Mathematics

Physics-informed neural networks (PINNs) formulate the solution of partial differential equations as residual minimization problems over neural network parameterizations. Although highly flexible, optimization of PINNs using modern variants…

Given a matrix $A$, a matrix nearness problem seeks an $X$ that most closely approximates $A$ in the sense of minimizing $\lVert A - X\rVert$ under a variety of constraints on $X$. A generalized matrix nearness problem seeks the same but…

We show that every stable UCT Kirchberg algebra has a principal \'etale groupoid model, and thus contains a C$^*$-diagonal. Every unital UCT Kirchberg algebra $A$ for which $[1_A]_0$ has infinite order in $K_0(A)$ is also covered by our…

We consider the numerical solution of partial differential equations with coefficients that are strongly heterogeneous in space. We provide an overview of higher-order localized orthogonal decomposition (LOD) methods for the elliptic…

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. It is known that, if $f$ is a function holomorphic on a…

A novel mixed spectral-Galerkin method based on generalized ball polynomials is proposed for solving the biharmonic equation on a unit ball. By introducing an auxiliary variable to decouple the biharmonic equation into a system of…

Developing high-order numerical schemes for two-phase flow in porous media that preserve key physical properties remains a significant challenge in numerical analysis. In this article, we propose a general framework to construct fully…

We consider a smooth compact manifold with boundary, $M$, embedded in a smooth manifold of the same dimension on which an amenable group $\Gamma$ acts by isometries. We do not assume $M$ to be invariant under $\Gamma$. This results in a…

We present a high-order implicit-explicit discontinuous Galerkin (IMEX-DG) solver for the compressible Euler equations to account for rotational effects within a fully compressible atmospheric framework. Time integration follows a…

In this paper, we present a quantum implicit-explicit (IMEX) scheme for multiscale ordinary and partial differential equations whose discretization parameters are independent of the scaling parameter $\varepsilon$. A key ingredient of our…

We give a simple example of a polynomial contraction automorphism of $\mathbb C^d$, $ d\ge 3 $, with unbounded degree growth. Combined with Poincar\'e-Dulac theorem it provides an algebraic automorphism of $\mathbb C^d$, $ d\ge 3 $, which…

We propose a high-order numerical methodology for computing the ground state and time evolution of the two-dimensional Gross-Pitaevskii equation with harmonic trapping potential. The ground state is obtained by combining normalized gradient…

This work presents a weighted quadrature (WQ) method to fast assemble Galerkin matrices based on unstructured spline surfaces. The method is developed upon a particular variant of unstructured splines, namely the bicubic analysis-suitable…

In this paper, we further investigate and refine the subspace-constrained preconditioning technique to enhance the theoretical and numerical convergence properties of randomized iterative methods for solving linear systems. In particular,…

This paper aims to employ the weak Galerkin method to solve a class of nonlinear eigenvalue problems. We proved the weak Galerkin scheme produces lower bound for the energy. Moreover, by the post-processing technique, we obtain lower bound…

This paper introduces a multifidelity formulation that reduces the computational cost of the proper orthogonal decomposition (POD) of a high-fidelity model by leveraging data from cheaper, lower-fidelity models. POD is a prevalent technique…

Centered finite-difference discretizations of convection--diffusion equations may oscillate when convection dominates at the mesh scale. For homogeneous Dirichlet problems with constant coefficients on uniform Cartesian grids, we derive…

In this paper, we investigate two subclasses of analytic and univalent functions associated with the exponential mapping $\varphi(z)=e^{\alpha z},\qquad 0<\alpha\le1,$ defined via the subordination conditions $\frac{zf'(z)}{f(z)}\prec…

The simulation of extreme Mach astrophysical flows is traditionally viewed through the lens of deterministic positivity-preserving schemes. However, due to Kelvin--Helmholtz instabilities and shock anomalies, the multi-dimensional Euler…

We introduce a new class of fractional backward orthogonal functions designed for the spectral approximation of weakly singular adjoint Volterra integral equations. These basis functions generate an approximation space that naturally…