Related papers: Projection-Based Finite Elements for Nonlinear Fun…

A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an $N$-dimensional hyperrectangle. In particular, it is shown that these functions can be expressed in terms of their highest-order mixed…

We develop commuting finite element projections over smooth Riemannian manifolds. This extension of finite element exterior calculus establishes the stability and convergence of finite element methods for the Hodge-Laplace equation on…

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,…

In this paper, we introduce a multiscale framework based on adaptive edge basis functions to solve second-order linear elliptic PDEs with rough coefficients. One of the main results is that we prove the proposed multiscale method achieves…

In this paper, we use a unified framework introduced in [3] to study two classes of nonconforming immersed finite element (IFE) spaces with integral value degrees of freedom. The shape functions on interface elements are piecewise…

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…

We develop a rigorous theoretical framework for principal manifold estimation that recovers a latent low-dimensional manifold from a point cloud observed in a high-dimensional ambient space. Our framework accommodates manifolds with…

For nonlinear reduced-order models, especially for those with non-polynomial nonlinearities, the computational complexity still depends on the dimension of the original dynamical system. As a result, the reduced-order model loses its…

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…

When modeling scientific and industrial problems, geometries are typically modeled by explicit boundary representations obtained from computer-aided design software. Unfitted (also known as embedded or immersed) finite element methods offer…

We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps of a Riemann surface, any holomorphic function can be approximated by a converging sequence of discrete holomorphic functions.

This paper is concerned with the ubiquitous inverse problem of recovering an unknown function u from finitely many measurements possibly affected by noise. In recent years, inversion methods based on linear approximation spaces were…

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…

We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and…

We derive $H_{\text{curl}}$-error estimates and improved $L^2$-error estimates for the Maxwell equations approximated using edge finite elements. These estimates only invoke the expected regularity pickup of the exact solution in the scale…

For conforming finite element approximations of the Laplacian eigenfunctions, a fully computable guaranteed error bound in the $L^2$ norm sense is proposed. The bound is based on the a priori error estimate for the Galerkin projection of…

We propose a new method for functional nonparametric regression with a predictor that resides on a finite-dimensional manifold but is only observable in an infinite-dimensional space. Contamination of the predictor due to discrete/noisy…

The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…

There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes…

We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…