Related papers: Exploring the locally low dimensional structure in…
In this paper, we address the numerical homogenization approximation of a free-boundary dam problem posed in a heterogeneous media. More precisely, we propose a generalized multiscale finite element (GMsFEM) method for the heterogeneous dam…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
We analyze the random fluctuations of several multi-scale algorithms such as the multi-scale finite element method (MsFEM) and the finite element heterogeneous multiscale method (HMM), that have been developed to solve partial differential…
We analyze the recent Multi-index Stochastic Collocation (MISC) method for computing statistics of the solution of a partial differential equation (PDEs) with random data, where the random coefficient is parametrized by means of a countable…
This paper presents a new finite difference method, called {\varphi}-FD, inspired by the {\phi}-FEM approach for solving elliptic partial differential equations (PDEs) on general geometries. The proposed method uses Cartesian grids,…
We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
We propose and study a scheme combining the finite element method and machine learning techniques for the numerical approximations of coupled nonlinear forward-backward stochastic partial differential equations (FBSPDEs) with homogeneous…
This work analyzes the overall computational complexity of the stochastic Galerkin finite element method (SGFEM) for approximating the solution of parameterized elliptic partial differential equations with both affine and non-affine random…
We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable…
Multiscale Finite Element Methods (MsFEMs) are now well-established finite element type approaches dedicated to multiscale problems. They first compute local, oscillatory, problem-dependent basis functions that generate a suitable…
The paper introduces an adaptive version of the stabilized Trace Finite Element Method (TraceFEM) designed to solve low-regularity elliptic problems on level-set surfaces using a shape-regular bulk mesh in the embedding space. Two…
We consider a sketched implementation of the finite element method for elliptic partial differential equations on high-dimensional models. Motivated by applications in real-time simulation and prediction we propose an algorithm that…
Partial differential equations (PDEs) with inputs that depend on infinitely many parameters pose serious theoretical and computational challenges. Sophisticated numerical algorithms that automatically determine which parameters need to be…
We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follow the…
The finite element method (FEM) is a cornerstone numerical technique for solving partial differential equations (PDEs). Here, we present $\textbf{Qu-FEM}$, a fault-tolerant era quantum algorithm for the finite element method. In contrast to…
The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the…
The combination of reduced basis and collocation methods enables efficient and accurate evaluation of the solutions to parameterized PDEs. In this paper, we study the stochastic collocation methods that can be combined with reduced basis…
In this paper, we propose a novel multiscale model reduction strategy tailored to address the Poisson equation within heterogeneous perforated domains. The numerical simulation of this intricate problem is impeded by its multiscale…
In this paper, we study a numerical method for the solution of partial differential equations on evolving surfaces. The numerical method is built on the stabilized trace finite element method (TraceFEM) for the spatial discretization and…