commonly…
In non-private stochastic convex optimization, stochastic gradient methods converge much faster on interpolation problems -- problems where there exists a solution that simultaneously minimizes all of the sample losses -- than on…
An efficient proximal-gradient-based method, called proximal extrapolated gradient method, is designed for solving monotone variational inequality in Hilbert space. The proposed method extends the acceptable range of parameters to obtain…
Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for…
In this paper we present a new multilevel quasi-interpolation algorithm for smooth periodic functions using scaled Gaussians as basis functions. Recent research in this area has focussed upon implementations using basis function with finite…
It's well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It's of the form$% | f(x)-s(x)| \leq (Cd)^{\frac{c}{d}}\left\Vert f\right\Vert_{h}$ where $C,c$ are constants, $h$ is the…
Fixed-point equations with Lipschitz operators have been studied for more than a century, and are central to problems in mathematical optimization, game theory, economics, and dynamical systems, among others. When the Lipschitz constant of…
We approximate the spectral data (eigenvalues and eigenfunctions) of compact Riemannian manifold by the spectral data of a sequence of (computable) discrete Laplace operators associated to some graphs immersed in the manifold. We give an…
Spectral graph bisections are a popular heuristic aimed at approximating the solution of the NP-complete graph bisection problem. This technique, however, does not always provide a robust tool for graph partitioning. Using a special class…
In this paper we obtain an approximation for the multivariate Laplace's integral with a large parameter and estimate error term for two cases, when the maximum of the exponent is in the interior of the domain and on the boundary. We are…
We derive two upper bounds for the probability of deviation of a vector-valued Lipschitz function of a collection of random variables from its expected value. The resulting upper bounds can be tighter than bounds obtained by a direct…
In this paper, we analyze the mirror descent algorithm for non-smooth optimization problems in which the objective function is relatively strongly convex, without relying on the standard Lipschitz continuity assumption commonly used in the…
This paper examines the asymptotic convergence properties of Lipschitz interpolation methods within the context of bounded stochastic noise. In the first part of the paper, we establish probabilistic consistency guarantees of the classical…
Three kinds of effective error bounds of the quadrature formulas with multiple nodes that are generalizations of the well known Micchelli-Rivlin quadrature formula, when the integrand is a function analytic in the regions bounded by…
This paper surveys hyperinterpolation, a quadrature-based approximation scheme. We cover classical results, provide examples on several domains, review recent progress on relaxed quadrature exactness, introduce methodological variants, and…