Related papers: Optimal Triangulation of Saddle Surfaces
This paper provides approximation orders for a class of nonlinear interpolation procedures for univariate data sampled over $\sigma$ quasi-uniform grids. The considered interpolation is built using both essentially nonoscillatory (ENO) and…
Explicit pointwise error bounds for the interpolation of a smooth function by piecewise exponential splines of order four are given. Estimates known for cubic splines are extended to a natural class of piecewise exponential splines which…
Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the…
We construct Monte Carlo methods for the $L^2$-approximation in Hilbert spaces of multivariate functions sampling no more than $n$ function values of the target function. Their errors catch up with the rate of convergence and the…
Given a function f defined on a bidimensional bounded domain and a positive integer N, we study the properties of the triangulation that minimizes the distance between f and its interpolation on the associated finite element space, over all…
Successive quadratic approximations, or second-order proximal methods, are useful for minimizing functions that are a sum of a smooth part and a convex, possibly nonsmooth part that promotes regularization. Most analyses of iteration…
We consider the problem of minimizing a convex function over the intersection of finitely many simple sets which are easy to project onto. This is an important problem arising in various domains such as machine learning. The main difficulty…
We present a non-conforming least squares method for approximating solutions of second order elliptic problems with discontinuous coefficients. The method is based on a general Saddle Point Least Squares (SPLS) method introduced in previous…
We introduce a novel type of approximation spaces for functions with values in a nonlinear manifold. The discrete functions are constructed by piecewise polynomial interpolation in a Euclidean embedding space, and then projecting pointwise…
We propose an iterative algorithm for low-rank matrix completion that can be interpreted as both an iteratively reweighted least squares (IRLS) algorithm and a saddle-escaping smoothing Newton method applied to a non-convex rank surrogate…
We study the existing algorithms that solve the multidimensional martingale optimal transport. Then we provide a new algorithm based on entropic regularization and Newton's method. Then we provide theoretical convergence rate results and we…
Triangulation refers to the problem of finding a 3D point from its 2D projections on multiple camera images. For solving this problem, it is the common practice to use so-called optimal triangulation method, which we call the L2 method in…
We propose to compute approximations to general invariant sets in dynamical systems by minimizing the distance between an appropriately selected finite set of points and its image under the dynamics. We demonstrate, through computational…
In this paper, several modifications are introduced to the functional approximation method iterLap to reduce the approximation error, including stopping rule adjustment, proposal of new residual function, starting point selection for…
In this paper,we present an inexact primal-dual method with correction step for a saddle point problem by introducing the notations of inexact extended proximal operators with symmetric positive definite matrix $D$. Relaxing requirement on…
Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first…
Unitary best approximation to the exponential function on an interval on the imaginary axis has been introduced recently. In the present work two algorithms are considered to compute this best approximant: an algorithm based on rational…
While the theory of operator approximation with any given accuracy is well elaborated, the theory of {best constrained} constructive operator approximation is still not so well developed. Despite increasing demands from applications this…
Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that…
We study in this paper the function approximation error of linear interpolation and extrapolation. Several upper bounds are presented along with the conditions under which they are sharp. All results are under the assumptions that the…