数值分析
In many real-world applications of data assimilation (DA), the strategic placement of observers is crucial for effective and efficient forecasting. Motivated by practical constraints in sensor deployment, we show that global recovery of the…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
We study a discrete analogue of the parametric Plateau problem in a non-polynomial tensor-product surface spaces generated by the generalized trigonometric (GT)--B\'ezier basis. Boundary interpolation is imposed by prescribing the boundary…
Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy…
We present two algorithms for constructing orthonormal bases of rational function vectors with respect to a discrete inner product, and discuss how to use them for a rational approximation problem. Building on the pencil-based formulation…
Physics-informed extreme learning machines (PIELMs) typically impose boundary and initial conditions through penalty terms, yielding only approximate satisfaction that is sensitive to user-specified weights and can propagate errors into the…
For low-frequency electromagnetic problems, where wave-propagation effects can be neglected, eddy current formulations are commonly used as a simplification of the full Maxwell's equations. In this setup, time-domain simulations, needed to…
A numerical framework is proposed for identifying partial differential equations (PDEs) governing dynamical systems directly from their observation data using Chebyshev polynomial approximation. In contrast to data-driven approaches such as…
We consider parabolic evolution equations with Lipschitz continuous and strongly monotone spatial operators. By introducing an additional variable, we construct an equivalent system where the operator is a Lipschitz continuous mapping from…
The Sinc convolution is an approximate formula for indefinite convolutions proposed by Stenger. The formula was derived based on the Sinc indefinite integration formula combined with the single-exponential transformation. Although its…
This paper presents a novel exact finite element formulation of quasi-3D beam for high-fidelity analysis of functionally graded sandwich beams. Unlike conventional displacement-based elements that rely on approximate interpolation functions…
This paper presents a novel finite volume mooring line model based on the geometrically exact Simo-Reissner beam model for analysing the interaction between a floating rigid body and its mooring lines. The coupled numerical model is…
In this work, we present a mathematical and computational framework to model the dynamics of open lipid bilayer membranes interacting with ambient Stokes flow. The model explicitly couples the three-dimensional viscous fluid, the…
We investigate the reconstruction of multivariate functions from samples using sparse recovery techniques. For Square Root Lasso, Orthogonal Matching Pursuit, and Compressive Sampling Matching Pursuit, we demonstrate both theoretically and…
Most Fredholm integral equations involve integrals with weakly singular kernels. Once the domain of integration is discretized into flat triangular elements, these weakly singular kernels become strongly singular or near-singular. Common…
We present a novel Asymptotic-Preserving Neural Network (APNN) approach utilizing even-odd decomposition to tackle the nonlinear gray radiative transfer equations (GRTEs). Our AP loss demonstrates consistent stability concerning the small…
We present a tensorization algorithm for constructing tensor train/matrix product state (MPS) representations of functions, drawing on sketching and cross interpolation ideas. The method only requires black-box access to the target function…
In this paper, we propose a subspace method based on neural networks for eigenvalue problems with high accuracy and low cost. We first construct a neural network-based orthogonal basis by some deep learning method and dimensionality…
We study the integration problem over the $s$-dimensional unit cube on four types of Banach spaces of integrands. First we consider Haar wavelet spaces, consisting of functions whose Haar wavelet coefficients exhibit a certain decay…
Given (orthonormal) approximations $\tilde{U}$ and $\tilde{V}$ to the left and right subspaces spanned by the leading singular vectors of a matrix $A$, we discuss methods to approximate the leading singular values of $A$ and study their…