Related papers: Adaptive guaranteed lower eigenvalue bounds with o…
We investigate the problem of certifying optimality for sparse generalized linear models (GLMs), where sparsity is enforced through a cardinality constraint. While Branch-and-Bound (BnB) frameworks can certify optimality using perspective…
Operator learning, the approximation of mappings between infinite-dimensional function spaces using machine learning, has gained increasing research attention in recent years. Approximate operators, learned from data, can serve as efficient…
This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation…
This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions…
We consider the numerical construction of minimal Lagrangian graphs, which is related to recent applications in materials science, molecular engineering, and theoretical physics. It is known that this problem can be formulated as an…
The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long…
In this work, we fully explore three refined convergence structures of the lowest-order rectangular Raviart-Thomas element in solving the Laplace eigenvalue problem. Firstly, the scheme possesses a property of supercloseness between the…
This paper presents a novel stochastic gradient descent algorithm for constrained optimization. The proposed algorithm randomly samples constraints and components of the finite sum objective function and relies on a relaxed logarithmic…
Most commonly used \emph{adaptive} algorithms for univariate real-valued function approximation and global minimization lack theoretical guarantees. Our new locally adaptive algorithms are guaranteed to provide answers that satisfy a…
A quasi-complementary sequence set (QCSS) refers to a set of two-dimensional matrices with low non-trivial aperiodic auto- and cross- correlation sums. For multicarrier code-division multiple-access applications, the availability of large…
This paper presents a parametric solution to piecewise linear regression through the Adaptive Block Gradient Descent (ABGD) algorithm. The heart of the method is the parametrization of piecewise linear functions as the difference of…
Maximization and minimization problems of the principle eigenvalue for divergence form second order elliptic operators with the Dirichlet boundary condition are considered. The principal eigen map of such elliptic operators is introduced…
The problem of developing an adaptive isogeometric method (AIGM) for solving elliptic second-order partial differential equations with truncated hierarchical B-splines of arbitrary degree and different order of continuity is addressed. The…
A scheme for locally adaptive bandwidth selection is proposed which sensitively shrinks the bandwidth of a kernel estimator at lowest density regions such as the support boundary which are unknown to the statistician. In case of a…
We consider the vertex-centered finite volume method with first-order conforming ansatz functions. The adaptive mesh-refinement is driven by the local contributions of the weighted-residual error estimator. We prove that the adaptive…
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $\R^d$, $d \geq 2$. In particular, we derive upper bounds on Riesz means of order $\sigma \geq 3/2$, that improve the sharp Berezin inequality…
The Laplace eigenvalue problem on circular sectors has eigenfunctions with corner singularities. Standard methods may produce suboptimal approximation results. To address this issue, a novel numerical algorithm that enhances standard…
Deep Ritz methods (DRM) have been proven numerically to be efficient in solving partial differential equations. In this paper, we present a convergence rate in $H^{1}$ norm for deep Ritz methods for Laplace equations with Dirichlet boundary…
The aim of this paper is to provide new perspectives on the relative finite elements accuracy. Starting from a geometrical interpretation of the error estimate which can be deduced from Bramble-Hilbert lemma, we derive a probability law…
In this paper, we study an adaptive finite element method for multiple eigenvalue problems of a class of second order elliptic equations. By using some eigenspace approximation technology and its crucial property which is also presented in…