相关论文: On $p$-robust saturation on quadrangulations

We consider the approximation of singularly perturbed linear second-order boundary value problems by $hp$-finite element methods. In particular, we include the case where the associated differential operator may not be coercive. Within this…

We consider the energy norm arising from elliptic problems with discontinuous piecewise constant diffusion. We prove that under the quasi-monotonicity property on the diffusion coefficient, the best approximation error with continuous…

We analyze and validate the virtual element method combined with a boundary correction similar to the one in [1,2], to solve problems on two dimensional domains with curved boundaries approximated by polygonal domains. We focus on the case…

We study a two-grid strategy for decoupling the time-dependent Poisson-Nernst-Planck equations describing the mass concentration of ions and the electrostatic potential. The computational system is decoupled to smaller systems by using…

This paper addresses a quadratic problem with assignment constraints, an NP-hard combinatorial optimization problem arisen from facility location, multiple-input multiple-output detection, and maximum mean discrepancy calculation et al. The…

We study the computational complexity of 2-local Hamiltonian problems generated by a positive-weight symmetric interaction term, encompassing many canonical problems in statistical mechanics and optimization. We show these problems belong…

Polygonal finite elements generally do not pass the patch test as a result of quadrature error in the evaluation of weak form integrals. In this work, we examine the consequences of lack of polynomial consistency and show that it can lead…

A number of non-standard finite element methods have been proposed in recent years, each of which derives from a specific class of PDE-constrained norm minimization problems. The most notable examples are $\mathcal{L}\mathcal{L}^*$ methods.…

We study an expansion method for high-dimensional parabolic PDEs which constructs accurate approximate solutions by decomposition into solutions to lower-dimensional PDEs, and which is particularly effective if there are a low number of…

In this paper, we study the polynomial optimization problem of multi-forms over the intersection of the multi-spheres and the nonnegative orthants. This class of problems is NP-hard in general, and includes the problem of finding the best…

This study presents a novel algorithm for identifying the set of extreme points that constitute the exact convex hull of a point set in high-dimensional Euclidean space. The proposed method iteratively solves a sequence of dynamically…

Several large-scale machine learning tasks, such as data summarization, can be approached by maximizing functions that satisfy submodularity. These optimization problems often involve complex side constraints, imposed by the underlying…

In this paper, we consider convex stochastic optimization problems arising in machine learning applications (e.g., risk minimization) and mathematical statistics (e.g., maximum likelihood estimation). There are two main approaches to solve…

This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy…

We describe a new, adaptive solver for the two-dimensional Poisson equation in complicated geometries. Using classical potential theory, we represent the solution as the sum of a volume potential and a double layer potential. Rather than…

An inexact framework for high-order adaptive regularization methods is presented, in which approximations may be used for the $p$th-order tensor, based on lower-order derivatives. Between each recalculation of the $p$th-order derivative…

In this paper we have derived explicitly computable bounds on the error in energy norms for the fully nonlinear Poisson-Boltzmann equation. Together with the computable bounds, we have also obtained efficient error indicators which can…

In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well…

We consider fourth order singularly perturbed eigenvalue problems in one-dimension and the approximation of their solution by the $h$ version of the Finite Element Method (FEM). In particular, we use piecewise Hermite polynomials of degree…

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…