相关论文: A Two-Level Direct Solver for the Hierarchical Poi…

This paper presents an efficient Krylov subspace iterative solver for the three-dimensional (3D) Helmholtz equation with non-constant coefficients and absorbing boundary conditions, combining high-resolution compact schemes with low-order…

We propose a fast fourth-order cut cell method for solving constant-coefficient elliptic equations in two-dimensional irregular domains. In our methodology, the key to dealing with irregular domains is the poised lattice generation (PLG)…

Solving partial differential equations (PDEs) has been a fundamental problem in computational science and of wide applications for both scientific and engineering research. Due to its universal approximation property, neural network is…

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems…

In this paper, a new iterative two-level algorithm is presented for solving the finite element discretization for nonsymmetric or indefinite elliptic problems. The iterative two-level algorithm uses the same coarse space as the traditional…

This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated…

We present differentially private (DP) algorithms for bilevel optimization, a problem class that received significant attention lately in various machine learning applications. These are the first algorithms for such problems under standard…

We study in this paper a multilayer discretization of second order elliptic problems, aimed at providing reliable multilayer discretizations of shallow fluid flow problems with diffusive effects. This discretization is based upon the…

In this paper, a Parallel Direct Eigensolver for Sequences of Hermitian Eigenvalue Problems with no tridiagonalization is proposed, denoted by \texttt{PDESHEP}, and it combines direct methods with iterative methods. \texttt{PDESHEP} first…

We describe and analyze preconditioned steepest descent (PSD) solvers for fourth and sixth-order nonlinear elliptic equations that include p-Laplacian terms on periodic domains in 2 and 3 dimensions. The highest and lowest order terms of…

This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional…

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization…

This paper is concerned with developing accurate and efficient numerical methods for fully nonlinear second order elliptic and parabolic partial differential equations (PDEs) in multiple spatial dimensions. It presents a general framework…

We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld, J. Sch\"oberl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an…

The least absolute shrinkage and selection operator (Lasso) is widely recognized across various fields of mathematics and engineering. Its variant, the generalized Lasso, finds extensive application in the fields of statistics, machine…

We study a low-rank iterative solver for the unsteady Navier-Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once…

Higher-order nonlinear time-evolution equations have widespread applications in science and engineering, such as in solid mechanics, materials science, and fluid mechanics. This paper mainly studies a direct time-parallel algorithm for…

In this paper, we propose and analyze an explicit time-stepping scheme for a spatial discretization of stochastic Cahn--Hilliard equation with additive noise. The fully discrete approximation combines a spectral Galerkin method in space…

In this article we present a general method to rigorously prove existence of strong solutions to a large class of autonomous semi-linear PDEs in a Hilbert space $H^{l}\subset H^{s}(\mathbb{R}^{m})$ ($s\geq1$) via computer-assisted proofs.…

We propose a stochastic multiscale finite element method (StoMsFEM) to solve random elliptic partial differential equations with a high stochastic dimension. The key idea is to simultaneously upscale the stochastic solutions in the physical…