相关论文: Duality between range and no-response tests and it…

We investigate 1) the rate at which refined properties of the empirical risk---in particular, gradients---converge to their population counterparts in standard non-convex learning tasks, and 2) the consequences of this convergence for…

We study an inverse boundary value problem for the nonlinear wave equation in $2 + 1$ dimensions. The objective is to recover an unknown potential $q(x, t)$ from the associated Dirichlet-to-Neumann map using real-valued waves. We propose a…

We show that a wide class of risk-constrained nonconvex functional optimization problems exhibit strong duality, regardless of nonconvexity. We develop two novel results under distinct sets of assumptions, establishing strong duality over…

In this note, we prove that for the Navier-Stokes equations, a pair of Dirichlet and Neumann data and pressure uniquely correspond to a pair of Dirichlet data and surface stress on the boundary. Hence the two inverse boundary value problems…

This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness…

We consider the Cauchy-Dirichlet problem to doubly nonlinear systems of the form \begin{align*} \partial_t \big( |u|^{q-1}u \big) - \operatorname{div} \big( D_\xi f(x,u,Du) \big) = - D_u f(x,u,Du) \end{align*} with $q \in (0, \infty)$ in a…

This paper explores hypothesis testing for the parametric forms of the mean and variance functions in regression models under diverging-dimension settings. To mitigate the curse of dimensionality, we introduce weighted residual empirical…

This paper analyzes the stability of a reactiondiffusion equation coupled with a finite-dimensional controller through Dirichlet boundary input and Neumann boundary output. Going against the flow, we intend to propose numerical certificates…

The comparative study of two globally convergent numerical methods for acoustic tomography is carried out in two dimensions. These are the boundary control method and the quasi-reversibility method. The novelty is that in the latter a…

In this paper we prove existence and uniqueness results for nonlinear parabolic problems with Dirichlet boundary values whose model is \[ \left\{ \begin{aligned} &b(u)_t-\Delta_{p}u=\mu\;\mbox{in }(0,T)\times\Omega,\\…

We examine the duality theory for a class of non-convex functions obtained by composing a convex function with a continuous one. Using Fenchel duality, we derive a dual problem that satisfies weak duality under general assumptions. To…

Deep learning methods operate in regimes that defy the traditional statistical mindset. Neural network architectures often contain more parameters than training samples, and are so rich that they can interpolate the observed labels, even if…

Recently Liu and Wang derived the likelihood ratio test (LRT) statistic and its asymptotic distribution for testing equality of two multinomial distributions vs. the alternative that the second distribution is larger in terms of increasing…

In this paper we study the asymptotic behaviour of the solutions of some minimization problems for integral functionals with convex integrands, in two-dimensional domains with cracks, under perturbations of the cracks in the Hausdorff…

In this paper, we study the deep Ritz method for solving the linear elasticity equation from a numerical analysis perspective. A modified Ritz formulation using the $H^{1/2}(\Gamma_D)$ norm is introduced and analyzed for linear elasticity…

We study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three nonzero solutions. When the reaction term is sublinear at infinity, we apply the second…

We consider an inverse problem arising in nonlinear ultrasound imaging. The propagation of ultrasound waves is modeled by a quasilinear wave equation. We make measurements at the boundary of the medium encoded in the Dirichlet-to-Neumann…

In this paper we are proving the existence of a nontrivial solution of the ${p}(x)$- Laplacian equation with Dirichlet boundary condition. We will use the variational method and concentration compactness principle involving positive radon…

We study the interior regularity of solutions to the Dirichlet problem $Lu=g$ in $\Omega$, $u=0$ in $\R^n\setminus\Omega$, for anisotropic operators of fractional type $$ Lu(x)= \int_{0}^{+\infty}\,d\rho \int_{S^{n-1}}\,da(\omega)\, \frac{…

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep…