Related papers: New Formulas and Methods for Interpolation, Numeri…

We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue…

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data…

Standard interpolation techniques are implicitly based on the assumption that the signal lies on a homogeneous domain. In this letter, the proposed interpolation method instead exploits prior information about domain inhomogeneity,…

We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization…

We propose a level set method to reconstruct unknown surfaces from point clouds, without assuming that the connections between points are known. We consider a variational formulation with a curvature constraint that minimizes the surface…

We consider linear iterative schemes for the time-discrete equations stemming from a class of nonlinear, doubly-degenerate parabolic equations. More precisely, the diffusion is nonlinear and may vanish or become multivalued for certain…

We consider a new splitting based on the Sherman-Morrison-Woodbury formula, which is particularly effective with iterative methods for the numerical solution of large linear systems. These systems involve matrices that are perturbations of…

This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations,…

The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied…

We provide elementary identities relating the three known types of non-symmetric interpolation Macdonald polynomials. In addition we derive a duality for non-symmetric interpolation Macdonald polynomials. We consider some applications of…

There has been a surge in Explainable-AI (XAI) methods that provide insights into the workings of Deep Neural Network (DNN) models. Integrated Gradients (IG) is a popular XAI algorithm that attributes relevance scores to input features…

In the context of the analysis of measured data, one is often faced with the task to differentiate data numerically. Typically, this occurs when measured data are concerned or data are evaluated numerically during the evolution of partial…

In this paper we present an estimate of accuracy for a piecewise polynomial approximation of a classical numerical solution to a non linear differential problem. We suppose the numerical solution U is computed using a grid with a small…

In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states,…

A fast multilevel algorithm based on directionally scaled tensor-product Gaussian kernels on structured sparse grids is proposed for interpolation of high-dimensional functions and for the numerical integration of high-dimensional…

The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when…

We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard…

We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.

Explicit formulas for the mean and variance of linear stochastic differential equations are derived in terms of an exponential matrix. This result improved a previous one by means of which the mean and variance are expressed in terms of a…

We introduce Interpolation Consistency Training (ICT), a simple and computation efficient algorithm for training Deep Neural Networks in the semi-supervised learning paradigm. ICT encourages the prediction at an interpolation of unlabeled…