Related papers: Local high-order regularization and applications t…
This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces.This operator gives optimal estimates of the…
Traditional measures of smoothness often fail to provide accurate $L_p$-error estimates for approximation by sampling or interpolation operators, especially for functions with low smoothness. To address this issue, we introduce a modified…
The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function…
We design a quasi-interpolation operator from the Sobolev space $H^1_0(\Omega)$ to its finite-dimensional finite element subspace formed by piecewise polynomials on a simplicial mesh with a computable approximation constant. The operator 1)…
For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting…
We study the local approximation properties in hierarchical spline spaces through multiscale quasi-interpolation operators. This construction suggests the analysis of a subspace of the classical hierarchical spline space (Vuong et al.,…
We study local approximation properties in hierarchical spline spaces through a twofold approach. First, we design and analyze a robust adaptive refinement algorithm to construct locally graded meshes. Second, we establish rigorous…
We consider a regularization concept for the solution of ill--posed operator equations, where the operator is composed of a continuous and a discontinuous operator. A particular application is level set regularization, where we develop a…
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…
We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator…
This paper extends and analyzes the high-order kernel regularization framework of Beale & Tlupova (arXiv:2510.13639) to all four on-surface boundary integral operators of the Helmholtz Calderon calculus in three dimensions: the…
This investigation seeks to establish the practicality of numerical frame approximations. Specifically, it develops a new method to approximate the inverse frame operator and analyzes its convergence properties. It is established that…
Solutions of partial differential equations can often be written as surface integrals having a kernel related to a singular fundamental solution. Special methods are needed to evaluate the integral accurately at points on or near the…
Based on the variable Hilbert scale interpolation inequality bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators…
We present a method for computing nearly singular integrals that occur when single or double layer surface integrals, for harmonic potentials or Stokes flow, are evaluated at points nearby. Such values could be needed in solving an integral…
The paper proposes a novel regularization procedure for machine learning. The proposed high-order regularization (HR) provides new insight into regularization, which is widely used to train a neural network that can be utilized to…
A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function \omega with a linear transformation. This setting includes Group Lasso methods,…
This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on the Cauchy formula, the Cauchy-Goursat…
The common graph Laplacian regularizer is well-established in semi-supervised learning and spectral dimensionality reduction. However, as a first-order regularizer, it can lead to degenerate functions in high-dimensional manifolds. The…
Numerical homogenization aims to efficiently and accurately approximate the solution space of an elliptic partial differential operator with arbitrarily rough coefficients in a $d$-dimensional domain. The application of the inverse operator…