数值分析
We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and…
We introduce an integral equation formulation of the surface Stokes equations, constructed using two-dimensional Stokeslets. The resulting integral equations are Fredholm integral equations of the second kind and can be discretized to high…
In this work, we discuss low-parametric approaches for approximating SimRank matrices, which estimate the similarity between pairs of nodes in a graph. Although SimRank matrices and their computation require a significant amount of memory,…
We derive parameter-robust quasi-optimal error estimates for mixed finite element methods for the nonlinear Darcy--Forchheimer equations with mixed boundary conditions. Using the framework of operator preconditioning, we also design…
We propose a novel class of neural network-like parametrized functions, i.e., general transformation neural networks (GTNNs), for high-dimensional approximation. Conventional deep neural networks sometimes perform less accurately on…
When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order…
In this work, a novel Eig-PIELM framework is proposed that extends physics-informed extreme learning machine for an efficient and accurate solution of linear eigenvalue problems. The method reformulates the governing differential equations…
The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among…
We establish Sobolev-Poincar\'e inequalities for piecewise $W^{1,p}$ functions over families of fairly general polytopic (thence also shape-regular simplicial and Cartesian) meshes in any dimension; amongst others, they cover the case of…
In this paper, we provide two algorithms based on the theory of multidimensional neural network (NN) operators activated by hyperbolic tangent sigmoidal functions. Theoretical results are recalled to justify the performance of the here…
This work introduces a generalized characteristic mapping method designed to handle non-linear advection with source terms. The semi-Lagrangian approach advances the flow map, incorporating the source term via the Duhamel integral. We…
Physics-Informed Neural Networks (PINNs) encounter accuracy limitations when solving the Allen--Cahn (AC) and Cahn--Hilliard (CH) partial differential equations (PDEs). To overcome this, we employ a novel loss function, Residuals-weighted…
Optimal experimental design (OED) aims to choose the observations in an experiment to be as informative as possible, according to certain statistical criteria. In the linear case (when the observations depend linearly on the unknown…
The flux-vector splitting scheme of Van Leer is a cornerstone of computational fluid dynamics, yet its original proof of the eigenvalue sign condition was presented in a condensed form. In this work, we provide a detailed and rigorous…
Retractions are the workhorse in Riemannian computing applications, where computational efficiency is of the essence. This work introduces a new retraction on the compact Stiefel manifold of orthogonal frames. The retraction is second-order…
In this paper, we derive an optimal $L^2$-norm error estimate of the multiphysics finite element method for the poroelasticity model by introducing an auxiliary problem. We show some numerical tests to verify the theoretical result and…
In this paper, we develop a new multiphysics finite element method for a nonlinear poroelastic model with Hencky-Mises stress tensor. By introducing some new notations, we reformulate the original model into a fluid-fluid coupling problem,…
The recent hardware trend towards reduced precision computing has reignited the interest in numerical techniques that can be used to enhance the accuracy of floating point operations beyond what is natively supported for basic arithmetic…
This paper develops a charge-conservative mixed finite element method with optimal convergence rates for the stationary incompressible inductionless MHD equations on three-dimensional curved domains. The discretization employs the…
We present a robust and efficient numerical framework based on a median filter scheme for solving a broad class of interface-related optimization problems, from image segmentation to topology optimization. A key innovation of our work is…